diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhgeu" "b/data_all_eng_slimpj/shuffled/split2/finalzzhgeu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhgeu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{intro}\n\nGiant extended emission-line regions (EELRs), with typical radii of 10--30 kpc, are found around roughly half of the low-redshift steep-radio-spectrum quasars with strong nuclear narrow-line emission \\citep{Sto87,Fu09}. Although the most luminous EELRs at low redshifts are exclusively associated with steep-spectrum radio-loud quasars, the distribution of the ionized gas typically shows no obvious connection with the radio jets or lobes, or with host galaxy morphologies. These EELRs most likely comprise gas that has been driven out of the host galaxies by superwinds \\citep{Sto02,Fu06,Fu07b,Fu08,Fu09}. But are these superwinds mostly produced by starbursts or by the quasars themselves? We have found some evidence in favor of quasar-driven outflows from momentum considerations \\citep{Fu06} and from the lack of any optical spectroscopic evidence for a very recent starburst in the radio galaxy 3C\\,79, which shows both scattered light from a hidden quasar and a strong EELR \\citep{Fu08}. However, to deal with this question more systematically, we need to work with a global star-formation indicator that we can apply to a larger sample of objects. In particular, for the present program, we have chosen carefully matched samples of steep-radio-spectrum quasars with and without luminous EELRs. Our main goal was to measure or to place strong upper limits to star formation rates (SFRs) in both subsamples of quasars.\n\nA number of measures of SFR are available for galaxies in general. These include optical emission lines, such as H$\\alpha$ and [O\\,{\\sc ii}]\\,$\\lambda$3727 \\citep[e.g.,][and references therein]{Kew04}, mid-infrared (MIR) fine structure lines, notably [Ne\\,{\\sc ii}]\\,12.8$\\mu$m\\ \\citep{Ho07}, polycyclic aromatic hydrocarbon (PAH) emission in the MIR \\citep{Sch06,Shi07}, the far-infrared (FIR) continuum \\citep[e.g.,][]{Row97}, and the radio continuum \\citep[e.g.,][]{Con92}. Some of these measures are difficult or impossible to apply to host galaxies of luminous, radio-loud QSOs (i.e.,\\ quasars). For example, photoionization by the quasar UV continuum produces emission lines that will mask those from star formation, and radio emission from the non-thermal radio source can swamp that due to star formation. The PAH emission and the FIR continuum appear to offer the most promise for use with quasar host galaxies, and they have the additional advantage over any optical approaches that they can reveal even deeply obscured starbursts.\n\nIn this {\\it Paper}, we present and discuss {\\it Spitzer}\\ MIR spectra and FIR photometry of 13 steep-radio-spectrum and\/or FR\\,II \\citep{Fan74} quasars at $z \\sim 0.3$. Throughout we assume a cosmological model with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\\Omega_m=0.3$, and $\\Omega_{\\Lambda}= 0.7$.\n\n\\begin{deluxetable*}{lcccccccccccc}\n\n\\setlength{\\tabcolsep}{.11cm}\n\\tablewidth{0pt}\n\\tabletypesize{\\footnotesize}\n\\tablecaption{Quasar Sample \\label{tab1}}\n\\tablehead{ \n\\colhead{} & \\colhead{} &\\colhead{} &\n\\colhead{log $L_{\\rm [O\\,III]}^{\\rm eelr}$} & \\colhead{log $\\lambda L_{4861}$} & \\colhead{} & \n\\colhead{log $P_{\\rm 178}^{\\rm tot}$} & \\colhead{Size} & \n\\colhead{$F_{\\rm 24 \\mu m}$} & \\colhead{$F_{\\rm 70 \\mu m}$} & \\colhead{$F_{\\rm 160 \\mu m}$} & \n\\colhead{log $\\lambda L_{\\rm 60 \\mu m}$} & \\colhead{log $L_{\\rm PAH}$}\n\\\\\n\\colhead{Name} & \\colhead{Designation} & \\colhead{$z$} & \n\\colhead{(erg s$^{-1}$)} & \\colhead{(erg s$^{-1}$)} & \\colhead{$\\alpha_{\\nu}$} & \n\\colhead{(W Hz$^{-1}$)} & \\colhead{(kpc)} & \n\\colhead{(mJy)} & \\colhead{(mJy)} & \\colhead{(mJy)} &\n\\colhead{(erg s$^{-1}$)} & \\colhead{(erg s$^{-1}$)}\n\\\\\n\\colhead{(1)} & \\colhead{(2)} & \\colhead{(3)} & \n\\colhead{(4)} & \\colhead{(5)} & \\colhead{(6)} & \n\\colhead{(7)} & \\colhead{(8)} & \\colhead{(9)} & \n\\colhead{(10)} & \\colhead{(11)} & \\colhead{(12)} &\n\\colhead{(13)} \n}\n\\startdata\n\\multicolumn{13}{c}{CSS Quasar with EELR} \\\\\n\\hline\n 3C\\,48&0137$+$3309&0.367&~~\\,42.75&45.11&0.82&28.41& \\hph3& 128.6& 754.6& ~475.9&46.11&~~\\,43.97\\\\\n\\hline\n\\multicolumn{13}{c}{FR\\,II Quasars with EELRs} \\\\\n\\hline\n 3C\\,249.1&1104$+$7658&0.312&~~\\,42.82&45.52&0.84&27.63&\\phn137&\\phn43.9&\\phn60.2& $<$14.0&44.84& $<$42.70\\\\\n Ton\\,616&1225$+$2458&0.268&~~\\,42.59&44.63&0.80&26.67&\\phn279&\\phn11.9&\\phn12.6& $<$13.5&44.04& $<$42.40\\\\\n Ton\\,202&1427$+$2632&0.366&~~\\,42.32&45.19&0.62&26.94& 1169&\\phn40.3&\\phn61.0& $<$11.5&44.99& $<$42.81\\\\\n 4C\\,37.43&1514$+$3650&0.371&~~\\,42.97&45.27&0.80&27.36&\\phn276&\\phn30.0&\\phn35.0& $<$13.0&44.76& $<$42.84\\\\\n 3C\\,323.1&1547$+$2052&0.264&~~\\,42.20&45.18&0.71&27.33&\\phn285&\\phn32.4&\\phn18.8& $<$14.1&44.19& $<$42.62\\\\\n PKS\\,2251+11&2254$+$1136&0.326&~~\\,42.47&45.17&0.73&27.37& \\tph47&\\phn42.2&\\phn30.1& $<$26.6&44.59& $<$42.72\\\\\n\\hline\n\\multicolumn{13}{c}{FR\\,II Quasars without EELRs} \\\\\n\\hline\n 4C\\,25.01&0019$+$2602&0.284& $<$40.95&45.11&0.66&26.82&\\phn204&\\phn67.8& 116.4&~~\\,53.6&45.06& $<$42.82\\\\\n 4C\\,13.41&1007$+$1248&0.241& $<$40.38&45.38&0.67&26.95&\\phn387&\\phn72.9&\\phn86.5& $<$22.1&44.78& $<$42.78\\\\\n 3C\\,246&1051$-$0918&0.344& $<$40.92&45.21&0.78&27.68&\\phn415&\\phn20.5&\\phn20.0& $<$15.8&44.46& $<$42.56\\\\\n 4C\\,49.22&1153$+$4931&0.334& $<$40.40&44.71&0.49&27.32& \\tph80&\\phn25.3&\\phn42.2& $<$13.6&44.75& $<$42.71\\\\\n 3C\\,351&1704$+$6044&0.372& $<$41.29&45.40&0.74&27.85&\\phn282&\\phn99.2& 173.7&~~\\,68.2&45.47& $<$43.08\\\\\n 4C\\,31.63&2203$+$3145&0.295& $<$40.94&45.41&0.16&26.99&\\phn352&\\phn67.2&\\phn83.3&~~\\,54.2&44.95& $<$42.81 \n\\enddata\n\\tablecomments{\nCol.\\,(1): Common name. \nCol.\\,(2): J2000.0 designation.\nCol.\\,(3): Redshift.\nCol.\\,(4): Extended {[O\\,{\\sc iii}]}\\ $\\lambda5007$ luminosity in logarithmic and\nupper limits for non-EELR quasars.\nCol.\\,(5): Optical continuum luminosity under H$\\beta$.\nCol.\\,(6): Radio spectral index ($f_{\\nu} \\propto \\nu^{-\\alpha_{\\nu}}$)\nbetween 0.4 and 2.7 GHz.\nCol.\\,(7): 178 MHz radio luminosity in logarithmic (The value for\nTon\\,202 was interpolated between 151 and 365 MHz fluxes).\nCol.\\,(8): Projected linear size of the radio source (References: \\citealt{Aku91,Bri94,Fen05,Gop00,Gow84,Kel94,Sto85}).\nCols.\\,(9)--(11): Observed flux densities in the MIPS 24, 70, and 160 $\\mu$m\\ bands.\nCol.\\,(12): Interpolated luminosity at rest wavelength 60 $\\mu$m\\ from MIPS photometry.\nCol.\\,(13): Observed luminosity and upper limits of the PAH 7.7$\\mu$m feature from IRS spectra.\nThe reference for Cols.\\,(4)--(6) is \\citet{Sto87}. Data in Col.\\,(7) are compiled from the NASA\/IPAC Extragalactic Database (NED). \n}\n\\end{deluxetable*}\n\n\\section{Sample Selection and {\\it Spitzer}\\ Observations}\n\nOur sample of quasars has been selected from the \\citet{Sto87} survey for extended emission around low-redshift QSOs. We have a principal sample of 12 FR\\,II quasars (which we will refer to as ``the FR\\,II sample''), half of which have luminous EELRs and half of which do not, to fairly strong upper limits. The EELR and non-EELR quasars have been paired as closely as possible in both redshift and rest-frame optical continuum luminosity. Characteristics of all of the objects are tabulated in Table~\\ref{tab1}. For the EELR and non-EELR quasars, the mean redshifts are 0.318 and 0.312, respectively. The continuum luminosities are measured at H$\\beta$, and the mean values for log $\\lambda L_{4861}$ (erg s$^{-1}$) are 45.23 and 45.26. The radio powers of the subsamples are also similar, with mean values for log $P_{178 {\\rm MHz}}$ (W Hz$^{-1}$) of 27.32 and 27.44. We also include 1 EELR quasar with a compact steep-spectrum (CSS) one-sided radio jet, 3C\\,48, which is a special object in a number of other ways (see, e.g., \\citealt{Can00a,Sto07}) and will be discussed apart from the FR\\,II sample.\n\nDeep MIR spectra of all 13 quasars were obtained with the {\\it Spitzer}\\ Infrared Spectrograph \\citep[IRS;][]{Hou04} as part of our Cycle 4 GO program (PID 40001). All of the objects were observed in the standard staring mode with the Short Low 1st order module (SL1, 7.5$-$14.2 $\\mu$m). The slit width was 3.7\\arcsec\\ or 16.5 kpc at $z = 0.3$. Each observation consisted of a 60 s ramp for 8$-$16 cycles with two nod positions. The total integration time was about 16$-$32 minutes. We used the IDL procedure IRSCLEAN\\footnote{All of the data reduction software mentioned in this section is available at the {\\it Spitzer}\\ Science Center website at http:\/\/ssc.spitzer.caltech.edu\/.} (ver. 1.9) to remove bad pixels in the pipeline post-BCD coadded nod-subtracted images and extracted the spectra with SPICE (ver. 2.1.2) using the optimal extraction method for point sources. \n\nFIR photometry at 24, 70, and 160 $\\mu$m\\ was obtained with the Multiband Imaging Photometer for {\\it Spitzer}\\ \\citep[MIPS;][]{Rie04} either as part of our GO program or from the archive (PIDs 00049, 00082, 20084). All of the observations were obtained using the small-field default resolution photometry mode. Typical integration time per pixel was 42 s at 24 $\\mu$m\\ (3 s exposure for 1 cycle or 3 s$\\times$1), 700 s at 70 $\\mu$m\\ (10 s$\\times$7), and 80 s at 160 $\\mu$m\\ (10 s$\\times$4). Our data reduction started from the pipeline BCD files. For 24 $\\mu$m\\ observations, final mosaic images were constructed with the MOPEX software after self-flat-fielding and background correction. At 70 and 160 $\\mu$m, we used the pipeline filtered BCDs to construct mosaics for most of the sources. 3C\\,48 and 3C\\,351 are so bright at 70 $\\mu$m\\ that they have to be masked out before applying the spatial and temporal filters; we thus used the offline IDL filtering procedures by D.~Fadda and D.~Frayer to generate filtered BCDs. The non-filtered 160 $\\mu$m\\ BCDs of 3C\\,48 were used because they result in a better mosaic image than the filtered BCDs. Point sources were extracted with APEX, and point response function (PRF)-fitting photometry was carried out. All of the sources were detected at 24 and 70 $\\mu$m, but only 4 were detected at 160 $\\mu$m. Upper limits at 160 $\\mu$m\\ were estimated from the standard deviation images associated with the mosaics using an aperture of 48\\arcsec\\ and corrected for the finite aperture size by multiplying the values by 1.6 (MIPS Data Handbook). The photometry results are listed in Table~\\ref{tab1}. The systematic uncertainties were estimated to be 4\\% at 24 $\\mu$m, 5\\% at 70 $\\mu$m, and 12\\% at 160 $\\mu$m\\ \\citep{Eng07,Gor07,Sta07}. Statistical errors from the PRF fitting are much smaller and therefore could be neglected.\n\n\\begin{figure*}[!tb]\n\\epsscale{1.1}\n\\plotone{f1.ps}\n\\caption{\nAverage IRS spectrum (thick black line) of ({\\it a}) six EELR quasars and ({\\it b}) six non-EELR quasars. Individual spectra have been normalized at 8.8 $\\mu$m\\ and are shown as light grey curves. Overplotted for comparison are the IRS spectrum of 3C\\,48 (blue), the average spectrum of 28 PG QSOs (red), and the average ``intrinsic\" QSO spectrum derived from 8 FIR-weak PG QSOs after subtraction of a starburst component (green; \\citealt{Net07}); all have been arbitrarily offset for clarity. Important emission features are labelled in panel {\\it b}.\n\\label{fig:irs}} \n\\end{figure*}\n\n\\begin{figure*}[!tb]\n\\epsscale{0.5}\n\\plotone{f2a.ps}\n\\hskip 0.2in\n\\plotone{f2b.ps}\n\\caption{\nMIR to FIR SEDs of FR\\,II quasars: ({\\it a}) EELR quasars and ({\\it b}) non-EELR quasars. The red curves are the IRS spectra and the data points (detections) and downward arrows (upper limits) are MIPS photometry. Overplotted for comparison is the average ``intrinsic\" QSO SED, scaled to the IRS luminosity of each source \\citep[grey dashed curve;][]{Net07}. In the panel for 3C\\,351, the most FIR luminous FR\\,II quasar in our sample, we also show the SED of 3C\\,48 as the blue dash-dot line (no scaling has been performed). \n\\label{fig:sed}} \n\\end{figure*}\n\n\\begin{figure*}[!tb]\n\\epsscale{1.1}\n\\plottwo{f3a.ps}{f3b.ps}\n\\caption{\n({\\it a}) PAH 7.7 $\\mu$m\\ luminosity vs. redshift for various\nsamples, as indicated. The solid curve indicates a constant flux limit of\n$6\\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$, approximating that of the PG\nsample. Note that the upper limits for our FR\\,II\nquasar sample (larger red and blue arrows) are below or comparable to\nthe detections for the PG QSO sample, in spite of the higher redshift\nof the FR\\,II sample. The symbols given in the legend for this panel\napply to panel $b$ as well.\n({\\it b}) PAH 7.7 $\\mu$m\\ luminosity vs. optical continuum luminosity.\nThe PG QSOs from \\citet{Sch06} are shown as filled black squares\n(detections) or downward black arrows (upper limits). Similarly, the\nhigh-redshift mm-bright QSOs from \\citet{Lut08} are shown as gray\nstars or large gray downward arrows, and FR\\,II 3CR quasars and radio\ngalaxies from \\citet{Shi07} are shown as orange diamonds or downward\narrows. Our FR\\,II quasars with EELRs are shown as downward red\narrows, and the ones without EELRs are downward blue arrows. 3C\\,48 is\nindicated by a green filled circle. The only FR\\,II quasar that is in the PG sample\nbut not in our sample, PG\\,2349$-$014, is marked by a filled black\nsquare with a white plus sign. The dotted and dashed lines show\nconstant ratios of star formation rates and black hole accretion\nrates. The three common objects between our sample and the PG QSOs\nare connected with grey solid lines. The $\\sim4\\times$ reduced upper\nlimits of $L_{\\rm PAH}$ for the three common objects result from our\nmuch longer IRS integrations.\nThe FR\\,II quasars are clearly under-luminious in the PAH emission for\ntheir optical luminosity (which can be considered to be a proxy for\nblack-hole accretion rate) with respect to both the PG QSOs that are\nnot FR\\,II quasars and the mm-bright QSOs.\n\\label{fig:opt}}\n\\end{figure*}\n\n\\section{Results}\\label{results}\n\nWe did not detect PAH emission in any of the FR\\,II quasars, although strong PAH features were clearly seen at 6.2, 7.7, and 8.6 $\\mu$m\\ in the spectrum of the CSS quasar 3C\\,48. It was already known that the host galaxy of 3C\\,48 is undergoing intense star formation, with evidence from both its warm {\\it IRAS} FIR color and the young stellar populations revealed by deep optical spectra \\citep{Can00a}. The luminosity of the PAH 7.7 $\\mu$m\\ feature ($L_{\\rm PAH}$) is about $2.4\\times10^{10}$ $L_{\\odot}$, implying a SFR of 680 $M_{\\odot}$\\ yr$^{-1}$ (SFR[$M_{\\odot}$\\ yr$^{-1}$] = 283 $L_{\\rm PAH}$[10$^{10}$ $L_{\\odot}$], assuming $L_{\\rm PAH}\/L_{\\rm IR}$ = 0.0061 and SFR[$M_{\\odot}$\\ yr$^{-1}$] = 1.73 $L_{\\rm IR}$[10$^{10}$ $L_{\\odot}$]; \\citealt{Vei09,Ken98}. Here, and throughout this paper, $L_{\\rm IR}$ refers to the {\\it star-forming} luminosity in the rest-frame 8--1000 $\\mu$m range). The aromatic features remain undetected even in the stacked spectrum of the FR\\,II sample. In Fig.~\\ref{fig:irs} we show average spectra of two subsamples---FR\\,II quasars with and without EELRs, each containing six objects. The average spectra of the two subsamples do not differ: basically, one sees only narrow high-ionization emission lines superposed on a featureless continuum. \\citet{Net07} provided a composite spectrum of 28 PG QSOs with IRS spectra, as well as an ``intrinsic\" QSO spectral energy distribution (SED) from 1.2 to 70 $\\mu$m, which was constructed by combining 8 FIR-weak PG QSOs (presumably with little star formation) and subtracting a scaled starburst component. The MIR portion of these two SEDs is also shown in Fig.~\\ref{fig:irs} for comparison. Broad PAH emission is clearly seen in the composite PG QSO spectrum, but by design it is absent from the intrinsic QSO spectrum. The average spectra of FR\\,II quasars are consistent with being entirely intrinsic QSO emission. \n\nWe compute the upper limits to the PAH 7.7$\\mu$m\\ flux based on the noise in the region between 6.5 and 8.5 $\\mu$m\\ excluding the narrow [Ne\\,{\\sc vi}] line. A low-order polynomial fit to the continuum is subtracted from each individual spectrum and the standard deviation (i.e.,\\ noise) of the residual is determined. To obtain the upper limits, we assume that the undetected PAH has a Lorentzian profile with the same width (0.6 $\\mu$m\\ FWHM) as our best fit to the 3C\\,48 PAH 7.7$\\mu$m\\ feature and that its peak amplitude cannot be greater than 5 times the noise level. This peak-to-noise ratio of 5 corresponds to a total S\/N of $\\sim$14 if we use the optimal aperture of 1.45$\\times$ FWHM. The PAH upper limits reported in Table~\\ref{tab1} are consistent with the upper limits obtained by \\citet{Sch06} for the 3 quasars that are in both samples, once one takes into account the $\\sim$18$\\times$ longer integrations of our observations. Furthermore, we found that the PAH feature would have been clearly seen in both stacked spectra and individual spectra if synthetic PAH emission with these upper limits were to be added to every object. \n\nFigure~\\ref{fig:sed} shows the MIR--FIR SEDs of our FR\\,II quasars. A scaled version of the \\citet{Net07} intrinsic QSO SED is included in each panel for comparison. Most of the SEDs probed by IRS show steady rises towards both ends of the spectrum and a dip near 8 $\\mu$m; these features are readily seen in the intrinsic QSO SED and can be attributed to a combination of hot dust blackbody emission and silicate emission that peaks near $\\sim$10 $\\mu$m. As an example of starburst quasars, the SED of 3C\\,48 is overplotted in the panel of 3C\\,351, the most FIR luminous FR\\,II quasar in our sample. The SEDs of the FR\\,II quasars can be mostly explained as intrinsic QSO emission without a starburst component. The SEDs of quasars with EELRs and the ones without EELRs are essentially the same. These results are consistent with our PAH measurements, as described in the two preceding paragraphs.\n\nA luminosity correlation between the optical continuum and the PAH emission has been observed in low-redshift PG QSOs and high-redshift millimeter-bright QSOs \\citep{Net07,Lut08}. This correlation implies an intimate connection between QSO nuclear activity, as indicated by the strength of the continuum at rest-frame 5100 \\AA, and star formation in their host galaxies, as traced by the PAH emission. While the median redshift of our sample is significantly above that of the PG sample, our IRS exposures, which are typically $\\sim18$ times those for the PG sample, are sufficient to give luminosity upper limits that are lower than the detections or upper limits for the bulk of the PG sample (see Fig.~\\ref{fig:opt}$a$). Figure~\\ref{fig:opt}$b$ examines whether our FR\\,II quasars are consistent with this optical-continuum\/PAH correlation. We converted the continuum luminosities under H$\\beta$ from \\citet{Sto87} to $\\lambda L_{5100}$ by multiplying the values by a factor of 1.74 (after the correction for cosmology), which was determined by comparing the two continuum luminosities of the three quasars in both our sample and the \\citet{Net07} sample. \nAlso plotted in Figure~\\ref{fig:opt} are the 32 FR\\,II radio galaxies and 15 FR\\,II quasars from the 3CR sample of \\citet{Shi07}, which have both 7.7 $\\mu$m\\ PAH measurements from IRS spectra and {[O\\,{\\sc iii}]}\\,$\\lambda5007$ and\/or {[O\\,{\\sc ii}]}\\,$\\lambda3727$ fluxes from the literature (see \\citealt{Jac97} and references therein\\footnote{An electronic version of the catalog is available at http:\/\/www.science.uottawa.ca\/$\\sim$cwillott\/3crr\/3crr.html}). We converted their {[O\\,{\\sc iii}]}\\ or {[O\\,{\\sc ii}]}\\ luminosities into $ \\lambda L_{5100}$ based on the observed {[O\\,{\\sc iii}]}-continuum correlation for type-1 QSOs \\citep[$ \\lambda L_{5100} \\simeq 320\\times L_{\\rm [O III]}$;][]{Zak03,Hec04} and assuming {[O\\,{\\sc iii}]}\/{[O\\,{\\sc ii}]}\\ = 3.3, as measured from sources where fluxes of both lines are available. \nIt is evident that the FR\\,II quasars\/radio galaxies fall systematically below the correlation established by the PG QSOs and the \\citet{Lut08} mm-bright QSOs.\n\nFive of the QSOs in the \\citet{Net07} sample are radio-loud, among which three are also in our sample (4C\\,13.41, 4C\\,31.63, and PKS\\,2251+11). The remaining two are PG\\,1302$-$102 and PG\\,2349$-$014. PG\\,1302$-$102 has a flat radio spectrum and PAH was not detected ($L_{\\rm PAH} < 5.3\\times10^9$ $L_{\\odot}$; \\citealt{Sch06}). PG\\,2349$-$014 has a steep radio spectrum and an FR\\,II radio morphology. Although the aromatic 7.7$\\mu$m\\ feature has been claimed to be detected in PG\\,2349$-$014 ($L_{\\rm PAH} = 3.6\\times10^9$ $L_{\\odot}$; \\citealt{Sch06}), the low PAH luminosity places it at the transitional region between the FR\\,II sources and the radio-quiet QSOs in Fig.~\\ref{fig:opt}{\\it b} (the black square with a white plus sign). \n\nThe optical continuum luminosity at 5100 \\AA\\ is widely used to estimate the bolometric luminosity for type-1 AGN ($L_{bol} \\sim 10 \\lambda L_{5100}$), which in turn provides the black hole accretion rate (BHAR) given a typical radiative efficiency of $\\eta = 0.054$ \\citep{Mart08}. The PAH luminosity can be converted into a SFR. As shown in Fig.~\\ref{fig:opt}$b$, most of the PG QSOs in the \\citet{Net07} sample and the mm-bright QSOs in the \\citet{Lut08} sample are within a narrow range of SFR\/BHAR ratios, 15 $<$ SFR\/BHAR $<$ 150; while our FR\\,II quasars show significantly lower SFR at any given BHAR (we have conservatively assumed that $L_{\\rm PAH} = 0.0061 L_{\\rm IR}$; see discussion of this ratio in \\S \\ref{sf_eelr}). \nAs the referee has pointed out, the observed separation in Figure~\\ref{fig:opt}{\\it b} could also be explained by an on-average 10$\\times$ higher radiative efficiency, $\\eta$, for FR\\,II sources than for radio-quiet QSOs. However, the theoretically allowed range of $\\eta$ is only from 0.054 to 0.42 (non-rotating and maximally rotating black holes, respectively; \\citealt{Sha83}), and, to make it worse, the presence of a jet can {\\it reduce} the efficiency \\citep{Jol08}. Therefore, a difference in $\\eta$ could, at best, partially explain the large offset that we observe in Figure~\\ref{fig:opt}{\\it b}. \nWe note that to explain the Magorrian bulge mass$-$black hole mass relation, under the assumption that star formation and black hole accretion are always coeval, a SFR\/BHAR $\\simeq$ 700 is required \\citep{Har04}. The measured low values have been used to argue for a much longer duration for star formation than that for the AGN activity \\citep{Net07}, making the existence of a SFR-BHAR correlation questionable.\n\n\\section{Implications}\n\n\\subsection{Can Quasar EELRs be Produced by Starburst Superwinds?}\\label{sf_eelr}\n\nMost of our EELR quasars have upper limits to $L_{\\rm PAH}$ of less than $2\\times10^9$ $L_{\\odot}$. The non-detection of PAH in the stacked spectrum of the entire sample of 12 FR\\,II quasars shows that such upper limits are quite conservative. These upper limits imply a SFR $<12$ $M_{\\odot}$\\ yr$^{-1}$. In the above calculation we used a higher $L_{\\rm PAH}\/L_{\\rm IR}$ ratio of 0.028 than in calculating the SFR of 3C\\,48 (0.0061, \\S~3), because it is both observed \\citep[e.g.,][]{Sch06} and theoretically predicted \\citep[e.g.,][]{Dal02} that $L_{\\rm PAH}\/L_{\\rm IR}$ increases with decreasing IR luminosity. Following \\citet{Shi07}, we determined $L_{\\rm PAH}\/L_{\\rm IR} \\simeq 0.028$ for $L_{\\rm PAH} = 2\\times10^9$ $L_{\\odot}$\\ from the star-forming templates of \\citet{Dal02}. However, as discussed in \\citet{Fu06}, to inject enough momentum into a typical EELR within a dynamical timescale of 10 Myr to explain the observed chaotic velocity field, the SFR of the central star burst must exceed $\\sim60$ $M_{\\odot}$\\ yr$^{-1}$ (eqs. [2]$-$[3] in \\citealt{Vei05}). More importantly, the absence of significant PAH emission in both FR\\,II quasars with EELRs and the ones without strongly suggests that starburst is not a critical factor in the formation of an EELR. This conclusion is also supported by the MIPS photometry, which indicates that the FIR SED is dominated by warm dust directly heated by the quasars (Fig.~\\ref{fig:sed}).\n\n\\subsection{Star Formation in FR\\,II Quasars}\n\nRelations among mergers, starbursts and nuclear activities have long been sought after by extragalactic researchers \\citep[e.g.,][]{San88}. Such connections are attractive because starbursts and AGN require many common triggering conditions, and gas-rich mergers can in principle provide these necessary conditions. Optical spectroscopy of QSO host galaxies, although technically challenging, has revealed diverse starforming properties. While active\/recent star formation ($< 300$ Myr) have been detected in the host galaxies of certain classes of QSOs, such as FIR-loud QSOs and post-starburst QSOs \\citep[e.g.,][]{Can01}, the majority of optically luminous QSOs show signs of past merger events and massive star formation that happened 1$-$2 Gyr ago (see \\citealt{Can06} for a review; also \\citealt{Can07,Ben08}). A similar result has been found for the FR\\,II radio galaxy 3C\\,79, which shows both a luminous EELR and evidence for a luminous quasar hidden from our direct view \\citep{Fu08}.\n\nAbout 40\\% of the PG QSOs studied by \\citet{Sch06} show PAH emission in individual {\\it Spitzer}\\ IRS spectra, and PAH is also present in the average spectrum of the remaining QSOs. In sharp contrast, our considerably deeper IRS spectra of steep-spectrum radio-loud quasars show no evidence for PAH emission, either individually or averaged, except in the case of the CSS quasar 3C\\,48. In fact, previous {\\it Spitzer}\\ observations have found that powerful radio galaxies and quasars from the 3CR catalog \\citep{Spi85} generally show FIR colors consistent with hot dust directly heated by the AGN \\citep{Shi05} and weak or undetectable PAH emission \\citep{Haa05,Ogl06,Cle07,Shi07}, suggesting a lower level of star formation compared to the QSOs from the PG and 2MASS samples \\citep{Sch83,Cut01}. The fraction of 3CR quasars\/radio galaxies that show detectable PAH features is only 18\\%, compared to $\\sim$48\\% for both PG and 2MASS QSOs in the same redshift range ($z < 0.5$) and observed with the same instrument.\n\nThere are substantial reasons for believing that radio-loud quasars, at least at low redshifts, are drawn from a population of massive elliptical galaxies that may have suffered mergers in the fairly recent past, while many radio-quiet QSOs are found either in disk galaxies or in ongoing gas-rich mergers \\citep{Sik07,Sik08,Wol08}. We can test this explanation by looking at the PG QSOs in the \\citet{Sch06} sample that also have morphological determinations by \\citet{Guy06} or elsewhere in the literature. Of the 9 objects which had individual PAH detections and for which morphological information is available (8 of which are classified by Guyon et al.), all are classified either as having a disk or as ``strongly interacting,'' a term that almost always implies a gas-rich interaction or merger. It is not at all surprising that such host galaxies should show substantial star formation. \n\n\\subsection{How Does 3C\\,48 Fit In?}\n\nAt first sight, the CSS quasar 3C\\,48 seems to be a counter-example to the trend both \\citet{Shi07} and we have found for steep-radio-spectrum quasars. In particular, 3C\\,48 has one of the most luminous EELRs among low-redshift quasars, yet its host galaxy is also undergoing star formation at a prodigious rate (\\S \\ref{results}). Is 3C\\,48 simply an anomaly, perhaps related to its status as a CSS source, or can it be understood within the framework we have described? Because of the small size of our sample and the rather large uncertainties in quasar lifetimes and duty cycles, we can do little more than explore some of the possibilities related to these options. First, however, we summarize some of the special characteristics of 3C\\,48 and its host galaxy (see \\citealt{Sto07} for more detail).\n\nBecause 3C\\,48 shows a one-sided jet, one might suspect that the jet is oriented almost along our line-of-sight and is strongly Doppler boosted; however, \\citet{Wil91} have argued from the weakness of the nuclear radio component that the jet is oriented closer to the plane of the sky and that the radio structure is intrinsically one sided; so the projected jet length of $\\sim3$ kpc is probably not too far from its actual length. 3C\\,48 is one of the only two CSS sources among powerful quasars with $z<0.5$. Such sources are rare, not because they are intrinsically uncommon, but because the CSS stage of an extended radio source likely lasts only a short time, typically $\\sim10^4$ years \\citep{deS99}. The host galaxy of 3C\\,48 is clearly undergoing a major merger \\citep{Sto87,Can00a,Sch04,Sto07}. The evidence suggests that the current episode of quasar activity has been triggered only relatively recently. \n\nWe consider first the possibility that 3C\\,48 is simply an earlier stage of the formation of an EELR like those in our FR\\,II EELR subsample. This picture has considerable appeal, because we do see a massive ($\\gtrsim10^9$ M$_{\\odot}$) outflow of ionized gas extending over a large solid angle from a region near the base of the CSS radio jet, with velocities ranging up to $\\sim1000$ {km s$^{-1}$} \\citep{Cha99,Can00a,Sto07}. This outflow is consistent with the sort of quasar-jet-driven wind we have supposed to be responsible for producing the large-scale EELRs seen in 3C\\,48 and other quasars. Nevertheless, there are difficulties with this simple evolutionary view. Since 3C\\,48 also has an EELR on scales much larger than that of the current radio jet, one would have to appeal to a previous episode of quasar activity in order to produce it according to the scenario we have outlined. It is also not clear that the respective lifetimes of the EELR and the starburst would allow evidence for the starburst to die out quickly enough. EELR lifetimes are likely on the order of $10^7$ years, both from their close association with radio sources and from dynamical estimates. Vigorous star formation in 3C\\,48 is taking place not only in the nuclear region, but also over most of the host galaxy \\citep{Can00a}. While we do not have similarly detailed spectroscopy for any of the host galaxies in our FR\\,II sample, our deep spectroscopy of the FR\\,II EELR radio galaxy 3C\\,79 has shown that its last significant episode of star formation was at least 1 Gyr ago \\citep{Fu08}. This EELR, at least, could not have been produced in conjunction with a starburst like that occurring in 3C\\,48. While it is clear that some QSOs are triggered more-or-less simultaneously with massive starbursts \\citep[e.g.,][]{Can01}, many showing similar levels of QSO activity seem to have been triggered $\\sim1$ Gyr {\\it after} major starbursts \\citep{Can06,Can07,Ben08}, although we do not yet have a clear understanding why this should be so.\n\nIt seems, then, that 3C\\,48 is not likely to be an example of a direct precursor to objects like those in our FR\\,II EELR subsample. Nevertheless, it does seem to be a close cousin. Its EELR has a luminosity similar to those of our other EELR quasars, and the massive outflow of ionized gas from the base of the radio jet seems a striking confirmation of the sort of process we have inferred, from less direct evidence, to have produced the massive EELRs in our FR\\,II subsample \\citep[][and references therein]{Fu09}. The crucial difference between 3C\\,48 and the rest is in whether a massive starburst is triggered approximately contemporaneously with the formation of the EELR. This difference, in turn, is likely to depend on the details of the mechanism by which gas is supplied to the system. In particular, the uniformly and abnormally low metallicities of the broad-line gas in the EELR quasars for which this measurement can be made \\citep{Fu07a} indicate an {\\it external} source for this gas and suggest the mergers of a gas-rich late-type galaxies with massive early-type galaxies with little cold gas of their own. Although we do not have direct metallicity information for the broad-line region in 3C\\,48, the morphology of the merger, the large amounts of molecular gas present \\citep{Win97}, and the fact that it has $L_{\\rm IR} > 10^{12}$ $L_{\\odot}$, qualifying it as an ``ultra-luminous infrared galaxy,'' all suggest that it is a merger between two roughly equal-mass gas-rich galaxies.\n\n\\subsection{Overview}\n\nOur deep {\\it Spitzer} IRS spectra and MIPS photometry of matched subsamples of FR\\,II quasars with and without luminous EELRs give tight upper limits to current SFRs for all of the quasars in both subsamples, supporting and extending earlier results on FR\\,II quasars and radio galaxies by \\citet{Shi07}. These upper limits indicate that SFRs, relative to BHARs, are generally much lower in FR\\,II quasars than they are in optically selected QSO samples. In the FR\\,II quasars, then, very little bulge mass is being added via star formation during the current episode of black-hole growth. Our star-formation upper limits are also sufficiently low that we can discount the possibility of galactic-scale superwinds resulting from star formation in the host galaxies of these quasars. As we mentioned in \\S \\ref{intro}, the most luminous EELRs are always associated with quasars with strong radio jets, yet there is little significant morphological correspondence between the EELRs and the radio structure. This fact indicates a mechanism connected with the production of the radio jet that is much less strongly collimated than the jet itself. We have suggested in this context that the initiation of an FR\\,II jet is accompanied by a nearly hemispherical blast wave \\citep{Fu07b}. In the usual case of two jets, these blast waves can be capable, at least in some cases, of clearing most of the interstellar medium from the host galaxy. Although, as we have emphasized, 3C\\,48 is atypical in several ways, it is not unreasonable to suppose that it gives us a fairly typical picture of a very young radio jet; and in this case we see a massive, high-velocity, wide-solid-angle outflow coming from a region near the base of the radio jet, tending to corroborate the picture we have suggested for the FR\\,II quasars with luminous EELRs.\n\n\\acknowledgments \nH. F. is grateful for the hospitality of the Purple Mountain Observatory, where part of this work was completed. We thank Luis Ho, David Rupke, Jong-Hak Woo, Yanling Wu, and Lin Yan for helpful discussions. We also thank the anonymous referee for helping us to improve the paper and clarify some obscure points. This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL\/Caltech and by NSF under grant AST-0807900. This research has made use of the NASA\/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nMassive stars, $M \\ga 8~M_{\\odot}$, can play a dominant role in the\nevolution of their host galaxies via feedback phenomena such as\nionizing radiation, strong stellar winds and supernova\nexplosions. These processes exert a large influence on the outcome of\nsubsequent star formation (SF). Massive SF can therefore be considered\nas an integral part of the overall SF process and a key ingredient for\na complete description of it. However, our comprehension of many\naspects of massive SF is still unsatisfactory. For example, it is\nstill not known whether massive SF is intrinsically different to the\ngenerally much better understood process of SF at low masses or not\n\\citep[see e.g.][]{ZinneckerandYorke2007}.\n\n\\smallskip\n\nThe uncertainties regarding massive SF are in part the result of the\nshort Kelvin-Helmholtz (KH) timescale associated with massive stars:\nthey contract rapidly to sustain a high luminosity while still\nacquiring their mass. If the accretion proceeds spherically, the\nensuing radiation pressure could limit the final stellar mass to\napproximately 40~$M_{\\odot}$\n\\citep[][]{Larson1971,Kahn1974,Wolfire1987}. This led to the\nsuggestion that massive SF is intrinsically different to the low mass\ncase. However, recent radiation-hydrodynamical simulations clearly\ndemonstrate that the radiation is actually channelled through\noptically thin polar cavities while accretion continues through an\nequatorial disc\n\\citep[][]{YorkeandSonnhalter2002,Krumholz2009,Kuiper2010}. Therefore,\ndeviations from spherical symmetry, which are suggested by the\nspectral energy distributions of young massive stars\n\\citep[][]{Harvey1977,Guertler1991}, remove the limit imposed by\nspherical accretion. Nonetheless, confirming that massive stars build\nup their final mass via disc accretion by direct observations is\nchallenging. Massive young stellar objects (MYSOs) are generally\nlocated at kpc distances and their AU-scale circumstellar environment\nremains unresolved with single dish telescopes at most wavelength\nregions \\citep[see e.g.][]{Cesaroni2007}.\n\n\\smallskip\n\nLong-baseline infrared interferometry is one of the few techniques\nwhich offers the milli-arcsecond (mas) resolution required to probe\nfor MYSO discs \\citep[see\ne.g.][]{Follert2010,Vehoff2010,deWit2011,Grellmann2011}. In\nparticular, an exemplary result is delivered by \\citet{Kraus2010}\nusing the Very Large Telescope Interferometer (VLTI) and the\nnear-infrared beam-combiner AMBER. These authors reconstruct a\n$K$-band synthesis image of a 20$M_{\\odot}$ MYSO with an angular\nresolution of 2.4~mas. The resultant image shows hot material in a\ndisc-like geometry of approximately 20~AU in size. As a result, a\nconsensus is beginning to emerge that the disc accretion scenario can\naccount for the formation of stars up to at least $\\sim\n30\\,M_{\\odot}$. This implies that the MYSO circumstellar environment\ndeviates significantly from spherical symmetry. In this scenario,\nrotation, accretion discs and outflow activity play a crucial role in\nproviding the asymmetric environment required to facilitate accretion\n\\citep[see e.g.][]{Beuther2002}.\n\n\n\\smallskip \n\nThe combination of a high stellar luminosity with accretion and\noutflow activity results in the circumstellar environment of MYSOs\nbeing shaped by several forces. Determining the relative importance of\nthese phenomena in sculpting the circumstellar environments of MYSOs\nrequires spatially resolved observations. In particular, the\nmid-infrared (MIR) wavelength region provides a wealth of information\non the geometry of the accretion environment. Furthermore, this\ninformation can be accessed by both interferometric techniques and\ndiffraction limited single-dish imaging with 8m class telescopes. MIR\ninterferometric observations reveal that the $N$-band emission of\nMYSOs on scales of 50 to 100~AU is dominated by warm dust located in\nthe envelope \\citep[][]{Dewit2007,Linz2009,Vehoff2010}. It is expected\nthat such envelopes will contain outflow cavities as there are many\ndetections of outflow activity associated with MYSOs\n(e.g. \\object{G35.20-0.74} in De Buizer 2006\\nocite{DeBuizer2006};\n\\object{IRAS 20126+4104} in De Buizer 2007\\nocite{DeBuizer2007};\n\\object{Cep\\,A HW2} in de Wit et al. 2009\\nocite{deWit2009}). In\n\\citet[][]{deWit2010}, we argue that, specifically, the warm dust\nresponsible for the $N-$band emission of W33A is located in the\nenvelope close to the walls of the cavities evacuated by the outflow.\n\n\\smallskip\n\nThe aim of this paper is twofold. We aim to assess the source of the\nMIR emission of a sample of MYSOs and thus test the premise that their\nMIR emission is dominated by warm dust located in outflow cavity\nwalls. This will then allow us to determine whether outflows play an\nimportant role in shaping the environments of MYSOs ({{as opposed to,\n for example, rotation}}). We address these goals by comparing\nspatially resolved images of a sample (20 objects) to appropriate\nmodels. Our approach is similar to the work presented previously in\n\\citet[][hereafter DW09]{deWit2009} with the addition of 2D\nmodelling. We present MIR VLT images observed at $20\\,\\mu$m for a\nsample of MYSOs. We then compare the spatially resolved images to\ntwo-dimensional, axis-symmetric dust radiative transfer models that\nfeature rotating, collapsing envelopes and outflow cavities \\citep[the\nmodels of][]{W22003,W12003}. The paper is structured as follows. We\ndescribe the sample selection, the observations and data reduction\nprocedures in Sect. \\ref{obs}. The MIR images are described in\nSect. \\ref{data} and the radiative transfer model and its use are\ndetailed in Sect. \\ref{model}. The results of the modelling are\npresented in Sect. \\ref{mod_res} and discussed in\nSect. \\ref{disc}. Finally, our conclusions are given in\nSect. \\ref{conc}.\n\n\\section{Selection, observations and data reduction}\n\\label{obs}\n\nWe aim to perform our analysis on a representative sample of MYSOs, in\norder to draw general conclusions. Selecting such a sample is not\ntrivial. Initial attempts employed $IRAS$ data \\citep[see\ne.g.][]{Palla1991,Molinari1996,Sridharan2002}. As a result, such\ncatalogues suffered from source confusion due to the large beam of\n$IRAS$. To rectify this problem and generate a Galaxy-wide, unbiased,\nMYSO sample, we have conducted a survey to detect and characterise\nMYSOs: the RMS survey \\citep[][]{Lumsden2002,James-RMS}. The RMS is\nbased on data of the $MSX$ survey \\citep[][]{Egan2003}. $MSX$ data\noffers a significant improvement over $IRAS$ data in terms of\nresolution (e.g. arcsecond rather than arcminute resolution), and thus\nenables the selection of a representative sample of MIR bright MYSOs.\n\n\\smallskip\n\nFrom the RMS database\\footnote{http:\/\/www.ast.leeds.ac.uk\/RMS\/} we\nselected objects with distance estimates within 3~kpc\\footnote{Some of\n distances estimates have since been revised to larger values.} and\nwith 21\\,$\\mu$m $MSX$ fluxes larger than 30\\,Jy. The flux limit was\nimposed in order to obtain a decent signal-to-noise ratio in the wings\nof the resolved profiles. In total, we observed 19 objects at\n20~$\\mu$m using the VLT Imager and Spectrometer for the Mid Infrared\n\\citep[VISIR,][]{VISIR} mounted at the Cassegrain focus of UT3 of the\nVLT (see Table\\ref{t1}). Observations were conducted using the imaging\nmode of the instrument and the Q3 filter which has a central\nwavelength of 19.5~$\\mu$m and a half-band-width of 0.4~$\\mu$m. During\nthe observations, VISIR was equipped with a DRS 256$\\times$256 Si:As\ndetector with an angular pixel size of 0.127\\arcsec. This\nconfiguration enabled us to obtain oversampled, diffraction limited\nimages, which with 8~m class telescopes results in an angular\nresolution of approximately 0.6\\arcsec.\n\n\\smallskip\n\nIn this paper, we also present the imaging observations and analysis\nof a key MYSO object \\object{W33A} (\\object{G012.9090-00.2607}). It\nwas observed at the slightly longer wavelength of 24.5~$\\mu$m with the\nCOMICS instrument \\citep[Cooled Mid Infrared Camera and\nSpectrometer][]{Kataza2000} mounted on the Cassegrain focus of the\nSubaru telescope. We employed the imaging facility of COMICS and used\na filter centred at 24.5~$\\mu$m (see DW09 for filter response\nfunctions). During the observations, COMICS was equipped with a\n320$\\times$240 Si:As IBC detector which provides oversampled\ndiffraction limited images with a pixel size of 0.13\\arcsec. A log of\nall the observations is presented in Table \\ref{t1}.\n\n\\smallskip\n\nFor both the VISIR and COMICS observations, standard stars were\nobserved to provide a reference point-spread-function (PSF). The\nstandard stars selected are MIR bright,$\\sim$100~Jy at 25~$\\mu$m ,\nisolated sources which are expected to be unresolved. Data reduction,\nconsisting of shifting and adding nodded and chopped images, was\nconducted using routines written in {\\sc{idl}}. The resultant images\nwere not astrometrically corrected. Therefore, the images are\npresented in terms of distance from the peak of the intensity\ndistribution of the target source. The azimuthally averaged intensity\nprofiles of the standards are displayed in Fig. \\ref{psfs}. The radial\nprofiles of the VISIR standard stars are generally consistent until a\nlevel of approximately 0.001 times the peak flux and appear to behave\nas point sources. The standard star of the COMICS data (HD 124897) is\nof lower signal-to-noise ratio but is similar out to 1 percent of the\npeak flux to the standards presented in DW09.\n\n\\begin{center}\n \\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=0.4\\textwidth]{fig_1_a.ps} \n \\caption{The azimuthally averaged intensity profiles of the PSF standard stars. HD\\,124897 is \n the PSF standard taken with Subaru\/COMICS. The solid line with square points is the average VISIR PSF.\\label{psfs}}\n \\end{center}\n \\end{figure}\n\\end{center}\n\n\n\\begin{table*}\n \\caption{The log of the observations.\\label{t1}}\n \\begin{small}\n \\begin{tabular}{llllccp{0.5cm} p{0.25cm} p{0.25cm} p{0.25cm}}\n \\hline\n \\vspace*{-2.5mm}\n \\rule{0pt}{0.5mm}\\\\\n Object & RA &DEC(J2000) & date & Integration & $\\overline{\\mathrm{AM}}$ & \\multicolumn{1}{c}{S\/N}& \\multicolumn{1}{c}{$d$} & \\multicolumn{1}{c}{$L$}&$F_{\\rm 21\\mu m}$ \\\\\n & (h:m:s) & (\\degr:\\arcmin:\\arcsec) & & (s) & & & \\multicolumn{1}{c}{(kpc)} & \\multicolumn{1}{c}{($\\mathrm{10^3}L_{\\odot}$)}&(Jy)\\\\\n \\hline\n \\hline\n {\\bf{MYSOs:}}\\\\\n {\\bf{VISIR}}\\\\\n G263.7434+00.1161 & 08 48 48.64 &$-$43 32 29.1 & 09\/05\/11 & 760 \t &1.1 & 900 & 2.0 & 4.1 & 93\\\\\n G263.7759$-$00.4281 & 08 46 34.84 &$-$43 54 29.9 & 09\/05\/11+05\/12 & 1500 &1.1 & 150 & 1.5 & 4.0 & 65\\\\\n G265.1438+01.4548 & 08 59 27.40 &$-$43 45 03.7 & 09\/06\/04+05\/05 & 1833 &1.4 & 150 & 2.5 & 8.6 & 42\\\\\n G268.3957$-$00.4842 & 09 03 25.08 &$-$47 28 27.6 & 09\/06\/05+06\/22 & $2\\times$ 1305 & 1.6\/1.7& 250 & 1.6 & 3.7 & 50\\\\ \n G269.1586$-$01.1383 & 09 03 31.76 &$-$48 28 45.5 & 09\/05\/12 & 700 &1.2 & 25 & 2.7 & 7.3 & 98\\\\\n G310.0135+00.3892 & 13 51 37.85 &$-$61 39 07.5 & 09\/05\/26 & 120 &1.4 & 250 & 3.3 & 57 & 259\\\\\n G314.3197+00.1125 & 14 26 26.28& $-$60 38 31.5 & 09\/05\/26 & 2000 &1.3 & 600 & 3.7 & 13 & 58\\\\\n G318.0489+00.0854 & 14 53 42.99 &$-$59 08 56.5 & 09\/06\/01+06\/07 & $2\\times$ 1996 &1.2\/1.2 & 50 & ? & ? & 41\\\\\n G318.9480$-$00.1969 & 15 00 55.31 &$-$58 58 52.6 & 09\/05\/10 & 2000 &1.2 & 650 & 2.6 & 11 & 55\\\\\n G326.4755+00.6947 & 15 43 18.97 &$-$54 07 35.6 & 09\/06\/07 & 1851 &1.2 & 650 & 2.8 & 8.8 & 42\\\\\n G332.2941+02.2799 & 16 05 41.84 &$-$49 11 29.5 & 09\/06\/07 & 3199 \t &1.2 & 75 & 1.9 & 0.8 & 32\\\\\n G332.9868$-$00.4871 & 16 20 37.81 &$-$50 43 49.6 & 09\/06\/07 & 1500 &1.1 & 500 & 3.5 & 26 & 64\\\\\n G333.1075$-$00.5020 & 16 21 14.22 &$-$50 39 12.6 & 09\/06\/07 & 1447 &1.4 & 50 & 3.5 & 2.9 & 48\\\\\n G339.6221$-$00.1209 & 16 46 05.99 &$-$45 36 43.9 & 09\/06\/07 & 2225 &1.4 & 150 & 13.1 & 520 & 39\\\\\n G341.1281$-$00.3466 & 16 52 33.19 &$-$44 36 10.8 & 09\/06\/08 & 2423 \t &1.1 & 400 & 3.5 & 6.2 &37\\\\\n G343.5024$-$00.0145 & 16 59 20.90 &$-$42 32 38.4 & 09\/06\/07 & 400 &1.4 & 25 & 3.0 & 18 & 131\\\\\n G343.5213$-$00.5171 & 17 01 34.04 &$-$42 50 19.7 & 09\/05\/10 & 1497 &1.1 & 75 & 3.0 & 13 & 47\\\\\n G345.0061+01.7944 & 16 56 46.37 &$-$40 14 26.7 & 09\/05\/10 & 280 &1.0 & 100 & 1.8 & 8.7 & 154\\\\\n G349.7215+00.1203 & 17 18 11.11& $-$37 28 23.4 & 09\/06\/07 & 3520 &1.6 & 200 & 24.4 & 310 & 31\\\\\n {\\bf{COMICS}}\\\\\n G012.9090$-$00.2607 & 18:14:39.56 &$-$17:52:02.3 & 09\/08\/30 & 1405 & 1.3 & 200& 3.8& 54& 144\\\\\n \\hline\n {\\bf{Standards:}}\\\\\n {\\bf{VISIR}}\\\\\n HD 108903 & 12 31 09.96 &$-$57 06 47.6 & 09\/05\/28+06\/07+06\/08 & 60\/60 & 2.0\/1.4\/1.4 & 200\/400\/700& \\\\\n HD 123139 & 14 06 40.95 &$-$36 22 11.8 & 09\/06\/23 & 2500 & 1.0 & 350&\\\\\n HD 150798 & 16 48 39.90 &$-$69 01 39.8 & 09\/05\/10+05\/26 & 3600\/3600 & 1.4\/1.4 & 1800\/600&\\\\\n HD 163376 & 17 57 47.80 &$-$41 42 58.7 & 09\/05\/26 & 3600 & 1.1 & 150&\\\\\n HD 81797 & 09 27 35.24 &$-$08 39 31.0 & 09\/05\/12 & 2500 & 1.2 & 850&\\\\\n {\\bf{COMICS}}\\\\\n HD 124897 & 14 15 39.67 &+19 10 56.7 & 09\/08\/30& 856& 1.9 & 50 \\\\\n \\hline\n \\end{tabular}\n \\tablefoot{The signal to noise ratio is defined as the ratio of the central peak to the root-mean-squared (rms) background noise. \n The distances and luminosities are taken from the RMS database.}\n \\end{small}\n\\end{table*}\n\n\\section{Observational results}\n\n\\label{data}\n\nWe assessed the MIR images of our target sources by comparing their\nazimuthally averaged intensity profiles to the instrumental PSF\ndetermined from the standard stars. We find that the majority of the\nMYSOs observed, 14 of 20, are spatially resolved. The images of the\ntarget MYSOs are presented in Figs. \\ref{unres_im} and \\ref{res_im},\nwhich contain the unresolved and resolved MYSOs respectively.\n\n\\smallskip\n\nThe resolved MYSOs generally appear as a single source within the\nVISIR field of view, $30\\arcsec\\times30\\arcsec$. Most display an\napproximately circular symmetric morphology at the resolution of the\nobservations ($\\sim$0.6\\arcsec), although some show clear signs of\nstructure (e.g G263.7759$-$00.4281, Fig. \\ref{res_im}: panel\nA). Several objects are associated with additional, distinct sources\nof localised emission, which often have a cometary type morphology\n(e.g. G269.1586$-$01.1385 \\& G349.7215+00.1203, Fig. \\ref{res_im}:\npanels D \\& M). This is suggestive of a H{\\sc{ii}} region\n\\citep[][]{Hoare2007PPV}. Indeed, both the objects mentioned above are\ndetected at radio wavelengths at coordinates consistent with the\nlocations of the cometary 20~$\\mu$m sources\n\\citep[see][]{Urquhart2007a}. It is of interest to note that\n24.5\\,$\\mu$m images of regions containing both MYSOs and ultracompact\nH{\\sc{ii}} regions show a clear morphological difference between the\ntwo types of source, i.e. compact vs extended (DW09).\n\n\\smallskip\n\nApproximately one third of the sample are unresolved. Using the\nKolmogorov-Smirnov test, we investigated whether the resolved and\nunresolved samples were drawn from intrinsically different\ndistributions of MYSOs in terms of distance and luminosity. There is\nno significant difference between the distance and luminosity\ndistributions of the two samples, as might be expected given that the\nobjects are drawn from a single catalogue. However, the angular size\nof a centrally heated source of flux is dependent upon both the\nluminosity of central object and the distance to it \\citep[angular\nsize $\\propto \\frac{\\sqrt{L}}{d}$, see e.g.][]{Vinkovic2007}. We find\nthat the resolved objects generally have a higher value of\n$\\frac{\\sqrt{L}}{d}$ than the unresolved objects. More specifically,\nthe hypothesis that the unresolved sources have the same\n$\\frac{\\sqrt{L}}{d}$ distribution as the resolved sources can be\ndiscarded at a level of 98 percent significance. Thus, we conclude\nthat resolved sources appear larger than the unresolved sources as the\nresolved sources are generally more luminous {\\emph{and}} less distant\nthan their unresolved counterparts.\n\n\\smallskip\n\n\\begin{center}\n \\begin{figure*}[!h]\n \\begin{center}\n \\begin{tabular}{l l l}\n \\includegraphics[width=0.3\\textwidth]{new_fig_2_a.ps} & \n \\includegraphics[width=0.3\\textwidth]{new_fig_2_b.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_2_c.ps}\\\\\n \\includegraphics[width=0.3\\textwidth]{new_fig_2_d.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_2_e.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_2_f.ps}\\\\\n \\end{tabular}\n \\caption{Unresolved sources\\label{unres_im}. The images are\n scaled logarithmically. North is to the top of the page and\n East is to the Left. {{The contours typically represent 1, 5,\n 25 and 75 percent of the peak flux.}}}\n \\end{center}\n \\end{figure*}\n\\end{center}\n\n\\begin{center}\n \\begin{figure*}[!h]\n \\begin{center}\n \\begin{tabular}{l l l}\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_a.ps} &\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_b.ps} &\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_c.ps} \\\\\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_d.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_e.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_f.ps}\\\\\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_g.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_h.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_i.ps}\\\\\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_j.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_k.ps}&\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_l.ps}\\\\\n \\end{tabular}\n \\caption{Resolved sources\\label{res_im}. The images are scaled\n logarithmically. North is to the top of the page and East is to the\n Left. {{The contours typically represent 2, 5, 10, 25 and 75 percent of\n the peak flux.}}}\n \\end{center}\n \\end{figure*}\n\\end{center}\n\n\\addtocounter{figure}{-1}\n\n\\begin{center}\n \\begin{figure*}[!h]\n \\begin{center}\n \\begin{tabular}{ l}\n \\includegraphics[width=0.3\\textwidth]{new_fig_3_m.ps} \\\\\n \\end{tabular}\n \\caption{continued.}\n \\end{center}\n \\end{figure*}\n\\end{center}\n\n\n\\section{Radiative transfer modelling}\n\\label{model}\n\nTo investigate the nature of the circumstellar structure of the\nresolved sources, we compare the observed images to 2D axis-symmetric\nradiative transfer models appropriate for massive stellar objects\nsurrounded by in-falling, rotating envelopes. By simultaneously\ncomparing the resolved MIR images and SEDs to the radiative transfer\ncalculations, we aim to understand the source of the 20\\,$\\mu$m MIR\nemission of MYSOs. The models and the modelling procedure are\ndescribed in detail in the following sections.\n\n\n\n\\subsection{The radiation transfer code}\nWe employ the 2D, axis-symmetric dust radiation transfer code of\nWhitney et al. \\citep[for details see][]{W22003,W12003}. The code\ncalculates radiation transfer through a dusty structure which consists\nof a proto-stellar envelope surrounding a central star. A low density\npolar cavity can be inserted into the overall envelope structure. The\ncode also allows for a dusty circumstellar disc. The density\ndistribution of the envelope is described by the treatment of a\ncollapsing, rotating envelope of \\citet{Ulrich1976} and\n\\citet[][]{Terebey1984}. The key parameters determining the\nenvelope density are the mass infall rate and the centrifugal\nradius. The dust sublimation radius is calculated self-consistently\nfrom the stellar luminosity and the dust sublimation temperature. The\nshape of the polar cavities can be described in two ways: using a\npolynomial function or following streamlines. The streamline is\nconical on large scales, while the apparent opening angle of the\npolynomial function can be specified (the opening angle at the stellar\nsurface is 180$\\degr$). If included, the dust disc is flared and\nfollows the $\\alpha$ prescription \\citep[see][]{Shakura1973} to\naccount for flux due to accretion. The reader is referred to\n\\citet{W22003,W12003} for more details on the code and the possible\ngeometries. We have used this code extensively and the reader is\nreferred to \\citet{deWit2010,deWit2011} for more details on the\napplication of the code to model MYSOs and their environments.\n\n\\subsection{Modelling methodology}\n\n\nOur previous use of the Whitney et al. code consisted of constructing\na series of dedicated models in order to reproduce the SEDs,\ninterferometric and auxiliary spatial information of MYSOs. We will\nexploit these customised models appropriate for MYSO environments in\nour approach here, partially motivated by the lack of such models in\nthe well-known SED grid \\citep[][]{Robitaille2007}. In particular, we\nwill use a model geometry that provides a successful fit to the source\nW33A \\citep[][]{deWit2010}. Its defining feature is that the MIR\nemission in the $N$-band originates in warm dust located in the\nwalls of outflow cavities. The model proto-stellar envelope extends\ninwards down to the dust sublimation radius and it has low density\noutflow cavities as observed in several MIR observations of MYSOs\n\\citep[see e.g.][]{DeBuizer2007}. We note that the model\ndoes not explicitely include a dust disc (according to the\nprescription of the Whitney et al. code). This is motivated by the\naforementioned MIR interferometric observations. Adding a disc\ngenerally results in high visibilities in the MIR whereas MYSOs\ntypically exhibit low visibilities, indicative of a more extended dust\ndistribution \\citep[see e.g.][]{deWit2010}.\n\n\\smallskip\n\nWe illustrate the model 24.5\\,$\\mu$m image for the W33A model in panel\na) of Fig.\\,\\ref{test_image}. Also shown are the PSF convolved image\n(panel b), the COMICS image of W33A and the corresponding radial\nintensity profile, panels c) and d) respectively. The shape of the\noutflow cavities can clearly be delineated in the model image in panel\na). This demonstrates that the dust located in the outflow\ncavity walls is warmed-up and emits a significant fraction of the MIR\nflux. W33A is know to drive an outflow in the SE\/NW direction\n\\citep[see][and references therein]{Davies2010}. The orientation of\nthe outflow is traced by nebulosity in the NIR. While the 24.5~$\\mu$m\nimage of this object is resolved (panel c), it appears relatively\nsymmetric. However, the image is slightly extended towards the SE, in\nthe direction of the blue lobe of its outflow. In panel d) we compare\nthe model's radial intensity profile with the observed one and find\nthey are essentially identical. We underline that we did not perform a\nmodel fit to the 24.5\\,$\\mu$m image, we simply used the final W33A\nmodel presented in de Wit et al. (2010) to create the corresponding\nimage.\n\n\n\\begin{figure}[t]\n \\includegraphics[width=0.5\\textwidth]{temp.ps} \n \\caption{{\\it Panel a:} Logarithmically scaled 24.5~$\\mu$m image of\n the standard W33A model. {\\it Panel b:} The same image as panel a)\n but convolved with the COMICS instrumental PSF. {\\it Panel c:} The\n COMICS image of W33A. {\\it Panel d:} The azimuthally averaged\n radial intensity profile at 24.5~$\\mu$m of W33A (points with error\n bars) alongside the model radial profile and that of a PSF\n standard (the lower line). The outflow axis of the model is\n aligned with the SE\/NW orientation of W33A's\n outflow.\\label{test_image}}\n\\end{figure}\n\n\nMotivated by the good match of the W33A model and image and the\nknowledge that the SEDs of MYSOs are known to be similar, we proceeded\nto fit the observations of the remaining MYSOs in the following\nway. The SEDs of the MYSOs observed were constructed from the\ncontinuum data assembled on the RMS database (which makes use of the\nwork of various authors) and the methodology of \\citet{M2011}. The SED\ncoverage differs from one source to the other. Generally one can\ndistinguish between sources with continuum flux measurements up to\n100\\,$\\mu$m and those with sub-mm observations. We note that we do not\nfit explicitly the 9.7\\,$\\mu$m silicate feature, contrary to the\nmodelling efforts in DW09. This is simply because this information is\nnot available for the majority of our sources.\n\n\\smallskip\n\nTo reproduce both the SEDs and 20\\,$\\mu$m intensity distributions of\nthe resolved sample, we used the W33A model geometry summarised\nabove. We only altered parameters that can be expected to vary between\nobjects: the inclination, the envelope infall rate (which scales the\ntotal dust mass in the envelope) and the opening angle of the outflow\ncavity (which may vary with time). We emphasise that we use one basic\nmodel to reproduce the observations. There may be differences between\nsources, for example the centrifugal radius and the properties of\ncircumstellar discs -- if any are present -- may vary from source to\nsource. The model will not account for such differences. However, it\nprovides a simple test of the hypothesis that the bulk of the $Q-$band\nemission of MYSOs can be attributed to warm dust in outflow cavity\nwalls. Here we outline the methodology followed in reproducing the\nobservations.\n\n\\smallskip\n\n\n\nFor a given MYSO, the stellar luminosity was initially set to the\nvalue in Table \\ref{t1} and the system inclination {{(the angle\n between the polar axis and the line of sight)}} was estimated from\nits MIR image. The stellar luminosity and infall rate were then varied\nuntil the model SED was consistent with that observed. This set the\nluminosity and provided first estimates of the other parameters. Once\nthe SED was satisfactorily reproduced, the inclination and outflow\nopening angle were varied in an attempt to recreate the observed\nspatial intensity profile. This had an affect on the reproduction of\nthe SED. Therefore, the infall rate was allowed to vary to enable a\nfit to both the SED and the radial profile. The model fitting was done\nby hand rather than using a $\\chi^2$ minimisation technique as the\ncomputational time required to make a grid of model images for each\nobject was prohibitively large.\n\n \\smallskip\n\n We demonstrate the effect of varying the free parameters in\n Fig. \\ref{vary}. In general, the more inclined models appear more\n extended. This might be expected if much of the MIR emission\n originates in outflow cavity walls. At large inclinations, the projected\n area of the outflow cavity walls is larger and thus the resultant flux\n distribution is more extended. The cavity opening angle has a similar\n effect on the intensity distribution. This is because increasing the\n opening angle increases the distance at which the majority of the\n cavity wall is exposed to direct irradiation from the central\n object. This also increases the projected area of the outflow\n cavities and the extension seen in the azimuthally averaged intensity\n distribution. Since both parameters affect the observed extension,\n they are slightly degenerate. However, the two parameters have\n different effects on the gradient of the radial profile (see\n Fig. \\ref{vary}).\n\n\\smallskip\n\n\n\nThe final SEDs are presented in\nFig. \\ref{seds} and the associated azimuthally averaged intensity\nprofiles are displayed in Fig. \\ref{rad_profs}. The parameters of the\nselected models are presented in Table \\ref{pars}. Since the sample\nexhibits a varied morphology (see Fig. \\ref{res_im}), we discuss the\nmodelling results for each object individually in Appendix\n\\ref{notes}. We discuss the general results in the following section.\n\n\n\n\\begin{center}\n \\begin{figure}\n \\begin{center}\n \\begin{tabular}{l l}\n \\includegraphics[width=0.235\\textwidth]{fig_5_a.ps} & \n \\includegraphics[width=0.235\\textwidth]{fig_5_b.ps}\\\\\n \\includegraphics[width=0.235\\textwidth]{fig_5_c.ps}&\n \\includegraphics[width=0.235\\textwidth]{fig_5_d.ps}\\\\\n \\includegraphics[width=0.235\\textwidth]{fig_5_e.ps}&\n \\includegraphics[width=0.235\\textwidth]{fig_5_f.ps} \\\\\n \\end{tabular}\n \\caption{An exploration of the parameter space immediately\n surrounding the model of G310.0135+00.3892. The plots show the\n effect of varying the free parameters (the inclination,\n opening angle and infall rate). For comparison, a PSF standard\n is also shown in the intensity distribution\n figures.\\label{vary}}\n \\end{center}\n \\end{figure}\n\\end{center}\n\n\n\n\\begin{table}\n \\begin{center}\n \\caption{The key parameters of the individual models.\\label{pars}}\n \\begin{tabular}{l c c c p{0.5cm} p{0.5cm} }\n \\hline\n Object & $\\frac{L_{\\mathrm{Model}}}{L_{\\mathrm{RMS}}}$ & $i$ & $\\dot{m}$ & \\multicolumn{1}{c}{$\\theta_{\\mathrm{op. ang.}}$} & \\multicolumn{1}{c}{$M_{\\rm{Env.}}$}\\\\\n & &$^{\\circ}$ & $M_{\\odot}{\\mathrm{yr^{-1}}}$ & \\multicolumn{1}{c}{$^{\\circ}$} & $10^3M_{\\odot}$\\\\\n \\hline\n \\hline\n\n \\multicolumn{4}{l}{{\\bf{SED and intensity profile}}}\\\\\n\n G263.7759$-$00.4281 & 1.0 & 87 & $\\mathrm{1.5\\times 10^{-4}}$ & \\multicolumn{1}{c}{25} & 15.5\\\\\n G265.1438+01.4548 & 1.2 & 32 & $\\mathrm{2.0\\times 10^{-4}}$ & \\multicolumn{1}{c}{25} & 15.5\\\\\n G268.3957$-$00.4842 & 0.5 & 30 & $\\mathrm{3.5\\times 10^{-4}}$ & \\multicolumn{1}{c}{10} & 9.4\\\\\n G310.0135+00.3892 & 1.0 & 32 & $\\mathrm{7.5\\times 10^{-4}}$ & \\multicolumn{1}{c}{10} & 9.8\\\\\n G332.2941+02.2799 & 1.4 & 87 & $\\mathrm{12.5\\times 10^{-5}}$ & \\multicolumn{1}{c}{20} & 12.8\\\\\n G332.9868$-$00.4871 & 1.0 & 15 & $\\mathrm{1.0\\times 10^{-4}}$ & \\multicolumn{1}{c}{10} & 9.3\\\\\n G339.6221$-$00.1209 & 1.2 & 60 & $\\mathrm{12.0\\times 10^{-4}}$ & \\multicolumn{1}{c}{12} & 11.4\\\\\n G345.0061+01.7944 & 1.0 & 63 & $\\mathrm{4.0\\times 10^{-4}}$ & \\multicolumn{1}{c}{10} & 9.4\\\\\n G349.7215+00.1203 & 1.4 & 65 & $\\mathrm{7.5\\times 10^{-4}}$ & \\multicolumn{1}{c}{15} & 11.3\\\\\n\n \\multicolumn{4}{l}{{\\bf{SED only}}}\\\\\n G269.1586$-$01.1383 & 0.8 & 60 & $\\mathrm{7.5\\times 10^{-4}}$ & \\multicolumn{1}{c}{45} & 32.2\\\\\n G343.5024$-$00.0145 & 1.3 & 85 & $\\mathrm{3.0\\times 10^{-4}}$ & \\multicolumn{1}{c}{10} & 9.3 \\\\\n G343.5213$-$00.5171 & 1.0 & 57 & $\\mathrm{2.5\\times 10^{-4}}$ & \\multicolumn{1}{c}{10} & 9.3\\\\\n\n \\hline\n \\end{tabular}\n \\tablefoot{We note that the output luminosity is affected by\n inclination angle, and is thus not necessarily identical to the\n source luminosity. Parameters not listed were not varied. In the\n case of several objects, particularly those associated with\n H{\\sc{ii}} regions, e.g. G343.5024$-$00.0145, the model only\n reproduces the objects' SED. {{$\\theta_{\\mathrm{op. ang.}}$\n denotes half the full opening angle}}.}\n \\end{center}\n\\end{table}\n\n\n\n\\begin{center}\n \\begin{figure*}[!h]\n \\begin{center}\n \\begin{tabular}{l l l}\n \\includegraphics[width=0.3\\textwidth]{fig_6_a} &\n \\includegraphics[width=0.3\\textwidth]{fig_6_b} &\n \\includegraphics[width=0.3\\textwidth]{fig_6_c} \\\\\n \\includegraphics[width=0.3\\textwidth]{fig_6_d}&\n \\includegraphics[width=0.3\\textwidth]{fig_6_e}&\n \\includegraphics[width=0.3\\textwidth]{fig_6_f}\\\\\n \\includegraphics[width=0.3\\textwidth]{fig_6_g}&\n \\includegraphics[width=0.3\\textwidth]{fig_6_h}&\n \\includegraphics[width=0.3\\textwidth]{fig_6_i}\\\\\n \\includegraphics[width=0.3\\textwidth]{fig_6_j}&\n \\includegraphics[width=0.3\\textwidth]{fig_6_k}&\n \\includegraphics[width=0.3\\textwidth]{fig_6_l} \\\\ \n \\end{tabular}\n \\caption{The SEDs of the resolved MYSOs compared to the final model SEDs\\label{seds}.}\n\n \\end{center}\n \\end{figure*}\n\\end{center}\n\n\\begin{center}\n \\begin{figure*}[!h]\n \\begin{center}\n \\begin{tabular}{l l l}\n \\includegraphics[width=0.3\\textwidth]{fig_7_a.ps} &\n \\includegraphics[width=0.3\\textwidth]{fig_7_b.ps} &\n \\includegraphics[width=0.3\\textwidth]{fig_7_c.ps} \\\\\n \\includegraphics[width=0.3\\textwidth]{fig_7_d.ps}&\n \\includegraphics[width=0.3\\textwidth]{fig_7_e.ps}&\n \\includegraphics[width=0.3\\textwidth]{fig_7_f.ps}\\\\\n \\includegraphics[width=0.3\\textwidth]{fig_7_g.ps}&\n \\includegraphics[width=0.3\\textwidth]{fig_7_h.ps}&\n \\includegraphics[width=0.3\\textwidth]{fig_7_i.ps}\\\\\n \\includegraphics[width=0.3\\textwidth]{fig_7_j.ps}&\n \\includegraphics[width=0.3\\textwidth]{fig_7_k.ps}&\n \\includegraphics[width=0.3\\textwidth]{fig_7_l.ps} \\\\ \n \\end{tabular}\n \\caption{The azimuthally averaged intensity profiles of the\n resolved MYSOs (upper circles) alongside the model radial\n profiles (squares) and the associated PSF standard (lower\n circles). The lower limit to the $y$ axis is set to the\n root-mean-square noise in the background of the MYSO\n images. The upper limit of the $x$ is axis is set to the\n distance at which the MYSO profiles fall to the level of the\n background noise. The radial profile of G263.7759-00.4281 is\n also shown averaged over the upper and lower half of its\n bipolar morphology separately (short and long dashed lines\n respectively). The error bars represent the rms within a given\n annuli and thus represent an upper limit on the uncertainty in\n the flux distribution.\\label{rad_profs}}\n \\end{center}\n \\end{figure*}\n\\end{center}\n\n\\section{Modelling results}\n\n\\label{mod_res}\n\n\nIn general, the observed SEDs could be reproduced relatively\nwell. There are several cases where the observed NIR fluxes could not\nbe reproduced exactly. However, we do not weight the NIR fluxes\nheavily as these can be strongly dependent on dust opacities and the\ncircumstellar geometry on small scales. In the majority of cases,\nbarring objects associated with H{\\sc{ii}} regions, the spatial\ndistribution of the 20\\,$\\mu$m emission of the MYSOs observed can also\nbe reproduced by our 2D axis-symmetric radiative transfer (RT)\nmodel. The individual model fitting results for the SEDs and the\n20\\,$\\mu$m intensity profiles are shown in Figs.\\,\\ref{seds} and\n\\ref{rad_profs}. Clearly, because of the unequal SED coverage from\nsource to source, the model fidelity differs. We do not attempt to\nprovide a unique model for each single source but evaluate whether the\nW33A model is capable of providing satisfactory fits, by changing a\nfew of the model's most basic parameters. From this starting point, we\nfind that in general, if the image exhibits a single source with an\napproximately symmetric morphology, both the object's SED and the\n20\\,$\\mu$m intensity profile are well reproduced. By extension of the\ncase for W33A, the majority of the 20\\,$\\mu$m emission in these\ninstances is very likely emission from warm dust in cavity walls.\n\n\\smallskip\n\nWhen an object's morphology appears either very complex or cometary, the\nmodel generally fails to reproduce the intensity profile. For example,\nthis is the case for G343.5024$-$00.0145, which is a known\nultracompact H{\\sc{ii}} region. For this object, a decent fit to the\nSED (panels I in Figs. \\,\\ref{seds} and \\ref{rad_profs}) predicts a\n``MYSO'' structure that is much more compact than the one observed,\neven with extreme values for the inclination and opening angle. A\ncometary appearance alone will often indicate an ionized nature for\nthe region, and this is generally confirmed via radio continuum\nobservations. That the model cannot reproduce the observations of\nMYSOs associated with H{\\sc{ii}} regions is perhaps to be expected as\nH{\\sc{ii}} regions represent a source of flux not considered by the\nmodel. Furthermore, the appearance of a H{\\sc{ii}} region likely\nsignifies the end of the embedded accretion phase of a MYSO, and it is\nthis phase the model is designed to represent.\n\n\\smallskip\n\nWe give an overview of the values for the varied parameters in\nTable\\,\\ref{pars}. The opening angles of the best-fitting models are\ngenerally narrow ($\\sim10 \\degr$). This implies that the outflows of\nthe MYSOs observed are relatively well collimated. The envelope infall\nrates of the models, a parametrisation of the density distribution,\nare generally of the order\n$\\mathrm{10^{-4}}M_{\\odot}\\mathrm{yr^{-1}}$. The corresponding values\nof visual extinction, $A_V$, are typically between 10 and 30, in\nagreement with the measured $A_V$s of MYSOs \\citep{Porter1998},\nalthough several of the more inclined models have $A_V$s in excess of\n100. The total masses of the envelopes are generally of the order\n10,000~$M_{\\odot}$. {\\color{black}We note that these are dependent on\n the envelope outer radii, which are set at $\\mathrm{5 \\times\n 10^5}$~AU. In the case of the W33A model, reducing the outer\n radius by a factor of 2 decreases the total mass by a factor of\n approximately 5. This is a result of adopting a low but constant\n density, $\\rho = 8 \\times 10 ^{-20} \\mathrm{g\\,cm^{-3}}$, within the\n outflow cavity.} The envelope masses calculated are comparable to\nthe clouds that the objects are located in \\citep[see][and references\ntherein]{Urquhart2011}. Even allowing for an uncertainty of a factor\nof a few, this indicates that these objects are still heavily embedded\nin their natal material and their environment still contains a large\nreservoir of material available to form further stars.\n\n\\smallskip\n\n\n\n\\section{Discussion}\n\\label{disc}\n\\subsection{On the mid-IR emission of MYSOs}\n\nOver the past decade, observational evidence indicating that the\nformation of massive stars, up to a mass of at least $\\sim\n30\\,M_{\\odot}$, is accompanied by phenomena characteristic\nof low mass SF has accumulated. Such phenomena include:\ncircumstellar discs \\citep[see\ne.g.][]{Kraus2010,Davies2010,Masque2011,Goddi2011}, molecular outflows\n\\citep[see e.g.][]{Beuther2002,Cyganowski2009} and jet activity\n\\citep[see e.g.][]{Rodriguez1994,Guzman2010}. \n\n\\smallskip\n\nDespite the many common characteristics of low and high mass SF, some\nobservational phenomena seem to be exclusively associated with young,\nembedded high-mass stars. One example is the Class II 6.7-GHz methanol\nmaser, whose activity is uniquely associated with massive star forming\nregions \\citep[e.g.][]{Walsh1997}. These sites evidently produce\nsufficient IR radiation in order to pump this transition, unlike sites\nof low mass SF. It was initially thought that this maser activity\noriginated in circumstellar discs. As a result, a correlation between\nthe extension of several MYSOs in the MIR and their maser emission led\nto the suggestion that the MIR emission of MYSOs traces discs\n\\citep[see e.g.][]{DeBuizer2000}. However, observations with 8m class\ntelescopes were able to spatially resolve the MIR emission of a number\nof masing sources to reveal that their MIR emission is aligned with,\nrather than perpendicular to, their large CO molecular outflows\n\\citep[see e.g.][]{DeBuizer2006,DeBuizer2007}. These findings\nassociate the MIR emission of massive YSOs with outflow cavities,\nrather than circumstellar discs. Nevertheless, the kinematic diversity\nof methanol masers in a large sample of massive outflow sources does\nnot support a uniquely identifiable source within the close MYSO\nenvironment \\citep[][]{Cyganowski2009,Moscadelli2011}. Therefore, the\nsource of the MIR emission of MYSOs has yet to be conclusively\nestablished.\n\n\\smallskip\n\nRecently, several massive young stellar objects with compact MIR\nemission were investigated using long-baseline interferometry. In\nseveral cases, the spatially resolved $N$-band emission on scales of\n100~AU was found to be consistent with thermal envelope emission (de\nWit et al. 2007; Linz et al. 2009\\nocite{Linz2009}) and in particular\nwith emission arising from the cavity walls (de Wit et al. 2010).\nHowever, in some other cases, the mas-scale MIR emission of MYSOs is\nsuggested to be (at least partially) located in a circumstellar disc\n\\citep[see e.g.][]{Follert2010,deWit2011,Grellmann2011}. In other\nwords, there are hints that the $N$-band emission, resolved with\ninterferometry, is not completely dominated by envelope emission. The\nanalysis of the $Q$-band emission presented in this paper favours the\ncavity wall emission interpretation for a sample of MYSOs. Upon first\ninspection, this appears at odds with the scenario of a disc\ndominating the $N-$band emission. Here, we consider the possible\ncontribution to the $Q-$band emission by a circumstellar disc in order\nto assess whether our finding agrees with theoretical expectations.\n\n\\smallskip\n\nThe recent study by \\citet{Zhang2011} discusses in detail the\nappearance of embedded young massive stars in the IR wavelength\nregion. The authors use a dust radiative transfer code, as we\ndo. {{However, their geometry differs from the one employed in this\n paper in that their code includes an accretion disc that extends\n from the stellar surface to the centrifugal radius}}. Therefore,\ntheir results allow us to assess the possible disc contribution to the\n$Q-$band emission of MYSOs. The RT calculations of \\citet{Zhang2011}\nreveal that outflow cavities have a large influence on the shape of\nthe SED. Furthermore, in their calculation for a $60^{o}$ inclination\n(see e.g. their Fig. 9), the outflow cavities are clearly significant\nfeatures in the images, up to a wavelength of 70\\,$\\mu$m. In the NIR,\nscattered light from the cavities makes a significant contribution to\nthe near-IR emission. In the $N$-band, thermal emission from the disc\nand cental object are also evident. In the $Q$-band however, the\nemission from the base of the cavity clearly dominates the flux.\n\n\\smallskip\n\nThe relative contributions of the disc and envelope to the modelled\nSED depend critically on exactly those parameters with which we fit\nour observations, i.e. opening angle, mass infall rate and\ninclination. We note that the opening angle applied by\n\\citet{Zhang2011} is rather wide ($2\\alpha \\approx 90^{o}$), which\nwould imply an object in a rather advanced phase of formation,\nfollowing the expected widening of the opening angle as function of\ntime \\citep[see e.g.][]{Kuiper2010}. Smaller opening angles increase\nthe extinction towards the central regions and thus favour the\ndominance of cavity wall emission in the total MIR emission. A similar\nreasoning applies for the mass infall rate. It is therefore not\nsurprising that the observed MYSOs, which are believed to be in a less\nevolved phase than the phase represented by the fiducial model of\n\\citet{Zhang2011}, are dominated by the cavity wall emission. This\nargues against a large effect, if any, by a circumstellar disc on our\nVISIR images. This conclusion can also be inferred from the spectral\nenergy distribution of an accretion disc, since it peaks at\nwavelengths much shorter than 20\\,$\\mu$m. Furthermore, the envelope\ncontains much more cool material than does any reasonable accretion\ndisc, whose sizes for MYSOs are estimated to be less than 500\\,AU\n\\citep[see\ne.g.][]{Patel2005,Hoare2006,Reid2007,Izas2007,deWit2011,Goddi_2011}. We\nconclude that it is reasonable to expect that dust emission from the\nenvelope (the cavity walls) dominates the continuum SED at\nwavelengths longward of the $N$-band. This is in agreement with our\nfinding that the observed extension of the $Q-$band flux can be\nmodelled as emission from outflow cavity walls.\n\n\n\\subsection{The structure of MYSO envelopes: rotation versus outflows}\n\nWe have shown that the morphology of the 20~$\\mu$m emission of MYSOs\nis consistent with that predicted by models incorporating outflow\ncavities. We followed a similar methodology to DW09 and have improved\non the modelling part of their analysis. DW09 compared their images to\n1-D radiative transfer models. This was motivated by the rather\nspherical appearance of most sources, which suggests that their MIR\nemission originates from a relatively symmetrical dusty envelope. They\nfound that, in general, their data were best fit by an envelope with a\ndensity gradient of $\\rho \\propto r^{-1}$. This density law is\nshallower than that obtained via a similar analysis performed in the\nsub-mm \\citep[see e.g.][]{Mueller2002}. It was therefore suggested\nthat this flattening of the density law towards smaller spatial scales\ncould be evidence for the onset of rotational support of the envelope\nat approximately 1000~AU. This suggestion is probably invalidated by\nthe results presented in this paper based on more realistic RT models.\n\n\\smallskip\n\nThermal emission from an envelope with evacuated outflow cavities fits\nthe SED and the $N$-band (de Wit et al. 2010) and $Q$-band (this\npaper) morphology of the luminous MYSO W33A. Additionally, in this\nparticular case, the envelope model also fits the observed\nmorphologies of the near-IR scattering nebula and the 350\\,$\\mu$m\nemission. This provides strong evidence in favour of this model and\nour explanation of the 20~$\\mu$m emission (warm dust in the cavity\nwalls). Moreover, the W33A image (Fig.\\,\\ref{test_image}) is slightly\nextended towards the SE which is the direction of the blue lobe of the\nlarge scale outflow, reinforcing this interpretation. Importantly, we\nnote that W33A is not the only case in our sample for which the\nextension at 20~$\\mu$m is in the direction of the outflow\nposition angle (see for example the sources G263.7759$-$00.4281 and\nG332.2941+02.2799). Therefore, our association of the observed MIR\nflux with outflow cavity walls can be substantiated.\n\n\\smallskip\n\nThe brightness of the cavity walls, as exemplified in panel a) of\nFig.\\,\\ref{test_image}, is a consequence of the density contrast\nbetween the in-falling envelope and the outflow cavities. The\nradiation of the central source cannot penetrate far in the dense\nenvelope, but can heat dust in the walls of the outflow cavity to\nlarge radii. Once the model images are convolved with an instrumental\nPSF, it is no longer obvious that the flux traces outflows, despite\nthe source being resolved. The flux from the outflow cavity walls can\nappear more extended than symmetric envelope emission. A centrally\nconcentrated density distribution (steep density law) in a spherical\nenvelope produces a small thermal emission region in the MIR. The\nshallower the density law the larger the (normalised) emission region\nbecomes. Therefore, the influence of the outflow cavities can mimic\nthe effect of a shallow density gradient. We verified that the sizes\nof the emission regions differ strongly between a $-2$ and $-1$\ndensity law, but the difference between a flat density law and a $-1$\ndensity law are relatively minor. Still the SEDs differ\nsignificantly. The steeper density law produces higher MIR fluxes. As\nan exercise, we fitted the standard 2D W33A model with a spherical\nmodel and find that the SED and intensity profiles are matched best by\na density law between $-0$ and $-1$. This explains the results\npresented in DW09. Given the more realistic nature of the 2D models we\nhave employed, we suggest that the preference for shallow density laws\nfound by DW09 is the result of much of the MIR emission of MYSO emanating from\nwarm dust in the cavity walls. An explanation in terms of rotational\nsupport on scales of $\\sim$1000~AU cannot be substantiated.\n\n\\subsection{SED fitting}\n\nFitting spectral energy distributions has long been used as a tool to\nprovide important constraints on the morphology of circumstellar\nenvironments \\citep[see e.g.][]{Guertler1991}. Still, the inclusion of\nspatially resolved observations at different wavelength regimes is\nindispensable for a correct interpretation of the observed emission\n\\citep[see e.g.][]{vdt2000}. This is because SED fitting can\nresult in degenerate solutions. To assess the added value provided by\nthe VISIR images presented in this paper, we use the results\npresented in \\citet{M2011}. These authors used the grid of models\nprovided by \\citet{Robitaille2007} to fit the SEDs of a large\npercentage of MYSOs from the RMS survey in order to determine their\nbolometric luminosity. The grid of \\citet{Robitaille2007} was created\nusing the same RT code of Whitney et al. which we use in this\npaper. Therefore, we used the code to create images for the best\nfitting models found in \\citet{M2011} (in the filter used) and assessed\nwhether the model images are consistent with our observations.\n\n\\smallskip\n\nTo illustrate the results of this exercise, we focus on the object\nG310.0135+00.3892. This is an object of particular interest as it is\nthe object studied by \\citet[][]{Kraus2010} via a high angular\nresolution VLTI aperture synthesis image in the $K$-band. We generate\n20\\,$\\mu$m images of the set of models that best fit the object's SED\nas found by \\citet{M2011}. The result is shown in Fig. \\ref{test}. The\nSED fitting tool preferentially returns models with large cavity\nopening angles ($\\sim30-50 \\degr$), but these all fail to reproduce\nthe observed intensity profile. The best fitting model of\n\\citet{Kraus2010}, which is also based on SED fitting with the model\ngrid of \\citet{Robitaille2007} and the code of Whitney et al.,\nprovides a better match. However, it still over-predicts the flux in\nthe central 1\\arcsec~from the source (see Fig. \\ref{test}). This is\nlikely due to the large opening angle of the Kraus et al. model\n($\\mathrm{40^{\\circ}}$) generating a MIR emission region which is too\nextended. The VISIR observations clearly demonstrate that, for the\nassumed intermediate inclination, a narrow opening angle is\nrequired. We conclude that the spatially resolved VLT-VISIR images at\n20~$\\mu$m provide valuable constraints on the nature of the envelopes\nsurrounding young massive stars.\n\n\n\n\\begin{center}\n \\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.4\\textwidth]{fig_8_a.ps}\n \\caption{The azimuthally averaged intensity profile of G310.0135\n (upper circles) compared to a PSF standard (lower circles) and\n the models that fit the SED of this object from\n \\citet[][dashed lines]{M2011}. The solid line is the radial\n profile of the model of \\citet{Kraus2010}. The error bars\n represent the rms within a given annuli and thus represent an\n upper limit on the uncertainty in the flux\n distribution.\\label{test}}\n \\end{center}\n \\end{figure}\n\\end{center}\n\n\n\n\\section{Conclusions}\n\n\\label{conc}\n\nIn this paper we present diffraction limited, MIR imaging of a sample\nof massive young stellar objects drawn from the RMS survey. By\ncomparing the spatially resolved images of the MYSOs to 2D radiative\ntransfer models, we constrain the structure of the envelopes that\nsurround them. Here we list the salient results.\n\n\\begin{itemize}\n\n\\item[--]{We spatially resolve the MIR (19.5 or 24.5~$\\mu$m) emission\n of 14 MYSOs. The remainder of the MYSOs observed (6) are unresolved\n at the current resolution ($\\sim$0.6\\arcsec).}\n\n\\item[--]{The model of the MYSO W33A that we have developed previously\n can, in most cases, reproduce the images and SEDs of the MYSOs. This\n is not the case for objects associated with a H{\\sc{ii}}\n region. That the model can reproduce the infrared emission of a\n sample of MYSOs suggests that the circumstellar environments\n of MYSOs are relatively uniform. The large\n envelope masses of the models are consistent with the\n identification of MYSOs as objects in their accretion phase.}\n\n\n\\item[--]{It is found that the extent of the MIR emission of the\n spatially resolved MYSOs can generally be attributed to warm dust\n located in the walls of outflow cavities. We suggest that the\n relatively shallow intensity profile gradients found by previous\n MIR diffraction limited imaging of MYSOs is indicative of outflow\n cavities (rather than alternative explanations such as rotational\n flattening).}\n\n\n\\item[--]{We show that the varied morphology observed can be\n attributed to outflow cavities seen at a variety of\n inclinations. This emphasises that outflows are an ubiquitous\n feature of massive star formation.}\n\n\\end{itemize}\n\n\n\\begin{acknowledgements}\n HEW acknowledges the financial support of the MPIfR. WJdW is grateful\n for the hospitality of G. Weigelt and the IR interferometry group at\n the MPIfR where this paper was finalised. J. Mottram is thanked for\n providing details of previous fits to the SEDs of many of the\n sources in this paper. This paper made use of information from the\n Red MSX Source survey database at www.ast.leeds.ac.uk\/RMS which was\n constructed with support from the Science and Technology Facilities\n Council of the UK.\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\label{introduction_section}\n\nThe syntactic dependency structure of a sentence can be defined as a network where vertices are words and connections indicate syntactic dependencies, e.g., the relationship between the subject of a sentence and its verb (Fig. \\ref{syntactic_dependency_trees_figure}). These networks are typically trees and directed \\cite{tesniere59,hays64,melcuk88}. However, link direction is irrelevant for the present article and therefore omitted. Syntactic dependency networks can be seen as \nspanning trees on a lattice \\cite{Manna1992a,Barthelemy2006a} and are indeed a particular case of geographically embedded or spatial networks \\cite{Reuven2010a_Chapter8, Gastner2006a, Guillier2017a}\nin one dimension, i.e. the dimension defined by the linear order of the words of the corresponding sentence \\cite{Ferrer2004b}. \n\n\n\\begin{figure*}\n\\includegraphics[scale = 0.8]{syntactic_dependency_trees}\n\\caption{\\label{syntactic_dependency_trees_figure} Syntactic dependency trees of sentences. Top: a tree without dependency crossings. \"John\" is the subject of the verbal form \"gave\". Center: a tree with one edge crossing (the crossing is formed by the edge linking \"ate\" and \"yesterday\" and the edge linking \"apple\" and \"which\"). Bottom: the same sentence after fronting \"yesterday\". Notice that the length of the dependency between \"ate\" and \"yesterday\" and the dependency between \"apple\" and \"which\" have reduced (while the length of other dependencies has remained constant). Top and center are adapted from \\cite{Ambati2008a}. }\n\\end{figure*}\n\nIn the context of syntactic dependency networks, the length of an edge is defined as an Euclidean distance, namely,\nthe linear distance between the words that are connected: adjacent words are at distance 1, words separated by one word are at distance 2, and so on \\cite{Ferrer2004b,Liu2017}. In the sentence at the top of Fig. \\ref{syntactic_dependency_trees_figure}, {\\em John} and {\\em gave} are at distance 1 while {\\em gave} and {\\em apple} are at distance 3. \n\nSyntactic dependency trees exhibit certain statistical patterns concerning the length of their dependencies and the variance of their degrees. First, edge lengths are biased towards low values \\cite{Ferrer2004b,Ferrer2003f} as it happens in other geographical networks \\cite{Gastner2006a}. The distribution of edge lengths decays exponentially \\cite{Ferrer2004b} as is the case of the distribution of projection lengths in real neural networks \\cite{Ercsey2013a}. Additionally, the mean edge length is smaller than expected by chance \\cite{Ferrer2004b, Ferrer2013c, Liu2008a, Futrell2015a, Liu2017}. The simplest null hypothesis assumes a uniformly random permutation of the words of a sentence and predicts that the expected edge length is $(n + 1)\/3$, where $n$ is the number of vertices of the tree (the length of the sentence in words) \\cite{Ferrer2004b,Zornig1984a}. \nSecond, their hubiness coefficient does not exceed $25\\%$ \\cite{Ferrer2017a}. $h$, the hubiness coefficient is a normalized variance of vertex degrees. $h$ is a number between 0 and 1 that is minimum for linear trees and maximum for star trees (Fig. \\ref{star_and_linear_trees_figure}). \nIndeed, the hubiness of real syntactic dependencies is close to trees from the ensemble of uniformly random trees, for which $h$ tends to zero as $n$ increases \\cite{Ferrer2017a}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale = 0.8]{star_and_linear_tree}\n\\caption{\\label{star_and_linear_trees_figure} Linear arrangements of trees with $n = 6$ vertices. Top: A star tree, a tree with a vertex of maximum degree \\cite{Ferrer2013d}. Bottom: A linear tree, a tree where vertex degrees do not exceed 2 \\cite{Ferrer2013d}. }\n\\end{center}\n\\end{figure}\n\nThe target of the present article are the edge crossings that can arise when drawing connections above the sentence. Fig. \\ref{syntactic_dependency_trees_figure} shows two planar sentences (a sentence is planar if it does not have crossings) and a sentence with one crossing. It is widely accepted that crossing dependencies are relatively uncommon in languages \\cite{lecerf60,hays64,melcuk88,Ferrer2006d,Park2009a,Liu2010a,Gildea2010a}. \nIndeed, the actual number of crossings per sentence does not reach $3.5$ across languages and is only above $1$ in a few of them \\cite{Ferrer2017a}. However, how small a number is depends on the scale of measurement and a null model is required. A rigorous demonstration that crossings are really scarce has been missing for decades. Recently, statistical evidence that crossings are significantly small has been provided \\cite{Ferrer2017a}. Furthermore, sentences where dependencies are shorter tend to have fewer crossings \\cite{Ferrer2015c}. Fig. \\ref{syntactic_dependency_trees_figure} (center and bottom) illustrates the tendency of crossings to reduce as dependency lengths reduce.\n\n\n\n\nResearch on syntactic dependency networks parallels research on non-spatial networks: as the statistical properties of many real networks have been compared against the predictions of null models, the Erd\\H{o}s-R\\'enyi graph being one of the most simple and popular examples \\cite{Newman2010a}, the statistical properties of syntactic dependency networks have been compared against the predictions of null models with increasing levels of complexity for the length of syntactic dependencies \\cite{Ferrer2004b, Liu2008a, Futrell2015a} or for the number of crossings \\cite{Ferrer2017a, Ferrer2014f, Ferrer2014c}.\n\nBeyond network theory, the issue of the presence and frequency of crossing dependencies in the syntax of natural languages has received considerable attention in the computational linguistics community, as supporting them makes parsing computationally harder \\cite{McDonald07,Hav07,GomCL2016}. Crossings are also relevant in biology, where they appear in networks of nucleotides whose vertices are occurrences of nucleotides $A$, $G$, $U$, and $C$ while edges are Watson-Crick ($A$-$U$, $G$-$C$) and $U$-$G$ base pairs \\cite{Chen2009a}. \n\nIn this context, a question naturally arises: what is the reason for the low frequency of crossing dependencies, consistently observed across languages? A traditional answer consists of postulating that there is some kind of grammatical ban on\ncrossing dependencies \\cite{sleator93,Tanaka97,KyotoCorpus,Starosta03,Lee04,hudson07,Ninio2017}.\nHowever, this position fails to explain many linguistic phenomena involving crossings\n\\cite{Versley2014,Levy2012a}. Another option is to assume that crossing dependencies can be grammatical, but only if they follow certain patterns or hard constraints. However, while some classes of non-crossing dependency structures have a very good empirical coverage of real sentences \\cite{kuhlmann06,gomez2011cl,GomNiv2013,GomCL2016}, these proposals still face counterexamples that fall outside the restrictions \\cite{chen2010unavoidable,Bhat2012,ChenMain2014}.\n\nFrom the perspective of theoretical linguistics, the grammatical ban on crossings can be interpreted:\n\\begin{itemize}\n\\item\nAs a ban set independently from performance considerations, e.g., requiring some hidden parameter to be turned. In this case the ban can be seen as avoidable (e.g., it depends on whether the parameter is on or off for each given language). \n\\item\nAs a consequence of performance constraints associated directly to crossing dependencies. The ban would be inevitable if the cognitive pressures were strong enough but then it would not be properly a ban (a norm added on top of human cognition) but rather a side-effect of cognitive constraints. This view is challenged by psychological and graph theoretic research indicating that crossing dependencies can be easier to process (\\cite{deVries2012a} and \\cite{Ferrer2014f} and references therein).\n\\end{itemize} \nSome researchers have adopted an apparently neutral position concerning the nature of the ban but assume that the low frequency of crossings derives from an independent and specific constraint on crossings: explicitly when postulating a principle of minimization of crossings \\cite{Liu2008a} or implicitly in a large body of research on dependency length minimization that takes for granted that syntactic dependencies should not cross \\cite{Liu2008a,Park2009a,GildeaTemperley10,Futrell2015a,Gulordava2015}. \n\nIf it turned out that non-crossing dependencies can be explained as a side-effect of some cognitive pressure that is not directly associated to crossings (e.g., dependency length minimization), could all these views be regarded as really neutral regarding the nature of the ban? \n\nIn this article, we explore a simpler hypothesis: that in order to explain the scarcity of crossing dependencies in language, it is not necessary to assume any underlying rule or principle of human languages that is responsible directly for this fact (including the possibility of some cognitive cost associated directly to crossings). \nInstead, the low frequency of crossings may naturally arise, indirectly, from the actual length of dependencies \\cite{Ferrer2015c}, which are constrained by a well-known psychological principle: dependency length minimization (see \\cite{Liu2017}, \\cite{Ferrer2013e} or \\cite{Tily2010a} for a review). That explanatory principle, which holds even in languages allowing for words to scramble freely \\cite{Futrell2015a}, could follow from more general constraints on language processing \\cite{Christiansen2015a}. \n\nAs dependency length minimization can be seen as particular case of minimization of the Euclidean distance between connected vertices in an $m$-dimensional space, our originally linguistic problem on crossings is related to the general problem of minimizing the cost of load transportation over a network in complex systems science \\cite{Guillier2017a} and the minimum linear arrangement problem of computer science \\cite{Ferrer2004b,Ferrer2016a}.\n\nTo investigate the origins of the scarcity of crossing dependencies, we use treebanks (collections of sentences with their corresponding syntactic dependency network) to provide statistical evidence that the amount of dependency crossings in a wide range of languages can be predicted with small error by a simple estimator based exclusively on dependency length information and information on which edges can potentially cross (edges that share a vertex cannot cross). \n\nWe will show that the estimator consistently delivers good predictions of the number of crossings, \nin two different collections of dependency treebanks with diverse annotations. An annotation is a set of criteria used to define the syntactic dependency structure of a sentence. We will argue that this is the best explanation for the low frequency of crossings when both psychological plausibility and parsimony at all levels (from a model of crossings to a general theory of language) are required. Our predictor is a null model in the sense that for every pair of edges that may potentially cross it assumes that the corresponding vertices take random positions in the linear sequence of the sentence. \n\nThe remainder of the article is organized as follows. Section \\ref{predictors_section} discusses various ways in which the crossings of a sentence could be predicted. \nSection \\ref{crossing_theory_section} presents the predictor of crossings chosen for this article and its theoretical background. The dependency trees used to test the predictor are presented in Section \\ref{methods_section}. Section \\ref{results_section} shows the results of the predictions, and Section \\ref{discussion_section} discusses some implications for computational linguistics and linguistic theory. \n\n\\section{Possible predictors}\n\n\\label{predictors_section}\n\nHere we will examine various possibilities to predict the number of dependency crossings in a sentence. \nThe problem of the origins of non-crossing dependencies can be recast as problem of modeling: we want to find the best model\nfor predicting the number of crossings in a sentence. According to modern model selection, the best model is the one that has the best trade-off between quality of fit (predictive power) and parsimony \\cite{Burnham2002a}.\nWe complement this view\ninvolving further requirements: \n\\begin{itemize}\n\\item\nThe model must be psychologically realistic. A model that assumes orderings of words that are hard to produce by the human brain should be penalized with respect to one that is based on orderings that real speakers produce (or can rather easily produce). We are not only simply concerned about predicting the low number of crossings of a sentence but also understanding why that number is that low. Hiding the problem under the carpet of grammar or an ad-hoc principle of planarity\ndoes not help.\n\\item \nIts assumptions must be valid. The predictions of a model may be compatible with real data and even be of high quality but its assumptions may not be supported by real data or inconsistent with the source that produced it. \n\\item \nWe are not only concerned about the best model in a local sense but one that leads to a general theory of word order or even a comprehensive theory of language that is compact. A real scientific theory is more than a collection of disconnected ideas or models \\cite{Bunge2001a_French}.\nModels that lead to an unnecessarily fat general theory when integrated into it should also be penalized. \nModels that exploit assumptions from successful models in other domains should be favored. \n\nFor instance, a model that allows one to understand not only the scarcity of crossings but also why adjectives tend to be placed \nbefore the noun in SOV languages is preferable to one that requires an independent solution to explain the placement of adjectives \\cite{Ferrer2014f}. SOV languages are languages that tend to put the subject (S) before the object (O) and in turn, O before the verb (V) under some general conditions \\cite{wals-81}.\n\\end{itemize}\n\nIn what follows, we will use $C$ to refer to the number of dependency crossings in the parse of a sentence (i.e., the number of pairs of syntactic dependencies that cross). Our goal is, therefore, to find a suitable predictor for $C$. Note that $C = 0$ for a star tree \\cite{Ferrer2013d}. The sum of the lengths of all dependencies in a sentence will be denoted by $D$. \n\n\\subsection{Minimization of crossings}\n\nA principle of minimization of crossings \\cite{Liu2008a} leads to a simple deterministic predictor: $C = 0$, reflecting a grammatical ban on crossings\n\\cite{sleator93,Tanaka97,KyotoCorpus,Starosta03,Lee04,hudson07}.\n\nThis predictor is problematic for various reasons:\n\\begin{itemize}\n\\item\nConcerning the validity of its assumptions, the model assumes that $C = 0$ independently from $D$, while $C$ and $D$ are positively correlated in many languages \\cite{Ferrer2015c}. \n\\item\nConcerning the accuracy of its predictions, this model fails because sentences with $C > 0$ are found in many languages \\cite{Ferrer2017a} and the likelihood of the model is minimum, which indicates that the model is among the worst possible models for crossing dependencies according to modern model selection \\cite{Burnham2002a} because its likelihood is zero. Furthermore, \nthe model fails to explain many linguistic phenomena involving crossings\n\\cite{Versley2014,Levy2012a}. \n\\item\nIts psychological validity is unclear. If the model is interpreted as arising from processing difficulties inherent to crossing dependencies \\cite{Levy2012a} or computational tractability (as reviewed in Section \\ref{introduction_section}) then it is challenged by psychological and graph theoretic research indicating that sentences with $C >0$ can be easier to process than sentences with $C= 0$ (see \\cite{deVries2012a}, \\cite{Ferrer2014f}, \\cite{Chung1978a} and references therein). Another problem is how a language generation process could warrant that $C=0$. If $C=0$ is determined before the sentence is produced, how is it possible that sentence production does not introduce (many) crossings? Crossing theory indicates that a star tree is needed to keep a low number of crossings \\cite{Ferrer2014f}. \nIf $C=0$ is determined while the sentence is produced (linearized), how are crossings avoided on the fly as real language production is not a batch process \\cite{Christiansen2015a}? It looks simpler to consider that non-crossing dependencies are a side-effect of a principle of dependency length minimization \\cite{Ferrer2006d,Ferrer2014c,Ferrer2014f}.\n\\item\nConcerning the compactness of the whole theory, the model $C = 0$ leads to a fatter theory of language because the scarcity of crossings and also the positive correlation between $D$ and $C$ could be explained to a large extent by recycling the highly predictive principle of dependency length minimization \\cite{Ferrer2013e}, as we will see below. \n\\end{itemize}\n\nAnother option is to assume that crossing dependencies can be grammatical, but only if they follow certain patterns or hard constraints. However, while some\nclasses of dependency structures tolerating certain crossings\nhave a very good empirical coverage \\cite{kuhlmann06,gomez2011cl,GomNiv2013,GomCL2016}, these proposals still face counterexamples that fall outside the restrictions \\cite{chen2010unavoidable,Bhat2012,ChenMain2014}.\n\nOne possibility is to relax the simple deterministic predictor above so that on average $C = \\gamma$, where $\\gamma$ is a constant, e.g., $\\gamma = 3.3$ as in ancient Greek \\cite{Ferrer2017a}. However, it has been shown that this is problematic because $C = \\gamma$ might be impossible to reach if $n$ is sufficiently small (see Appendix of \\cite{Ferrer2015c}). Therefore, a proper relaxation of this deterministic predictor is $C = \\gamma(n)$, where $\\gamma$ is a function that only depends on $n$ \\cite{Ferrer2015c}.\nThis allows one to capture the variation in the number of crossings across languages, but adding extra parameters, and it is still problematic for the reasons of the case $\\gamma(n) = 0$ that we have examined above. Further arguments can be found in Section 4.3 of \\cite{Ferrer2014f}. \n\n\\subsection{Minimum linear arrangement}\n\nA minimum linear arrangement of a sentence is an ordering of the words of the sentence that minimizes the sum of dependency lengths.\nOne may predict the assumed number of crossings by calculating the minimum linear arrangements of a sentence \\cite{Ferrer2006d}. A possible predictor could be the mean number of crossings over all those arrangements. \n\nThe predictive power of the model is supported by the fact that solving the minimum linear arrangement problem reduces crossings to practically zero \\cite{Ferrer2006d}, as in many languages. A potential problem of this model is that it has never been checked whether it predicts the actual number of crossings of real sentences, as far as we know. \n\nPerhaps the major challenge for this predictor is the validity of the assumption of a minimum linear arrangement because:\n\\begin{itemize}\n\\item\nThe actual value of $D$ in real sentences is located between the minimum and that of a random ordering of vertices \\cite{Ferrer2004b,Ferrer2013c}. The ratio $\\Gamma = D\/D_{min}$ (where $D_{min}$ is the minimum value of $D$) is greater than 1.2 in Romanian for sufficiently long sentences \\cite{Ferrer2004b} and a similar lower bound on language efficiency has been found in English across centuries \\cite{Tily2010a}.\n\\item\nIt may not be valid also for theoretical reasons: word order is a multiconstraint satisfaction problem where the principle of dependency length minimization is in conflict with other word order constraints \\cite{Ferrer2014a,Futrell2015a}.\nThus, a model based on minimum linear arrangements is not that simple: it has to explain why dependency length minimization dominates fully over other principles or provide evidence that the distortion caused by other principles can be neglected. Below we will present a model that does not have this problem because it works on true dependency lengths, which are expected to be determined by the interplay between dependency length minimization and other principles. \n\\item\nThe full minimization of $D$ is cognitively unrealistic, as it is incompatible with the predictions of the now-or-never bottleneck \\cite{Christiansen2015a}. As for the latter, notice that the minimization of $D$ implies that the whole sentence must be available as input for some minimum linear arrangement algorithm, whereas actual language generation and processing is intrinsically online and heavily constrained by our fleeting memory \\cite{Christiansen2015a}.\n\\end{itemize}\n\n\\subsection{Random linear arrangement}\n\nIf the minimum linear arrangement is too restrictive, one could consider the opposite: \npredicting the number of crossings assuming a random ordering of the words of the sentence \\cite{Ferrer2014f}. However, a random linear arrangement cannot explain the low numbers of crossings observed in real sentences. Empirically, the number of crossings of sentences is much smaller than the number of crossings expected by random linear arrangement \\cite{Ferrer2017a}. Theoretically, a constant low number of crossings requires a star tree \\cite{Ferrer2014f}. \n\nThe failure of a random linear arrangement is not surprising. First, it \nis cognitively unrealistic: even in languages with high word order flexibility, word order is constrained \\cite{wals-81,Lester2015a}. Second, the assumption that the ordering of sentences is arbitrary (unconstrained) is easily rejected by the fact that dependency lengths are below chance in real languages \\cite{Ferrer2004b,Ferrer2013c,Futrell2015a}. \nThus, this predictor is only useful as a random baseline for other predictors. Here we will compare it against a better predictor that is introduced next. \n \n\\subsection{Random linear arrangement with some knowledge about dependency lengths}\n\nA stronger predictor can be built by focusing on the set of pairs of edges that may potentially cross and basing predictions on the actual length of the edges under the assumption of a random linear arrangement of the four vertices that are potentially involved in an edge crossing \\cite{Ferrer2014c}. So far, its predictive power is supported by its capacity to predict the actual number of crossings in random trees with an error of about $5\\%$ \\cite{Ferrer2014c}.\nA crucial goal of the present article is to test the accuracy of its predictions on real sentences. \nThis predictor is promising because actual dependency lengths are below chance, i.e. below $(n+1)\/3$ \\cite{Ferrer2004b,Ferrer2013c}, a domain where the probability theory behind this model indicates that shortening a dependency yields a reduction in the probability that it crosses a dependency of unknown length in a random linear arrangement of the two edges (Section 5 of \\cite{Ferrer2014f}).\n\nFor the reason above,\nthis predictor is fully compatible with the positive correlation between $D$ and $C$ \\cite{Ferrer2015c,Ferrer2014f}, in contrast with the deterministic predictor ($C=0$) and its generalization. \nConcerning assumptions, this model is simpler than the model based on minimum linear arrangements: this model does not assume an unrealistic ordering of the elements of the sentence but the true ordering. \nIts psychological validity is greater than that of the minimum linear arrangement predictor because it can base its prediction on information from sentences that have actually been produced by a speaker or a writer.\nContrary to the minimum linear arrangement predictor, this model bases its prediction on true dependency lengths instead of ideal values of $D$.\n\nHowever, it can be argued that a fundamental assumption of the model, namely that vertices are arranged linearly at random, is not supported empirically, following the arguments against the random linear arrangement predictor. This is a fair criticism, but for this reason this model should be regarded as a null hypothesis rather than as a fully realistic model. \n\nHaving said that,\nmodeling requires a compromise between quality of fit, adequacy and parsimony. If this null hypothesis model provides predictions of sufficient quality on real sentences, do we really need to worry about providing a more realistic but also more complicated model? \nPut differently, suppose that the information provided by the lengths of edges suffices to predict reasonably well the low number of crossings of real sentences, even without assuming any additional constraint on the linear arrangement of the involved vertices that could help to minimize crossings, but instead modeling it under the weakest possible assumption (namely placing vertices at random). Then considering more fine-grained information or more realistic orderings is secondary to our particular goal. In the worst case, this predictor would be an inevitable baseline for an alternative model. \n\nBefore we proceed, it is worth noting that our article is not a mere application of an established model or theory to a concrete dataset, but the first test of a novel theory on a massive collection of networks from different languages and different annotation criteria, which has implications to our understanding of the faculty of language as such. \nThe result of such a test is far from trivial, and thus its success is a relevant contribution, for two reasons.\nFirst, our model, which is a null model rather than a realistic model, assumes that vertices are arranged purely at random in a sequence (preserving the original edge lengths). However, real sentences are not random sequences of words, as research on long correlations in physics has been showing for more than a decade, e.g. \\cite{Montemurro2001b,Altmann2012a}. \nSecond, although such null model predictor has been tested previously on uniformly random trees \\cite{Ferrer2014c}, one cannot assume the predictor will work on real sentences given the substantial statistical differences between uniformly random trees and real syntactic dependency trees \\cite{Ferrer2017a}.\n\nThe next section introduces the mathematical definition of the promising predictor above and its theoretical background. \n\n\\section{Crossing theory}\n\n\\label{crossing_theory_section}\n\nHere we provide a quick overview of a crossing theory developed in a series of articles \\cite{Ferrer2013b,Ferrer2013d,Ferrer2014c,Ferrer2014f, Ferrer2017a}. \nIt is correct to state that $C$ cannot exceed the number of pairs of different edges, namely\n\\begin{equation}\nC \\leq {n - 1 \\choose 2}\n\\label{naive_maximum_number_of_crossings_equation}\n\\end{equation} \nHowever, the truth is that \n\\begin{equation}\nC \\leq {n - 2 \\choose 2}\n\\label{maximum_number_of_crossings_equation}\n\\end{equation}\nwith equality in case of a linear tree (see \\cite{Ferrer2017a} for linear arrangements of linear tree that maximize $C$). The upper bound above is defined based only on knowledge of the size of the tree. Adding further properties of the tree, the upper bound can be refined.\n\nA central concept of crossing theory is $Q$, the set of pairs of edges of a tree that can potentially cross when their vertices are arranged linearly in some arbitrary order (edges sharing a vertex cannot cross). \n$|Q|$, the cardinality of $Q$, is the potential number of crossings, i.e. \n\\begin{equation}\nC \\leq |Q| \\leq {n - 2 \\choose 2}.\n\\end{equation}\nWe have \n\\begin{equation}\n|Q| = \\frac{n}{2} \\left(\\left< k^2 \\right>_{star} - \\left< k^2 \\right>\\right),\n\\label{potential_number_of_crossings_equation}\n\\end{equation}\nwhere \n$\\left< k^2 \\right>$ is the mean of the squared degrees of its vertices and $\\left< k^2 \\right>_{star}= n - 1$ is the value of $\\left< k^2 \\right>$ in a star tree of size $n$ \\cite{Ferrer2014c,Ferrer2013d}. $|Q| = 0$ if and only if the tree is a star tree \\cite{Ferrer2013d}.\n\nWith the theoretical background above, it is easy to see why $C$ cannot exceed the number of different pairs that can be formed out of $n-2$ elements (Eq. \\ref{maximum_number_of_crossings_equation}) instead of $n-1$, that coincides with the number of edges (Eq. \\ref{naive_maximum_number_of_crossings_equation}): that is the conclusion of computing the value of $\\left< k^2 \\right>$ for a linear tree, i.e. $\\left< k^2 \\right>_{linear} = 4 - 6\/n$, and then applying \n$\\left< k^2 \\right> \\geq 4 - 6\/n$ to Eq. \\ref{potential_number_of_crossings_equation} \n\\cite{Ferrer2017a}.\n\n$C$ denotes the number of crossings of the linear arrangement of a graph in general while \n$C_{true}$ denotes the number of crossings of the syntactic dependencies of a real sentence. The relative number of crossings is $\\bar{C} = C\/|Q|$ or \n$\\bar{C}_{true} = C_{true}\/|Q|$ \\cite{Ferrer2014c}. $C$ can be expressed as a sum over $Q$, i.e. \n\\begin{equation}\nC = \\sum_{(e_1, e_2) \\in Q} C(e_1, e_2),\n\\label{number_of_crossings_equation}\n\\end{equation} \nwhere $C(e_1, e_2)$ is an indicator variable, $C(e_1, e_2)$ = 1 if the edges $e_1$ and $e_2$ cross and $C(e_1, e_2) = 0$ otherwise. \nThe simplest prediction about $C$ than can be made departs from the null hypothesis that the vertices are arranged linearly at random (all possible orderings are equally likely). Following Eq. \\ref{number_of_crossings_equation}, the expected number of crossings under that null hypothesis turns out to be \n\\begin{eqnarray}\nE_0[C] & = & \\sum_{(e_1, e_2) \\in Q} E[C(e_1, e_2)]\\\\\n & = & \\sum_{(e_1, e_2) \\in Q} p(C(e_1, e_2) = 1), \n\\label{expected_number_of_crossings_equation}\n\\end{eqnarray}\nwhere $p(C(e_1, e_2) = 1)$ is the probability that the edges $e_1$ and $e_2$ cross knowing that they belong to $Q$. \nUnder that null hypothesis, the probability that two edges of $Q$ cross is constant, i.e. $p(C(e_1, e_2) = 1) = 1\/3$, yielding \\cite{Ferrer2013d} $E_0[C] = |Q|\/3$.\n\n\\begin{figure*}\n\\includegraphics[scale = 0.8]{probability_given_two_lengths_4_panels}\n\\caption{\\label{probability_given_two_lengths_figure} $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$\nas a function of $d(e_1)$ and $d(e_2)$ for different values of $n$ (tree size in vertices). \n$d(e)$ is the length of the edge $e$ and $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$ is the probability that $e_1$ and $e_2$ (a pair of edges of $Q$) cross in a random linear arrangement of their vertices, knowing their lengths. Brightness is proportional to $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$ (black for $p(C(e_1, e_2) = 1 | d(e_1), d(e_2)) = 0$ and white for $p(C(e_1, e_2) = 1 | d(e_1), d(e_2)) = 1$). The two panels on top show the value of the probability.} \n\\end{figure*}\n\nThe prediction offered by $E_0[C]$ can be improved by introducing knowledge about the length of the dependencies (e.g., edges of length 1 or $n - 1$ are not crossable). Suppose that $d(e)$ is the length of the edge $e$ and that $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$ is the probability that $e_1$ and $e_2$ (two arbitrary edges of $Q$) cross in a random linear arrangement of their vertices knowing their lengths.\nThe predictor $E_2[C]$ is obtained when $p(C(e_1, e_2) = 1)$ is replaced by $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$\nin Eq. \\ref{expected_number_of_crossings_equation}, yielding \n\\begin{equation}\nE_2[C] = \\sum_{(e_1, e_2) \\in Q} p(C(e_1, e_2) = 1 | d(e_1), d(e_2)),\n\\label{predicted_number_of_crossings_equation}\n\\end{equation}\n$p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$ depends only on $n$, $d(e_1)$ and $d(e_2)$ and\nis defined as \n\\begin{equation}\np(C(e_1, e_2) = 1 | d(e_1), d(e_2)) = \\frac{|\\alpha(d(e_1),d(e_2))|}{|\\beta(d(e_1),d(e_2))|},\n\\label{probability_of_crossing_given_two_lengths_equation}\n\\end{equation}\nwhere here $|..|$ is the cardinality operator, $\\alpha(d_1,d_2)$ is the set of valid pairs of initial positions of two edges of lengths $d_1$ and $d_2$ that involve a crossing and $\\beta(d_1,d_2)$ is the set of valid pairs of initial positions of edges of lengths $d_1$ and $d_2$, thus $\\alpha(d_1,d_2) \\subseteq \\beta(d_1,d_2)$. Fig. \\ref{probability_given_two_lengths_figure} shows a two-dimensional map of $p(C(e_1, e_2) = 1 | d(e_1), d(e_2))$. The perimeter of the map contains zeroes because an edge of minimum length ($1$) or maximum length ($n-1$) cannot cross any other edge. The map is symmetric with respect to the diagonal that crosses the top-left corner and the bottom-right corner by symmetry, namely\n\\begin{eqnarray}\np(C(e_1, e_2) = 1 | d(e_1), d(e_2)) = \\notag \\\\ \n p(C(e_2, e_1) = 1 | d(e_2), d(e_1)). \n\\end{eqnarray}\nThe map for $n = 4$, the minimum value of $n$ needed to have $|Q| > 0$, shows that only edges of length 2 can cross. The maps for $n = 8$, $n = 12$ and $n=16$ show that a reduction of the length of one of the edges causes the probability of crossing to reduce if edge lengths are sufficiently small. This reduction of the probability of crossings is likely to occur in real languages, where the mean length of dependencies is on average smaller than the random baseline $(n+1)\/3$ \\cite{Ferrer2004b, Ferrer2013c} and long edges would imply a cognitive cost that may not be afforded \\cite{Liu2017,Christiansen2015a,Ferrer2013e}.\n\nAlthough $E_0[C]$ and $E_2[C]$ are predictors of $C$ that have the same mathematical structure (they are sums of probabilities over pairs of edges of $Q$), $E_0[C]$ is a true expectation while $E_2[C]$ is not. \n\nThe relative error of a predictor is defined as \\cite{Ferrer2014c}\n\\begin{equation}\n\\Delta_x = E_x\\left[\\bar{C}\\right] - \\bar{C}_{true} = \\frac{1}{|Q|}\\left(E_x[C]-C_{true}\\right).\n\\label{error_equation}\n\\end{equation}\n$\\Delta_0$ will be used as a baseline for $\\Delta_2$. Interestingly,\n$\\Delta_0$ converges to $1\/3$ for sufficiently long sentences when $C_{true}$ is bounded by a constant and $|Q|$ is large enough. The reason is that $E_0[C]\/|Q| = 1\/3$ and then \n\\begin{equation}\n\\Delta_0 = \\frac{1}{3} - C_{true}\/|Q|.\n\\end{equation}\nThat explains why $\\Delta_0$ converges to $1\/3$ for sufficiently large $n$ in uniformly random trees where $C$ is bounded by a small constant \\cite{Ferrer2014c} because uniformly random trees have a high $|Q|$, or equivalently, a low hubiness \\cite{Ferrer2017a}. We also expect $\\Delta_0$ to converge to $1\/3$ in real syntactic dependency trees because $C_{true}$ is small and their hubiness is also low \\cite{Ferrer2017a}. \n\n\\section{Materials and methods}\n\n\\label{methods_section}\n\nWe considered the corpora in version 2.0 of the HamleDT collection of treebanks \\cite{HamleDTJournal,HamledTStanford}. This collection is a harmonization of existing treebanks for 30 different languages into two well-known annotation styles: Prague dependencies \\cite{PDT20} and Universal Stanford dependencies \\cite{UniversalStanford}. Therefore, this collection allows us to evaluate our predictions of crossings both across a wide range of languages and two popular annotation schemes. The latter is useful because observations like the number of dependency crossings present in treebank sentences do not only depend on the properties of languages themselves, but also on annotation criteria (\\cite{Ferrer2015c} lists some examples of how annotation criteria may affect $C$).\n\nEach of the syntactic dependency structures in the treebanks was preprocessed by removing nodes corresponding to punctuation tokens, as it is standard in research related to dependency length (e.g., \\cite{Ferrer2004b,Ferrer2015c,Futrell2015a}), which is only concerned with dependencies between actual words. To preserve the syntactic structure of the rest of the nodes, non-punctuation nodes that had a punctuation node as their head were attached as dependents of their nearest non-punctuation ancestor. Null elements, which appear in the Bengali, Hindi and Telugu corpora, were also subject to the same treatment as punctuation.\n\nAfter this preprocessing, a syntactic dependency structure was included in our analyses if (1) it defined a tree and (2) the tree was not a star tree. \nThe reason for (1) is that our theory (e.g., $Q$) assumes a tree structure \\cite{Ferrer2013d,Ferrer2014c} and that we wanted to avoid the statistical\nproblem of mixing trees with other kinds of graphs, e.g., the potential number\nof crossings depends on the number of edges \\cite{Ferrer2013b,Ferrer2014f,Ferrer2013d}. \nThe reason for (2) is that crossings are impossible in a star tree \\cite{Ferrer2013b}. Condition (2) implies that the syntactic dependency structure has at least four vertices (otherwise all the possible trees are star trees). By excluding star trees we are discarding trees where the prediction cannot fail. \nAn additional reason for excluding star trees is that their relative number of crossings, $C\/|Q|$, is not defined because\n$C = |Q| = 0$.\n\nTable \\ref{relative_table} shows the number of sentences in the original treebanks and the number of sentences actually included in our analyses, after filtering by the criteria (1) and (2) above. The average number of crossings per sentence does not exceed 1 for most of the treebanks. See \\cite{Ferrer2017a} for further details on the statistical properties of crossings in our collections of dependency treebanks. \n\nHere we adopt the convention of sorting languages in tables not alphabetically but decreasingly by number of crossings, measured according to the average number of crossings (the average $C_{true}$) with Stanford dependencies. It can be observed that languages that are known for their word order freedom, e.g., Latin or Ancient Greek, stand out on top of Table \\ref{relative_table}. On the other hand, agglutinating languages like Basque, Japanese, Turkish, the Uralic languages Estonian and Finnish, and the Dravidian languages Tamil and Telugu, are placed rather to the bottom of the table. \nAgglutinating languages are languages where certain information is often integrated into words as morphemes (not leading to new vertices in the tree, except in the Turkish treebank) while non-agglutinating languages would instead place it in separate words (leading to separate vertices). Therefore, one expects fewer chances for dependency crossings in agglutinating languages, as equivalent information is expressed with fewer vertices, and the number of crossings tends to increase with the length of the sentence \\cite{Ferrer2017a}. \n\nOur ordering by crossings should be taken as an approximation. For the sake of space, we only employ an ordering by crossings based on Stanford dependencies. Furthermore, the potential number of crossings may depend on factors such as genre, topic, sentence length or treebank size (number of sentences) and other biases \\cite{Ferrer2015c,Ferrer2017a}. The collection of treebanks is heterogeneous in this respect. Therefore, the fact that one treebank has more crossings than another does not imply that the language of the former exhibits higher word order freedom than that of the latter. Other variables should be controlled for a more accurate ordering. Therefore, the focus of our article is on the power of the predictors in spite of the heterogeneity of the treebank collection. Linguistic distinctions such as agglutinating versus non-agglutinating languages are made to illustrate the potential of future linguistic research.\n \n\\section{Results}\n\n\\label{results_section}\n\n\\begin{figure*}\n\\includegraphics[scale = 0.8]{error2}\n\\caption{\\label{relative2_figure} $\\Delta_2$, the error of the predictor based on the probability that two edges cross in a random linear arrangement preserving their original length, as a function of $n$, the size of the tree. Points and error bars indicate, respectively, mean values and $\\pm 1$ standard deviation over proportions in a collection of treebanks. Tree sizes represented by less than two treebanks are excluded. Top: Stanford annotations. Bottom: Prague annotations.}\n\\end{figure*}\n\nFigure \\ref{relative2_figure} shows that, on average across treebanks, $\\Delta_2$ increases as $n$ increases till $n = 10$ and decreases from that point onwards in both annotations. The maximum average $\\Delta_2$ that is reached at $n = 10$ is $0.052$ for Stanford annotations and $0.038$ for Prague annotations. \nThe predictor never fails for $n = 4$ and $n = 5$ ($\\Delta_2 = 0$ in both cases) and from $n = 6$ onwards it always overestimates (on average) the actual number of crossings (recall Eq. \\ref{error_equation}). Figure \\ref{relative0_figure} shows that, on average across treebanks, $\\Delta_0$ converges to $1\/3$ as expected. \n\n\\begin{figure*}\n\\includegraphics[scale = 0.8]{error0}\n\\caption{\\label{relative0_figure} $\\Delta_0$, the error of the random linear arrangement predictor, as a function of $n$, the size of the tree. The format is the same as in Fig. \\ref{relative2_figure}. A control line has been added to indicate 1\/3, namely, the $\\Delta_0$ that is expected when $C_{true}$ is bounded above by a small constant and $n$ goes to infinity. }\n\\end{figure*}\n\nTable \\ref{relative_table} shows that the average $\\Delta_2$, the relative error of the predictor $E_2[C]$,\nis small: it does not exceed $5\\%$. Thus, the average $\\Delta_2$ is at least 6 times smaller than the baseline $\\Delta_0 \\approx 33\\%$. \nThe averages presented in Table \\ref{relative_table} have been produced mixing measurements from sentences of different lengths. This is potentially problematic because the results might be heavily determined by the distribution of sentence lengths \\cite{Ferrer2013c}. \n\nTo control for sentence length, sentences were grouped by length and the average $\\Delta_2$ was computed for the sentences within each group. Table \\ref{relative_grouping_by_sentence_length_table} summarizes the statistical properties over the average $\\Delta_2$ of each group. Interestingly, the average over group averages of $\\Delta_2$ decreases with respect to the previous analysis: it does not exceed $4.3\\%$. Thus, the average $\\Delta_2$ is at least 7 times smaller than the baseline error, again $\\Delta_0 \\approx 33\\%$. The minimum size of a group is one sentence; the qualitative results are very similar if the minimum size is set to 2.\n \n\\newcommand{\\specialcell}[2][c]{%\n \\begin{tabular}[#1]{@{}c@{}}#2\\end{tabular}}\n\n\\begin{turnpage}\n\\begin{table*}\n\\caption{\\label{relative_table} Summary of results for each treebank: number of sentences before and after filtering, average values of $C_{true}$ and $\\Delta_0$, and average, median and standard deviation of the relative error $\\Delta_2$, over the trees of each treebank. \nRomanian (Prague) is the only treebank with no crossing dependencies. Languages are sorted decreasingly by average $C_{true}$ according to Stanford dependencies.}\n\n\\begin{small}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{lr|rccccc|rccccc} \n\\phantom{x} & \\phantom{x} & \\multicolumn{2}{l}{\\emph{Stanford annotation}} & \\phantom{x} & \\phantom{x} & \\phantom{x} & \\phantom{x} & \\multicolumn{2}{l}{\\emph{Prague annotation}} & \\phantom{x} & \\phantom{x} & \\phantom{x} & \\phantom{x} \\\\ \\hline\nTreebank & \\#Sent & \\specialcell{\\#Sent\\\\ \\footnotesize{(filtered)}} &\n\\specialcell{$C_{true}$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_0$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(median)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(st. dev.)}} \n& \\specialcell{\\#Sent\\\\ \\footnotesize{(filtered)}} &\n\\specialcell{$C_{true}$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_0$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(median)}} & \\specialcell{$\\Delta_2$\\\\ \\footnotesize{(st. dev.)}}\\\\ \\hline\nAnc. Greek & 21173 & 18713 & 3.2621 & 0.244 & 0.030 & 0.027 & 0.058 & 16237 & 3.3528 & 0.243 & 0.025 & 0.020 & 0.058 \\\\\nLatin & 3473 & 3036 & 2.1785 & 0.282 & 0.034 & 0.031 & 0.046 & 2833 & 1.8503 & 0.286 & 0.036 & 0.032 & 0.047 \\\\\nDutch & 13735 & 10974 & 1.3980 & 0.311 & 0.046 & 0.041 & 0.051 & 11131 & 0.9898 & 0.315 & 0.034 & 0.027 & 0.041 \\\\\nHungarian & 6424 & 6103 & 0.9720 & 0.326 & 0.036 & 0.031 & 0.033 & 5047 & 0.8675 & 0.326 & 0.034 & 0.030 & 0.032 \\\\\nArabic & 7547 & 2280 & 0.9807 & 0.328 & 0.019 & 0.016 & 0.021 & 2248 & 0.0881 & 0.333 & 0.013 & 0.010 & 0.016 \\\\\nGerman & 38020 & 33492 & 0.7826 & 0.325 & 0.050 & 0.046 & 0.036 & 32443 & 0.7230 & 0.326 & 0.043 & 0.039 & 0.033 \\\\ \nSlovenian & 1936 & 1719 & 0.7749 & 0.322 & 0.047 & 0.039 & 0.046 & 1581 & 0.3125 & 0.327 & 0.035 & 0.027 & 0.038 \\\\ \nDanish & 5512 & 4894 & 0.6800 & 0.324 & 0.047 & 0.040 & 0.038 & 4840 & 0.1643 & 0.331 & 0.027 & 0.022 & 0.027 \\\\ \nGreek & 2902 & 2584 & 0.6540 & 0.330 & 0.039 & 0.033 & 0.028 & 2543 & 0.2057 & 0.332 & 0.030 & 0.024 & 0.023 \\\\ \nCatalan & 14924 & 14520 & 0.6419 & 0.331 & 0.034 & 0.029 & 0.023 & 14556 & 0.0873 & 0.333 & 0.020 & 0.017 & 0.016 \\\\\nPortuguese & 9359 & 8621 & 0.6336 & 0.328 & 0.039 & 0.033 & 0.032 & 8596 & 0.2465 & 0.331 & 0.021 & 0.016 & 0.021 \\\\\nSpanish & 15984 & 15354 & 0.6218 & 0.331 & 0.034 & 0.029 & 0.024 & 15424 & 0.1105 & 0.333 & 0.020 & 0.016 & 0.017 \\\\\nPersian & 12455 & 11579 & 0.5914 & 0.326 & 0.027 & 0.023 & 0.031 & 11632 & 0.4024 & 0.329 & 0.030 & 0.024 & 0.033 \\\\ \nCzech & 87913 & 74843 & 0.5277 & 0.326 & 0.040 & 0.035 & 0.035 & 70023 & 0.3729 & 0.327 & 0.031 & 0.025 & 0.031 \\\\ \nEnglish & 18791 & 18275 & 0.5241 & 0.330 & 0.049 & 0.043 & 0.031 & 18369 & 0.1072 & 0.333 & 0.034 & 0.029 & 0.024 \\\\\nSwedish & 11431 & 10714 & 0.4871 & 0.328 & 0.043 & 0.039 & 0.034 & 10207 & 0.1946 & 0.332 & 0.034 & 0.029 & 0.029 \\\\ \nSlovak & 57408 & 47727 & 0.4559 & 0.324 & 0.044 & 0.036 & 0.044 & 44297 & 0.2688 & 0.326 & 0.034 & 0.026 & 0.039 \\\\ \nRussian & 34895 & 31581 & 0.4171 & 0.326 & 0.038 & 0.032 & 0.035 & 31900 & 0.1570 & 0.330 & 0.027 & 0.021 & 0.028 \\\\ \nItalian & 3359 & 2502 & 0.4153 & 0.329 & 0.035 & 0.029 & 0.032 & 2398 & 0.0621 & 0.333 & 0.020 & 0.014 & 0.024 \\\\ \nBulgarian & 13221 & 12119 & 0.3598 & 0.326 & 0.045 & 0.039 & 0.042 & 11947 & 0.1248 & 0.329 & 0.023 & 0.017 & 0.029 \\\\ \nFinnish & 4307 & 4078 & 0.3183 & 0.326 & 0.034 & 0.028 & 0.038 & 4011 & 0.1279 & 0.330 & 0.028 & 0.024 & 0.031 \\\\ \nHindi & 13274 & 12417 & 0.3043 & 0.332 & 0.027 & 0.025 & 0.017 & 12334 & 0.3875 & 0.330 & 0.015 & 0.012 & 0.015 \\\\ \nJapanese & 17753 & 4614 & 0.1641 & 0.326 & 0.024 & 0.019 & 0.032 & 4792 & 0.0002 & 0.333 & 0.006 & 0.000 & 0.013 \\\\ \nBasque & 11225 & 9072 & 0.1391 & 0.330 & 0.028 & 0.022 & 0.033 & 8717 & 0.1252 & 0.330 & 0.026 & 0.021 & 0.029 \\\\ \nRomanian & 4042 & 3145 & 0.1021 & 0.331 & 0.028 & 0.021 & 0.036 & 3193 & 0.0000 & 0.333 & 0.015 & 0.005 & 0.026 \\\\ \nBengali & 1129 & 678 & 0.1062 & 0.321 & 0.027 & 0.000 & 0.051 & 651 & 0.1244 & 0.320 & 0.025 & 0.000 & 0.052 \\\\ \nTurkish & 5935 & 3862 & 0.0984 & 0.330 & 0.031 & 0.025 & 0.038 & 3518 & 0.1373 & 0.327 & 0.015 & 0.000 & 0.026 \\\\\nEstonian & 1315 & 851 & 0.0376 & 0.331 & 0.016 & 0.000 & 0.037 & 843 & 0.0130 & 0.332 & 0.013 & 0.000 & 0.031 \\\\ \nTamil & 600 & 584 & 0.0240 & 0.333 & 0.026 & 0.022 & 0.025 & 585 & 0.0137 & 0.333 & 0.023 & 0.019 & 0.023 \\\\ \nTelugu & 1450 & 429 & 0.0140 & 0.322 & 0.016 & 0.000 & 0.045 & 373 & 0.0080 & 0.325 & 0.014 & 0.000 & 0.043 \\\\ \n\n\\end{tabular}\n\n\\end{ruledtabular}\n\\end{center}\n\\end{small}\n\\end{table*}\n\\end{turnpage}\n\n\n\\begin{table*}\n\\caption{\\label{relative_grouping_by_sentence_length_table} \nSummary of results for each treebank: number of distinct sentence lengths, \naverage $\\Delta_0$, and average, median and standard deviation of the average values of $\\Delta_2$ over the groups of sentences with the same length. Languages are sorted decreasingly by average \n$C_{true}$ according to Stanford dependencies as in Table \\ref{relative_table}.}\n\\begin{scriptsize}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{l|rcccc|rcccc} \n\\phantom{x} & \\multicolumn{2}{l}{\\emph{Stanford annotation}} & \\phantom{x} & \\phantom{x} & \\phantom{x} & \\multicolumn{2}{l}{\\emph{Prague annotation}} & \\phantom{x} & \\phantom{x} & \\phantom{x} \\\\ \\hline\nTreebank & \\#Lengths & \\specialcell{$\\Delta_0$ \\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(median)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(st. dev.)}} \n& \\#Lengths & \\specialcell{$\\Delta_0$ \\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(avg.)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(median)}} & \\specialcell{$\\Delta_2$ \\\\ \\footnotesize{(st. dev.)}} \\\\ \\hline\n\nAnc. Greek & 66 & 0.293 & 0.025 & 0.024 & 0.019 & 65 & 0.292 & 0.021 & 0.021 & 0.020 \\\\ \nLatin & 59 & 0.309 & 0.031 & 0.029 & 0.018 & 59 & 0.313 & 0.031 & 0.030 & 0.017 \\\\ \nDutch & 54 & 0.319 & 0.037 & 0.035 & 0.019 & 54 & 0.323 & 0.027 & 0.024 & 0.016 \\\\ \nHungarian & 65 & 0.328 & 0.027 & 0.025 & 0.014 & 65 & 0.329 & 0.026 & 0.023 & 0.013 \\\\ \nArabic & 109 & 0.331 & 0.014 & 0.013 & 0.006 & 109 & 0.333 & 0.010 & 0.008 & 0.005 \\\\ \nGerman & 85 & 0.328 & 0.033 & 0.032 & 0.012 & 85 & 0.329 & 0.029 & 0.027 & 0.011 \\\\ \nSlovenian & 57 & 0.326 & 0.034 & 0.032 & 0.016 & 50 & 0.329 & 0.027 & 0.024 & 0.014 \\\\ \nDanish & 66 & 0.328 & 0.031 & 0.029 & 0.013 & 66 & 0.332 & 0.019 & 0.017 & 0.010 \\\\ \nGreek & 75 & 0.331 & 0.027 & 0.024 & 0.010 & 74 & 0.333 & 0.021 & 0.019 & 0.008 \\\\ \nCatalan & 98 & 0.332 & 0.023 & 0.021 & 0.008 & 98 & 0.333 & 0.014 & 0.012 & 0.006 \\\\ \nPortuguese & 88 & 0.331 & 0.024 & 0.023 & 0.009 & 88 & 0.332 & 0.013 & 0.012 & 0.006 \\\\ \nSpanish & 95 & 0.332 & 0.023 & 0.022 & 0.009 & 95 & 0.333 & 0.014 & 0.013 & 0.006 \\\\ \nPersian & 93 & 0.329 & 0.023 & 0.021 & 0.009 & 93 & 0.331 & 0.022 & 0.021 & 0.009 \\\\ \nCzech & 88 & 0.330 & 0.024 & 0.022 & 0.010 & 87 & 0.331 & 0.019 & 0.017 & 0.009 \\\\ \nEnglish & 74 & 0.331 & 0.033 & 0.031 & 0.013 & 75 & 0.333 & 0.023 & 0.021 & 0.010 \\\\ \nSwedish & 74 & 0.329 & 0.028 & 0.026 & 0.011 & 73 & 0.331 & 0.022 & 0.021 & 0.010 \\\\ \nSlovak & 92 & 0.330 & 0.024 & 0.022 & 0.010 & 87 & 0.331 & 0.020 & 0.018 & 0.010 \\\\ \nRussian & 80 & 0.330 & 0.024 & 0.022 & 0.010 & 80 & 0.332 & 0.017 & 0.016 & 0.008 \\\\ \nItalian & 69 & 0.331 & 0.024 & 0.022 & 0.010 & 68 & 0.333 & 0.014 & 0.012 & 0.008 \\\\ \nBulgarian & 64 & 0.330 & 0.029 & 0.027 & 0.012 & 63 & 0.332 & 0.016 & 0.014 & 0.009 \\\\ \nHindi & 69 & 0.332 & 0.020 & 0.018 & 0.008 & 69 & 0.331 & 0.012 & 0.010 & 0.007 \\\\ \nJapanese & 44 & 0.330 & 0.021 & 0.020 & 0.010 & 44 & 0.333 & 0.008 & 0.006 & 0.007 \\\\ \nFinnish & 41 & 0.329 & 0.028 & 0.025 & 0.016 & 41 & 0.331 & 0.024 & 0.021 & 0.013 \\\\ \nBasque & 35 & 0.331 & 0.026 & 0.022 & 0.017 & 35 & 0.331 & 0.024 & 0.021 & 0.015 \\\\ \nRomanian & 46 & 0.332 & 0.023 & 0.021 & 0.011 & 46 & 0.333 & 0.012 & 0.010 & 0.008 \\\\ \nBengali & 18 & 0.322 & 0.034 & 0.028 & 0.026 & 17 & 0.321 & 0.034 & 0.027 & 0.030 \\\\ \nTurkish & 51 & 0.332 & 0.030 & 0.027 & 0.013 & 49 & 0.331 & 0.015 & 0.013 & 0.010 \\\\ \nEstonian & 25 & 0.331 & 0.036 & 0.032 & 0.020 & 25 & 0.332 & 0.032 & 0.028 & 0.019 \\\\ \nTamil & 40 & 0.333 & 0.023 & 0.020 & 0.011 & 40 & 0.333 & 0.018 & 0.016 & 0.010 \\\\ \nTelugu & 10 & 0.330 & 0.043 & 0.030 & 0.030 & 10 & 0.331 & 0.037 & 0.029 & 0.033 \\\\ \n\\end{tabular}\n\\end{ruledtabular}\n\\end{center}\n\\end{scriptsize}\n\\end{table*}\n\n\\section{Discussion}\n\n\\label{discussion_section}\n\nWe have shown that $E_2[C]$ predicts $C_{true}$ with small error, much better than the baseline. \nThe positive results are not surprising given the previous success of $E_2[C]$ predicting crossings on uniformly random trees, where $\\Delta_2$ is about $5\\%$, i.e. about 6 times smaller than the baseline $\\Delta_0$, for sufficiently long sentences \\cite{Ferrer2014c}. \nIt is also worth noting that $E_2[C]$ behaves well even in the treebanks with the lowest proportion of crossings, where one could argue that grammar would impose the heaviest constraints against crossings. For example, it achieves a particularly low relative error in the Romanian and Japanese Prague treebanks although they contain no or almost no crossings (Table \\ref{relative_table}).\n\n\nFrom a linguistic standpoint (recall Section \\ref{methods_section}), notice that $E_2[C]$ does not achieve its worst performance in languages known for their high word order freedom such as Ancient Greek and Latin (which are also the ones with the highest number of crossings according to Table \\ref{relative_table}) based on Stanford dependencies; however, its relative performance worsens for these languages when Prague dependencies are employed. \nIn Table \\ref{relative_table}, the average $\\Delta_2$ with Stanford dependencies indicates that $E_2[C]$ is able to make its best predictions in Estonian and Telugu, two agglutinating languages, with other agglutinating languages like Japanese or Tamil also showing better predictions than average. \nHowever, this may be an effect of the shorter sentences observed in these languages (Tables 5 and 6 of \\cite{Ferrer2017a}) and the tendency of the errors of the predictor to be smaller in sufficiently short sentences (Figure \\ref{relative2_figure}).\n\nIf we instead look at the table obtained by grouping by sentence lengths (Table \\ref{relative_grouping_by_sentence_length_table}), we observe that the predictor is remarkably robust across very dissimilar language types and families. As a representative example, if we focus on Stanford dependencies, the best prediction (average $\\Delta_2 = 0.014$) is obtained for Arabic: an Afro-Asiatic, non-agglutinating language whose treebank contains long sentences with a relatively high number of crossings; while the third best (average $\\Delta_2 = 0.021$) corresponds to Japanese: a Japonic, agglutinating language with short sentences and little observed crossings. The situation is very similar with Prague annotations, with Japanese exhibiting the best prediction, and Arabic the second best.\nThese simple and partial linguistic analyses are just reported to illustrate the potential of future linguistic research that explores in more depth the relationship between language traits and annotation criteria on the one hand, and crossings and predictions on the other.\n\nIt could be argued that the good predictions of $E_2[C]$ are not surprising at all \nbecause the syntactic dependency structures that we have analyzed could be the result of some sophisticated apparatus: a complex language faculty or external grammatical knowledge which could have produced, indirectly, a distribution of dependency lengths and vertex degrees that is favorable for $E_2[C]$. \nThen the input with which the predictor yields good predictions, e.g., dependency lengths, would be an indirect result of that complex device.\nHowever, $E_2[C]$ does not require such a device: $E_2[C]$ also makes accurate predictions on uniformly random trees with a small number of crossings \\cite{Ferrer2014c}. Therefore, the need of external grammatical knowledge to explain the origins of non-crossing dependencies is seriously challenged. \n\nThe high precision of $E_2[C]$ suggests that the actual number of crossings in sentences might be a side effect of the dependency lengths, which are in turn constrained by a general principle of dependency length minimization (see \\cite{Ferrer2013e,Liu2017} for a review of the empirical and theoretical backup of that principle).\nA ban on crossings by grammar (e.g., \\cite{hudson07,Tanaka97}),\na principle of minimization of crossings \\cite{Liu2008a} or a competence-plus \\cite{Hurford2012_Chapter3} limiting the number of crossings, may not be necessary to explain the low frequency of crossings in world languages. \n\n\nIn spite of the arguments in favor of a model predicting crossings based on dependency lengths reviewed and expanded in this article, other factors must be considered. First, chunks, i.e. subsequences of words that work as a unit, could also contribute to explain the scarcity of crossings: the number of crossings has been shown to reduce when chunks are sufficiently small in computer experiments \\cite{Lu2016a}. \nSecond, it looks difficult to rule out \nsome principle of minimization of crossings or planarity constraint. The reason is the positive correlation between crossings and dependency lengths that has been unveiled by this article and previous research combining both theory and experiment (see \\cite{Ferrer2014c}, \\cite{Ferrer2014f}, \\cite{Ferrer2015c} and references therein). The question is: what is the causal force for the scarcity of crossings: (a) a principle of minimization of crossings that explains why dependency lengths are short or (b) a principle of dependency length minimization that explains the scarcity of crossings? \\cite{Ferrer2014f}. A temporary solution to this dilemma is straightforward if we are seriously concerned about the construction of a general theory of language that is not only highly predictive but also parsimonious: a theory of language based on (b) is more parsimonious than one based on (a) \\cite{Ferrer2014f}.\n\nFrom a higher perspective, dependency length minimization follows from \nthe now-or-never bottleneck, a fundamental constraint on language processing \\cite{Christiansen2015a}, and then the scarcity of crossings could be a further prediction of such a fundamental constraint. The latter would imply that the now-or-never bottleneck and the theory of spatial\/geographical networks \\cite{Reuven2010a_Chapter8,Gastner2006b,Ferrer2004b,Chung1984} are the key for the development of a parsimonious theory of language.\n\nDespite the focus of this article on language, the article is relevant for research on other spatial networks. As researchers on dependency networks have been assuming that syntactic dependency trees tend to be planar or should be planar \\cite{sleator93,Tanaka97,KyotoCorpus,Starosta03,Lee04,hudson07,Ninio2017}, research on infrastructure networks, e.g. road networks, has been assuming that road networks are planar (see \\cite{Viana2013a} and references therein) while indeed crossings in road networks cannot be neglected \\cite{Eppstein2008a}. In the domain of infraestructure networks, we could borrow questions that have been formulated for syntactic dependency networks: is the number of crossings actually small? \\cite{Ferrer2017a}. Can the number of crossings of real infraestructure networks be explained as a result of pressure to reduce crossings directly or indirectly as a result of some principle of dependency length minimization? (this article). These are questions that may not have the same answer as in syntactic dependency trees and that could be illuminated with extensions or generalizations of the theoretical framework reviewed in this article for the two dimensional continuous case. We hope that our work stimulates further research in the field of spatial networks.\n\n\n\\begin{acknowledgments}\n\nWe thank Morten Christiansen for helpful discussions, Wolfgang Maier for comments on an earlier version of this manuscript, and Dan Zeman for help with data conversion.\n\nRFC is funded by the grants 2014SGR 890 (MACDA) from AGAUR (Generalitat de Catalunya) and also\nthe APCOM project (TIN2014-57226-P) from MINECO (Ministerio de Economia y Competitividad).\nCGR has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714150 - FASTPARSE) and from the TELEPARES-UDC project (FFI2014-51978-C2-2-R) from MINECO.\n\n\\end{acknowledgments}\n\n\n\n\n\\newcommand{\\beeksort}[1]{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Background}\n\\subsection{X-ray Binaries}\nX-ray binaries are among the brightest X-ray sources in the sky. An \nX-ray binary consists of a compact object (neutron star or black hole) \nand a normal companion star, orbiting around each other. At certain \nstage of the binary evolution, the separation of the two stars becomes \nso close that material from the upper atmosphere of the companion star \nbegins to flow toward the compact object under the influence of the \nlatter's gravity. This process is known\nas mass accretion due to ``Roche-lobe overflow''. The accreted \nmatter, carrying large amount of angular momentum from the orbital\nmotion, circulates around the compact object and forms an accretion \ndisk. As the matter spirals in toward the compact object, due to \nangular momentum losses from viscous processes, its gravitational \nenergy is converted into heat. Depending upon the mass accretion rate,\nthe temperature of the inner accretion disk can reach more than one \nmillion degrees so that X-rays are produced. Therefore, the X-ray \nobservation of such \nsources provides a valuable tool to probe regions very close \nto the compact object, where relativistic effects are strong thus \nimportant. It should be noted, however, that Roche-lobe overflow \ndoes not always occur. For some sources (especially \nthose with a massive companion star), the accretion process may simply \ninvolve the capture of stellar wind from the companion star by the \ncompact object. A small accretion disk can still form in such ``wind-fed'' \nsystems, due to any residual angular momentum of the captured matter. \n\n\\subsection{Black Hole Binaries}\nIn some cases, the companion star is visible optically. By carefully\nmonitoring its orbital motion, we can derive the so-called mass function, \nwhich is defined as\n\\begin{equation}\nf_c(M) \\equiv \\frac{M_x^3 \\sin^3 i}{(M_x + M_c)^2} = \\frac{P_{orb} K_c^3}{2\\pi\nG},\n\\label{eq:mf}\n\\end{equation}\nwhere $M_x$ and $M_c$ are the mass of the compact object and its\ncompanion star respectively, $i$ is the inclination angle of the\nbinary orbit with respect to the line-of-sight (with $i=0$ for face-on\nsystems), $P_{orb}$ is the binary orbital period, and $K_c$ is the\nsemi-amplitude of the optical radial velocity curve. Both $M_x\/M_c$ \nand $i$ can be further constrained by modeling ellipsoidal variation \nin the optical light curves~\\cite{av78}, and $M_c$ by the spectral \ntype of the companion star. For roughly a dozen X-ray binaries, \n$M_x$ is estimated to be greater than $\\sim 3 M_{\\odot}$, which is \nconsidered to be a reliable upper limit to the mass of a neutron\nstar~\\cite{rr74}, for any values of $M_c$ and $i$ within their\nrespective allowed parameter spaces. These sources are, therefore,\ngood candidates for black hole binaries (BHBs). Table~\\ref{tab:bh} \nlists all such systems known to date. It is clear from Eq.~\\ref{eq:mf} \nthat without making any assumptions the mass function sets a firm \nlower limit to the mass of the compact object. Therefore, those with \nthe measured mass function greater than $3 M_{\\odot}$ are the best \nBHB candidates (BHBCs).\n\\begin{table}[btp]\n\\caption{Stellar-Mass Black Hole Candidates$^{\\dag}$}\n\\vspace{0.4cm}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{|l|cccc|}\n\\hline\nSources & $M_x$ & $M_c$ & $i$ & $f_c(M)$ \\\\ \n & ($M_{\\odot}$) & ($M_{\\odot}$) & (degrees) & ($M_{\\odot}$) \\\\ \\hline \nCyg X-1~\\cite{her95} & 4.7--14.7 & 11.7--19.2 & 27--67& 0.25 \\\\ \nLMC X-3~\\cite{cow83} & 7--14 & 4--8 & 50--70 & 2.3 \\\\ \nLMC X-1~\\cite{hut87} & 4--8 & 17-24 & 40--63 & 0.14 \\\\ \nGS 2023+338~\\cite{sha94} & 10--15 & 0.5--1.0 & 52--60 & 6.26 \\\\ \nA 0620-00~\\cite{snc94} & 5.1--17.1 & 0.2--0.7 & 31--54 & 3.18 \\\\ \nGS 1124-68~\\cite{snc97} & 4--11 & 0.5--0.8 & 39--74 & 3.1 \\\\ \nGS 2000+25~\\cite{bee96} & 4.8--14.4 & $<$ 0.7 & 43--85 & 4.97 \\\\ \nGRO J1655-40~\\cite{ob97} & 6.8--7.2 & 2.2--2.5 & 69.5 & 3.24 \\\\ \nH 1705-250~\\cite{mar95,har97} & 4.9--7.9 & 0.07--0.42 & 48--51 &4.65 \\\\ \nGRO J0422+32~\\cite{bee97} & $>$ 9 & 0.2--0.4 & 13--31 & 1.21 \\\\ \n4U 1543-47~\\cite{oro98} & 2.7--7.5 & 2.3--2.6 & 24--36 & 0.22 \\\\ \\hline\n\\multicolumn{5}{l}{$^{\\dag}$In case of multiple measurements, the\nnumbers are taken from the} \\\\\n\\multicolumn{5}{l}{most recent papers. See references in these papers \nfor proper credits} \\\\\n\\multicolumn{5}{l}{on individual results.}\n\\end{tabular}\n\\end{center}\n\\label{tab:bh}\n\\end{table}\n\n\\subsection{Spin of Black Hole}\nBHBs are thought to be formed by the core collapse of very massive\nstars which are usually fast rotators. The conservation of stellar \nangular momentum would lead to extremely rotating black holes, if \nthe progenitor stars behave like rigid bodies. In reality, however, \nthe stellar core and envelop are likely decoupled, with the ejected \nenvelop carrying away significant amount of angular momentum. There \nmay also be other processes that can cause additional loss of angular \nmomentum during the formation. Therefore, while we expect that all\nblack holes rotate to some extent, it is not clear whether any can \nbe formed rapidly spinning.\n\nIn a binary configuration, the process of mass accretion can, in \nprinciple, result in the \naccretion of angular momentum by the black hole and, thus, the hole \ncan be subsequently spun up~\\cite{ba70,th74}. However, most \nBHBs are transient sources (see Table~\\ref{tab:bh}, except for the \ntop three sources) with a very low duty cycle of X-ray outbursts \n(when the mass accretion rate surges by orders of magnitude), \nso the lifetime-averaged accretion rate is extremely low. Consequently, \nthe accretion-induced spin-up of black holes is probably not effective \nin these sources (unlike in active galactic nuclei~\\cite{ba70}). Other \ntheoretical considerations, such as the presence of sub-Keplerian \naccretion flows in the vicinity of black holes~\\cite{ny95,cht95}, would \nonly make the mechanism less effective. \n\nRecently, evidence has been found for the presence of rapidly spinning \nblack holes in GRO J1655-40 and GRS 1915+105~\\cite{zcc97}, the only\ntwo Galactic sources that occasionally display spectacular radio jets \nwith superluminal motion, analogous to radio-loud quasars, which is\nwhy they are sometimes referred to as ``microquasars''. Also\ndiscovered are slowly or moderately spinning black holes in several\n``normal'' BHBCs. Not only has the study raised questions \nabout the formation of rapidly rotating black holes, it has also\nbrought into new focus possible\nobservational consequences of black hole rotation in BHBs. Subsequently, \na lot of excitement has been generated about the prospect of using\nBHBs to probe relativistic effects in strong gravity \nregime~\\cite{no97,zcc97,czc98}.\n\n\\section{GRO J1655-40: A Test Case}\n\\subsection{Existence of A Spinning Black Hole?}\n\\label{ssec:spin}\nThe binary parameters are known to such a high accuracy for GRO\nJ1655-40 (with uncertainties typically only a few percent; see \nTable~\\ref{tab:bh}) that makes the interpretation of the X-ray data\nfor this source much more reliable than for any other sources. Despite\nbeing a microquasar, GRO J1655-40 appears as a normal BHBC in terms of\nthe observed spectral properties. The X-ray spectrum can be well\ndescribed by an ultra-soft component at low energies and a power law \nat high energies~\\cite{zh97}, which is canonical for\nBHBCs~\\cite{tl95}. The ultra-soft component is generally attributed to \nthe emission from the hot inner region of an accretion disk, while \nthe power-law tail is thought to be the product of soft photons being \nCompton upscattered by energetic electrons in the region.\n\nIn practice, the ultra-soft component is often modeled by a\nmulti-color blackbody~\\cite{mi84}. In this model, the accretion disk \nis assumed to be geometrically thin, optically thick (i.e., the \nstandard $\\alpha$ disk~\\cite{ss73,nt73}). The emission from such \na disk is a blackbody, but the temperature of the blackbody is a\nfunction of the distance to the black hole. There are only two free \nparameters in the model: the radial distance of the inner disk edge \nto the black hole and the effective temperature at the inner edge, \nonce the distance of the source and the inclination angle of the disk \n(which is usually assumed to be equal to the inclination of the binary \norbit) are known. The model has worked remarkably well for a number of \nBHBCs (see a review by Tanaka and Lewin~\\cite{tl95}). One of the most \ninteresting results is that the inferred radius of the inner disk edge \nremains fairly stable and constant as the observed fluxes vary \ngreatly~\\cite{tl95}, strongly implying that the last (marginally) \nstable orbit is reached.\n\nSince the model assumes Newtonian gravity, appropriate corrections\nneed to be made to include relativistic\neffects~\\cite{pt74,ha89}. Also, gravitational shift and \ngravitational focusing cause both the observed color temperature \nand integrated flux to deviate from the local values, depending on \nthe inclination angle of the disk and the spin of the black \nhole~\\cite{cu75}. Furthermore, in the hot inner region of an \naccretion disk, electron scattering is likely to dominate over \nfree-free absorption, so the inner disk actually radiates \napproximately as a ``diluted'' blackbody with the peak effective \ntemperature lower than that derived from fitting the\nspectrum~\\cite{ehs84}. Unfortunately, this ``color correction factor'' \nis still poorly determined, although it appears to depend only weakly \non the properties of the black hole, mass accretion rate, or the \nlocation on the disk~\\cite{st95}. \n\nTaking into account these factors, we found that for several BHBCs the\ninferred radius of the inner disk edge is consistent with that of the \nlast stable orbit around non-rotating black holes~\\cite{zcc97}. When \nwe applied the same model to the spectrum of GRO J1655-40 during an \noutburst, however, we found that the inner edge of the disk is only\nabout 1.2 Schwarzschild radii away from the black hole~\\cite{zcc97}, \nwhich is, of course, impossible if the black hole does not rotate. \nTherefore, we were forced to conclude that GRO J1655-40 contains a \n{\\it rapidly} spinning black hole. The spin of the black hole was \ndetermined to be $\\sim$93\\% maximal rotation for this source.\n\nThe ``ordinary'' nature of GRO J1655-40 as a BHBC (in terms of its \nspectral properties) is strongly supported by \nrecent observations with the {\\it Rossi X-ray Timing Explorer}\n(RXTE). The source was monitored by RXTE extensively throughout its \nrecent (1996--1997) X-ray outburst. From these observations, Sobczak \net al.~\\cite{sob98} found that the observed X-ray spectrum can be \nwell characterized by the canonical spectral shape of BHBCs (soft \nmulti-color blackbody plus hard power law) throughout the entire\nperiod, which confirms the previous findings~\\cite{zh97} (although \nthe actual values of the parameters differ slightly). In addition, \nthey found that the inferred radius of the inner disk edge remained \nroughly constant (with occasional low points) as the X-ray flux varied \nby a factor of 3--4, similar to normal BHBCs~\\cite{tl95}. \n\nHowever, these authors mistakenly concluded that the corrections for\nrelativistic effects are negligible for this source (actually they\nmis-quoted Zhang et al.~\\cite{zcc97} on this issue). As a matter of\nfact, the general relativistic effects are very significant in this \ncase~\\cite{zcc97}. In particular, the inner boundary condition (i.e.,\ntorque free at the last stable orbit) can drastically change the \ntemperature profile of the inner portion of the accretion disk under \nNewtonian gravity~\\cite{pt74,ha89}. For GRO J1655-40, the\ncorrection amounts to a factor of $\\sim$2, after taking into account \nall the effects. Consequently, the radius of the inner disk edge in\nSobczak et al. was over-estimated by the same amount~\\cite{sob98}. \n\nOur interpretation of the same data would, therefore, put the last \nstable orbit roughly at 1.5 Schwarzschild radii, which confirms the \nneed for the presence of a rapidly rotating black hole in this system \n(although the spin of the black hole is $\\sim$78\\% maximal rotation \nin this case). We take note of occasional low values of the inferred \nradius when the measured X-ray luminosity is relatively\nhigh~\\cite{sob98}. We interpret them as the times when the adopted \nspectral model breaks down. It is likely that when accretion rate is \nhigh the inner portion of a standard $\\alpha$ disk may become unstable \nto radiation pressure and fragmentation~\\cite{le74,kro98}. During the \noutburst, GRO J1655-40 might be situated close to the critical point \nfor such instability, as in the case of Cyg X-1 (A. Zdziarski, private\ncommunication). \n\nIn contrast, Sobczak et al. proposed that the last stable orbit was\nreached by the accretion disk {\\it only} at the {\\it smallest} radius \n(i.e., the lowest point) obtained from their spectral analysis. To \nexplain why the inner edge of the disk stays far away from the last \nstable orbit during nearly the entire outburst, they invoked strong\nradiation drag. Intuitively, however, the radiation drag should be\nstronger in the higher luminosity state (when the ``low points'' \noccur), which would be opposite to their line of arguments. Moreover, \nit seems unlikely that the radiation drag is capable of significantly \naffecting gas dynamics in an {\\it optically thick} disk (which is \nthe most fundamental assumption in the model), since any external\nradiation fields (if present) can hardly penetrate the surface of the \ndisk and the internal emission percolates through the surface\npreferentially in the direction perpendicular to the thin\ndisk. Finally, from a\npractical point of view, it seems risky to base the entire argument on \na single data point which is barely allowed physically~\\cite{sob98}, \nespecially considering the fact that not all the ``low points'' reach \nthe same level.\n\nIn summary, while further studies are required to reveal important \ndetails about accretion processes in BHBCs, it seems highly plausible, \nas suggested by \nobservations~\\cite{tl95}, that during an X-ray outburst the accretion \ndisk is of the standard form~\\cite{ss73,nt73} in transient BHBCs \nand the inner disk edge reaches and remains at the last stable orbit. \nGRO J1655-40 is no exception. Quantitative modeling of its X-ray \nspectrum has led to the conclusion that a rapidly rotating black hole \nis likely to be present in this source. \n\n\\subsection{Gravitomagnetic Precession}\nA natural consequence of curved spacetime in the vicinity of a spinning \nblack hole is the gravitomagnetic precession of orbits that are tilted \nwith respect to the equatorial plane perpendicular to the spin\naxis~\\cite{lt18}. For circular orbits, which are probably most\nrelevant to accretion processes in X-ray binaries, the precession\nfrequency can be expressed as \n$\\nu_{FD} = \\nu_{\\theta} \\Delta \\Omega\/2\\pi$, \nwhere $\\nu_{\\theta}$ is the orbital frequency of the polar (or $\\theta$)\nmotion, and $\\Delta \\Omega$ is the angle by which the line of nodes of a\ncircular orbit are dragged during each orbital period. Note that this\ndefinition is slightly different form the one adopted by Cui et \nal.~\\cite{czc98}. As correctly pointed out by Merloni et al.~\\cite{mer98}, \nwith the new definition, $\\nu_{FD}$ approaches the rotation frequency \nof the black hole at the event horizon, which is physically expected. \nFig.~\\ref{fig:fd} shows $\\nu_{FD}$ as a function of black hole spin. \nComparing it with Fig. 1 of Cui et al.~\\cite{czc98}, we find that, as \nexpected, the difference becomes significant only in the case of \nextremely spinning black holes when the last stable orbit approaches \nthe event horizon. Therefore, the results of Cui et al.~\\cite{czc98}\nhardly change, even for microquasars, neither do any of the\nconclusions that they reached (see also discussion in the following \nsection). \n\n\\subsection{``Stable'' QPOs}\nQuasi-periodic oscillations (QPOs) have been observed in the X-ray light\ncurves of BHBCs over a wide frequency range, from mHz to roughly a few \nhundred Hz (reviews by van der Klis~\\cite{van95} and Cui~\\cite{cui98}). \nWhile it is clear that no single process can account for all QPOs, the \norigins of QPOs are not known. \n\\begin{figure}[t]\n\\centerline{\\epsfig{figure=freq.ps,width=3.0in}}\n\\caption{Gravitomagnetic precession frequency (multiplied by black\nhole mass), as a function of the distance from the last stable orbit, \nfor different black hole spin. The weak-field limits are also shown \nin dash-dotted line for comparison. }\n\\label{fig:fd}\n\\end{figure}\n\nOne type of QPOs in BHBCs is of particular interest here. Such QPO was \nfirst reliably established in GRS 1915+105 with the detection of a \n67 Hz QPO~\\cite{mrg97}. One unique characteristic of this QPO \nis the stability of its frequency against any variation in the X-ray flux. \nThe QPO appears to represent a transient event, and seems to be\npresent only in certain spectral states. Shortly after this discovery, \na similar \nsignal was detected at $\\sim$300 Hz for GRO J1655-40~\\cite{rem98},\n{\\it only} when the X-ray luminosity was relatively high~\\cite{sob98}\n(i.e., at the ``low points'', as discussed in \\S~\\ref{ssec:spin}). \nWe subsequently proposed that these ``stable'' QPOs are perhaps the\nobservational manifestation of gravitomagnetic precession of accreted,\norbiting matter around spinning black holes~\\cite{czc98}.\n\nTo facilitate comparison with observations, we have computed the\nexpected gravitomagnetic precession frequency of a disk annulus at\nwhich the effective disk temperature peaks. The results are plotted in\nFig.~\\ref{fig:fdspin}, as a function of black hole spin for three\nvalues of black hole mass.\nComparing this figure to Fig. 2 of Cui et al.~\\cite{czc98}, we find,\nagain as expected, significant difference exists only for large (and\npositive) black hole spin. Note that the results depend little on the\nvalue of Q~\\cite{czc98,mer98}, which is a constant of motion \nthat specifies the tilt of precessing orbits~\\cite{wil72}.\n\nFor GRO J1655-40 (where $M_{bh} = 7 M_{\\odot}$), the spin of the black \nhole would be about 97\\% maximal rotation were the observed 300\nHz QPO to be attributed to the gravitomagnetic precession (as shown \nin Fig.~\\ref{fig:fdspin}). This result assumes that the QPO originates \nin the modulation of the X-ray emission from the disk, which does not\nnecessarily have to be the case, as will be discussed in\n\\S~\\ref{ssec:xmm}. To\nderive a lower limit to the black hole spin, we computed the frequency\nof gravitomagnetic precession right at the last stable orbit, as a\nfunction of the spin, and set the result equal to the QPO\nfrequency. We found that in this case the spin would be $\\sim$87\\% \nmaximal rotation. The results are consistent with those obtained \nspectroscopically (see also Fig.~\\ref{fig:ms}). \n\n\\section{Application to Other Sources}\n\\label{sec:aos}\nAttempts have been made to apply the model to QPOs observed in ``normal''\nBHBCs~\\cite{czc98}. The closest examples are perhaps \nthe so-called ``very-high-state'' QPOs that were observed only twice, \nin GX 339-4~\\cite{miy91} ($\\sim$6 Hz) and GS 1124-68~\\cite{bel97} \n(5--8 Hz), in a high luminosity state (i.e., the very-high state).\nThe frequency of such QPOs is fairly stable against flux variation, \nand any change in the QPO frequency can be explain by the variation \nin radiation pressure~\\cite{czc98}. The low values of the QPO\nfrequency would imply the presence of only slowly rotating black holes \nin these sources were these QPOs to be produced by the gravitomagnetic \nprecession of accreted matter. This would in turn support the\nspeculation that the relativistic jets observed in microquasars might\nsomehow related to the spin of black holes~\\cite{zcc97}.\n\\begin{figure}[t]\n\\centerline{\\epsfig{figure=fvsa.ps,width=3.0in}}\n\\caption{Gravitomagnetic precession frequency of a disk annulus, at\nwhich the effective disk temperature peaks, as a function of black \nhole spin (assuming $Q=1$). }\n\\label{fig:fdspin}\n\\end{figure}\n\nSimilar QPOs have also been observed in GS 1124-68~\\cite{bel97} and Cyg\nX-1~\\cite{cui97} during the transition between the low and high state.\nHowever, in these cases, the situation is much more complicated and,\nthus, the interpretation is much more uncertain, because of the\nvariable nature of the sources and the transient nature of the QPO\nphenomenon in general~\\cite{cui98}. For a particular source, a \ndifferent set of QPOs are often present at different times (or\nfluxes). It is, therefore, difficult to associate one QPO (if any)\nwith the gravitomagnetic precession of accreted matter. For instance, \na lower-frequency QPO (6--8 Hz) was often present in GRO J1655-40 \nduring the same outburst when the 300 Hz QPO was\ndiscovered~\\cite{men98}. To make further progress, an systematic \neffort is imperative to reliably characterize and classify QPOs \nin BHBCs. \n\n\\section{Unsettled Issues}\n\\label{sec:ui}\n\\subsection{Origin of Tilted Orbits} \nAccreted matter must be in a non-equatorial orbit, with respect to \nthe spin axis of the black hole, in order for the orbit to undergo \ngravitomagnetic precession. In principle, such orbits could be \nproduced during the formation of a black hole binary, due to the \nmisalignment of the spin axis of the black hole with the axis of the \nbinary motion~\\cite{hil83}. \n\nIf mass accretion proceeds through a flat viscous disk, the disk can\nstill be warped, due to irradiation by a central X-ray \nsource~\\cite{pri96}. However, in this case, differential \ngravitomagnetic precession generates twists in the disk and the inner \nregion of the disk tends to gradually realign with the equatorial \nplane due to the viscous torque~\\cite{bp75}. Once the realignment\ntakes place, gravitomagnetic precession modes are damped out and,\nthus, the QPO should simply disappear. On the other hand, this process \ncan help explain why the QPOs seem to be present {\\it only} during \ntransitional or unstable periods, when the inner region of the disk \nexperiences significant changes.\n\nAs mentioned in \\S~\\ref{ssec:spin}, the inner portion of the disk may\nbecome unstable~\\cite{le74,kro98}, when the accretion rate is\nrelatively high. Once this occurs, the disk might become fragmented. \nThe gravitomagnetic precession of the fragments is probably only \nweakly damped, so it might be capable of producing the observed QPOs.\n\n\\subsection{Excitation of Gravitomagnetic Modes}\nA recent theoretical development involves the discovery of a class of\nhigh-frequency gravitomagnetic modes that are only weakly damped, in \naddition to the strongly-damped low-frequency ones~\\cite{ml98}. These\nhigh-frequency modes are very localized spiral corrugations of the\ninner portion of the disk. They are perhaps most relevant to the\nobserved QPOs, because X-ray emission from BHBCs is\nlikely to originate very close to the central black hole. However, it\nremains to be seen whether the modes can be excited easily and whether\nthey are capable of modulating X-ray emission~\\cite{ml98}.\n\nIt should be pointed out that two critical assumptions are made in the\nstudy: small tilt angle and (pseudo) Newtonian potential. The former \nsimplifies the equation by ignoring the non-linear aspect of the\nproblem, which might affect the damping time scales of gravitomagnetic \nmodes in a significant way; non-linear calculations are required to \nshed light on this issue. The latter might still represent a good \napproximation for systems that contain a relatively slowly rotating \nneutron star, but is certainly invalid for rapidly rotating black\nholes in which the stable QPOs are observed. These assumptions,\ntherefore, tend to limit the scope of the applicability of the\nresults. At present, it would seem premature to draw any definitive \nconclusions about confirming or rejecting the gravitomagnetic origin \nof the stable QPOs in microquasars.\n\n\\subsection{X-ray Modulation Mechanisms}\n\\label{ssec:xmm} \nThe gravitomagnetic precession of accreted matter only provides a \nnatural frequency for the QPOs. To actually see them, certain physical \nprocesses are required to produce X-ray modulations. The processes \nare entirely unknown at present. Possibilities include (by no means\nexclusively): (1) variation in the X-ray emitting area of the \ndisk, due to precession; (2) Doppler or gravitational shift of photon \nenergy, due to the Keplerian motion of self-illuminating clumps or \n``hot spots'' in the accretion disk; (3) oscillation that modulates \nthe emission from the disk; and (4) occultation of a highly compact \nX-ray emitting region by disk fragments or the inner part of the \ndisk itself.\n\nThe transient nature of the QPOs seems to imply that the\nfirst possibility, if viable at all, must be only a small effect. Any\nprocesses that involve discrete clumps would unavoidably also produce \nX-ray modulation at the orbital (both polar and azimuthal)\nfrequencies~\\cite{ms98,mer98}. These QPOs have not been seen \nin BHBCs, despite the high quality of RXTE data (the QPOs of comparable \nstrength have been detected at around kHz, using the RXTE data, in \nmore than a dozen neutron star systems~\\cite{van97}). Therefore, such \nprocesses also seem unlikely to be\nresponsible for the observed QPOs. Hot spots or oscillations that\nmodulate the emission from the accretion disk can also modulate\nComptonized hard X-ray emission, if the seed photons originate in \nthe disk emission. However, it seems difficult for these processes \nto amplify the fractional amplitude of the QPOs toward high photon \nenergies, which is observed~\\cite{mrg97,rem98}. Such energy\ndependence of the QPO amplitude might, in our view, hold the key to\nour understanding of the origin of the QPOs.\n\nThe only remaining possibility in the above list is the occultation of\na highly compact hard X-ray emitting region by detached disk annuli \nor the inner disk itself. Interestingly, both\nobservations~\\cite{gro98} and theories~\\cite{na97,lt98} support the \npresence of such X-ray emitting \nregions in transient BHBCs during an X-ray outburst. As an\nexample, Laurent \\& Titarchuk~\\cite{lt98} recently showed that the \nobserved power-law type hard X-ray spectrum of BHBCs may be the result \nof soft photons from the accretion disk being inverse-Compton\nscattered by the bulk motion of relativistic electrons in the vicinity\nof central black holes. The Comptonizing electrons are spatially confined \nalmost entirely within the inner edge of the disk. In the context of \nthis model, harder X-rays should, on average, originate closer to the \nblack hole. Therefore, a precessing ring of matter occults a larger \nfraction of the harder X-ray emitting region, which might explain the \nobserved energy dependence of the QPO amplitude, although a definitive \nanswer still awaits detailed calculations based on this model.\n\n\\subsection{Alternative Models} \nThe stable QPOs in GRS 1915+105 and GRO J1655-40 can conceivably be \nassociated with the Keplarian motion of self-illuminating clumps \nright at the last stable orbit around the black \nhole~\\cite{mrg97,rem98}. In the case of GRO J1655-40 whose\nblack hole mass is accurately known, the model would be able to\nexplain the observed QPO frequency (300 Hz) only if the black hole \ndoes not rotate~\\cite{rem98}. This is clearly incompatible with the\nrequirement for the presence of a rapidly rotating black hole in this\nsource from modeling the observed X-ray spectrum. A more serious (and\nmodel independent) problem with this interpretation is due to the \nfact that general relativistic effects cause the temperature of the \naccretion disk to vanish at the last stable\norbit~\\cite{pt74,ha89}. Consequently, no X-ray emission is expected \nfrom the inner edge of the disk.\n\nAnother proposal has been made to associate the stable QPOs to \nepicyclic oscillation modes in the accretion disk~\\cite{now97}. Not \nonly can such modes be supported by the disk, they also become trapped \nin the innermost portion of the disk, purely due to relativistic\neffects~\\cite{kf80,nw92,nw93,ct95,per97}. It is also \nrealized that g-mode oscillations affect the largest area of a region \nin the disk where most X-rays are emitted and are, therefore, perhaps \nmost relevant to observations~\\cite{now97}. This model can explain \nthe measured frequencies of the stable QPOs in microquasars. However,\nlike other disk-based processes, it may have difficulty in explaining\nthe observed energy dependence of the QPO amplitude. Furthermore, it \ncannot account for the very-high-state QPOs or other relatively \nlow-frequency QPOs in BHBCs (see \\S~\\ref{sec:aos}), although arguments \ncan be made that these are fundamentally different from the stable \nQPOs in microquasars.\n\nTo quantify the comparison among the three possible models\n(gravitomagnetic precession, Keplerian motion, and g-mode\noscillation), we have applied the models to both GRO J1655-40 and GRS\n1915+105. For each model, a relationship between the mass and spin of\nthe black hole is derived from the measured frequency of the QPO. The\nresults are shown in Fig.~\\ref{fig:ms}, for both sources. In the\n\\begin{figure}[th]\n\\centerline{\\epsfig{figure=summary.1655.ps,width=2.8in}}\n\\centerline{\\epsfig{figure=summary.1915.ps,width=2.8in}}\n\\caption{(top) Allowed parameter space for the mass and spin of the \nblack hole in GRO J1655-40. The solid line shows the spectroscopic\nresults of Zhang et al. (see \\S~2.1), with the \nlightly-shaded region indicating the estimated uncertainty. The \ndot-dashed, short-dashed, and long-dashed lines show, respectively, the \npredictions of three proposed models: Keplerian motion, g-mode\noscillation, and gravitomagnetic precession. For the last model,\ntwo limiting cases are shown for frequencies of gravitomagnetic \nprecession at the inner edge of the disk (upper curve) and at where \nthe effective disk temperature peaks (lower curve). The heavily-shaded \narea represents the confidence region of the measured black hole \nmass. (bottom) Similar to the top panel, but for GRS 1915+105. In \nthis case, the mass of the black hole is not known. }\n\\label{fig:ms}\n\\end{figure}\ngravitomagnetic precession model, two limiting cases are shown for \nfrequencies at the inner edge of the disk and at where the effective\ndisk temperature peaks, since we do not know the actual X-ray\nmodulation mechanism (see \\S~\\ref{ssec:xmm}). Also shown in\nthe figure is the same relationship derived from modeling the observed\nX-ray spectrum for each source~\\cite{zcc97}. The intersection between \nthe latter\nand each of the three model curves, therefore, yields a solution to\nthe mass and spin of the black hole, for a particular source, as \npredicted by that model. For GRO J1655-40, the mass of the black hole\nis known quite accurately, so it can be used to determine which models\nare compatible with the observed properties.\n\nThe uncertainty of the spectroscopic results\nis likely dominated by that of the color correction factor (up to\n$\\sim$12\\%~\\cite{st95}), which can contribute as much a uncertainty\nas $\\sim$24\\% in the derived radius of the last stable\norbit~\\cite{zcc97}. The next major source of uncertainty comes from \ndetermining the source distance, to which the radius of the last stable\norbit is directly proportional. It amounts to $\\sim$6\\% for \nGRO J1655-40~\\cite{hr95} and $\\sim$12\\% for GRS 1915+105~\\cite{mr94}. \nLikely minor contributions include uncertainties in the binary \ninclination angle, in the derived X-ray flux and the peak effective \ntemperature of the disk, and in various relativistic correction \nfactors~\\cite{zcc97}. To be relatively conservative, we have adopted \nan overall uncertainty of 40\\% on the radius of the last stable orbit\nfor both sources. The lightly-shaded area in Fig.~\\ref{fig:ms}\nreflects this measurement uncertainty.\n\nClearly, the Keplerian motion model is ruled out, since it is far \nfrom being consistent with the spectroscopic results. While the \ng-mode oscillation model seems to work for GRO J1655-40, it fails to \nyield any physical solutions for GRS 1915+105. The difference is so\nlarge that any reconciliation between the model and the data seems\nunlikely, unless the spectroscopic method has severely underestimated \nthe radius of the last stable orbit (or the method does not apply in \nthis case). On the other hand, the gravitomagnetic precession model\nseems to explain the observations well for both sources. For GRS\n1915+105, the model constrains the mass ($\\sim$3--14 $M_{\\odot}$) and \nspin ($\\sim$37--90\\% maximal rotation) of the black hole. Future\ninfrared observations of the source might be able to test the mass\nconstraint. \n\n\\section*{Acknowledgments} We would like to thank Andrea Merloni for\nsharing with us the results from his work prior to publication. W. Cui \nis also grateful to Ron Remillard for keeping him updated of the\nresults on GRO J1655-40 and for many helpful discussions, despite \nsome difference in their interpretation of the data. This work is \nsupported in part by NASA through Contract NAS5-30612.\n\n\\section*{References}\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nCommunity structure \\cite{Mucha2010} is widely observed in various types of real networks such as social networks \\cite{Zachary1977}-\\cite{Chouchani2020}, biological networks \\cite{Jonsson2006}-\\cite{Cui2020}, technological networks \\cite{Newman2004a}, and so on. A community is a group of nodes with more edges connecting nodes within the group and less edges connecting nodes to different groups \\cite{Girvan2002}. Community detection is of significant importance in understanding organization principles and predicting dynamical behaviors of a network \\cite{Fortunato2010}, which has already found wide applications in interdisciplinary domains, such as to find the pathways between diseases and drugs \\cite{Pham2019}, to reveal the role of each part in a layered neural network \\cite{Watanabe2019}, to mine user opinions from social networks \\cite{Li2019}, and so on. \n\nSince the past decades, many methods of community detection have been proposed, such as graph partitioning \\cite{Shi2011}, spectral clustering \\cite{Donetti2004}-\\cite{Jin2015}, similarity expansion \\cite{Xiang2009}-\\cite{Pan2010}, statistical inference \\cite{Newman2007}, dynamic methods \\cite{Zhou2003}-\\cite{Hu2008}, and integer programming model \\cite{Srinivas2019}. Among them, greedy algorithms based on quality function optimization have attracted a lot of attention \\cite{Newman2004b}-\\cite{Shao2020}. \nIn a typical greedy algorithm, some nodes are selected as central nodes of communities (others are non-central nodes), from which the nearest neighbor search is usually adopted to expand provisional communities \\cite{Saha2016}. Obviously, the selection of central nodes are vital for the accuracy of a greedy algorithm. In an early work \\cite{Chen2013}, nodes either with global maximal degrees (GMD) or with local maximal degrees (LMD) are regarded as central nodes. The GMD takes nodes of the top-$k$ highest degrees as central nodes. However, the selection of the value of $k$ is often arbitrary. On the other hand, nodes with maximal degrees are not always central nodes \\cite{Blondel2008b,Lu2016}. In contrast, the LMD designates nodes with higher degree than all neighbors as central nodes. Empirical evidences \\cite{Chen2013} suggest that the LMD performs overall better than the GMD in identifying central nodes of provisional communities. \n\nIn this paper, we propose a novel local centrality index called the local superiority index (LSI) and combine it to the greedy algorithm to detect communities. In our proposed algorithm, the community expansion, aiming to maximize the value of the quality function, starts from central nodes selected by the LSI. After expansion, we allot residual nodes to suitable communities. Allocation of residual nodes may decrease the quality function, while it is found that the optimal community structure is not always with the largest quality function \\cite{Zhang2014}. Our algorithm is evaluated by normalized mutual information (NMI) \\cite{Kuncheva2004} on real networks with known communities and benchmark networks generated artificially \\cite{Lancichinetti2008}. \n\n\n\\section{Methods}\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.9\\textwidth]{fig1_new2_rev.eps\n \\caption{Comparison among the three considered indices. The illustrated network is constructed by connecting two complete networks through a single edge. (a) According to the GMD, the red node (node $1$) is the top-$1$ highest-degree node, the pink nodes (nodes $2,3,4,5$) and node $1$ are the top-$5$ highest-degree nodes. (b) According to the LMD, only node $1$ is a central node, other nodes are all non-central nodes. (c) According to the LSI, nodes $1$ and $2$ are central nodes.}\n\\label{fig_Illustration}\n\\end{figure}\n\nAs shown in figure 1, we construct an illustration network by connecting two complete subnetworks through a single edge. Intuitively, this network has two communities (\\{1,3,4,5\\} and \\{2,6,7\\}) and nodes 1 and 2 should be treated as central nodes. However, neither GMD nor LMD can exactly identify those two central nodes. The top-$k$ highest-degree nodes may indeed belong to $\\textless k$ communities and thus are not all central nodes, and the LMD can't identify central nodes with relatively smaller degrees.\n\nInspired by these shortcomings, we propose the so-called LSI index. The LSI of an arbitrary node $i$ reads\n\n\\begin{equation}\nLSI_i = ({{k_i} - \\frac{1}{{{k_i}}}\\sum\\limits_{j \\in {\\Gamma _i}} {{k_j}} })\/ ({{k_i} + \\frac{1}{{{k_i}}}\\sum\\limits_{j \\in {\\Gamma _i}} {{k_j}} }),\n\\end{equation}\nwhere $k_i$ and $k_j$ represent the degree of node $i$ and node $j$, and $\\Gamma_i$ denotes the set of $i$'s neighbors. Obviously, the value of LSI lies between -1 and 1, and the larger LSI corresponds to the higher local centrality. Without loss of generality, in this paper we treat nodes with $LSI_i \\geq 0$ as central nodes for community detection, while other nodes are non-central nodes. As shown in Fig. 1(c), LSI identifies nodes $1$ and $2$ as central nodes.\n\nWe adopt the fitness function $F$ \\cite{Lancichinetti2009,Zhou2007} as the quality function. For a community $C_i$, its $F$ value is\n\n\\begin{equation}\nF(C_i) = \\frac{d_{in}(C_i)}{d_{in}(C_i) + d_{out}(C_i)}, \n\\end{equation}\nwhere $d_{in}(C_i)$ is the internal degree (i.e., the twice number of edges in $C_i$), and $d_{out}(C_i)$ is the external degree (i.e., the number of edges connecting nodes in $C_i$ with nodes outside $C_i$). The fitness function $F$ for the whole network is defined as the sum of $F(C_i)$ of all communities, as\n\\begin{equation}\nF = \\sum_{i=1}^m{F(C_i)},\n\\end{equation}\nwhere $m$ is the total number of communities.\n\n\nDetailed procedures of the greedy algorithm are described below. Firstly, we identify all central nodes with LSI values no less than zero. Then we start the community expansion procedure. Initially, we choose one central node randomly as the seed of a provisional community $C_1$. At each step, we search all unassigned nodes that are neighboring to at least one node in $C_1$. Our goal is to find one node whose addition can maximize $F(C_1)$ and add this node to $C_1$. In case more than one node can maximize $F(C_1)$, we randomly choose one of them. We repeat this step until any addition of one neighbor can't increase $F(C_1)$. If $F(C_1)\\leq 0.5$, the provisional community $C_1$ is disbanded and the current expansion becomes invalid, because the condition of weak community (i.e., $F(C_1)>0.5$) is not satisfied. At the same time, the central node is converted to an unassigned non-central node. We repeat this process until all provisional communities stop expanding. After the expansion, some nodes may still be unassigned and form one or more connected subgraphs. For each subgraph, we define its potential neighbors as all assigned nodes neighboring to at least one node of this subgraph. Then each subgraph is allotted to a provisional community who contains the potential neighbor with the highest LSI value. In case more than one provisional community satisfies such condition, we randomly choose one of them. Although the allocation of unassigned nodes may decrease the quality function, it is reasonable that the optimal community structure is not always with the largest quality function \\cite{Zhang2014}. In the above procedure, the different order in choosing the central nodes may lead to different results, so we carry out the algorithm many times and adopt the result with the highest $F$. \n\n\n\n\\begin{table} \n\\small\n\\centering \n\\caption{Structural features of the six real networks under consideration. $N$, $L$ and $m$ represent the number of nodes, edges and communities respectively, and $\\langle k\\rangle$ represents the average degree. }\n\\begin{indented}\n\\item[]\\begin{tabular} \n{>{\\columncolor{white}}rccccc} \n\\hline\n\\rowcolor[gray]{0.9} Networks &$N$ &$L$ &$m$ &$\\langle k \\rangle$ &references \\\\ \n\\hline\nkarate &34 &78 &2 &4.5882 &\\cite{Zachary1977} \\\\ \ndolphins &62 &159 &4 &5.129 &\\cite{Lusseau2004} \\\\ \nfootball &115 &613 &12&10.6609 &\\cite{Girvan2002} \\\\ \npolblogs &1222 &16714 &2&27.3552 &\\cite{Lada2005} \\\\ \npolbooks &105 &441 &3&8.4 &\\cite{Krebs0000},\\cite{Newman0000} \\\\ \nhighschool &248 &1004 &8 &8.0968 &\\cite{Moody2001} \\\\ \n\\hline \n\\end{tabular} \n\\end{indented} \n\\end{table} \n\nIn this paper, we use two kinds of networks. One is the real networks with known ground truth (i.e., recognized community structure), the other is the LFR benchmark networks generated artificially \\cite{Lancichinetti2008}. We consider six real networks, with detailed description as follows. (i) Karate network \\cite{Zachary1977}. A network of friendship in a karate club in an American university. The club split into two parts after a dispute between the coach and the treasurer. (ii) Dolphins network \\cite{Lusseau2003}. This network contains 62 dolphins living in Doubtful Sound, New Zealand. An edge exists between two dolphins if they are observed together more often than expected by chance from 1994 to 2001. This network contains two communities \\cite{Lusseau2004}. (iii) Football network \\cite{Girvan2002}. A network of American football games between Division IA colleges during regular season Fall 2000. The nodes denote the 115 teams and the edges represent 613 games played in the course of the year. Each conference form a community and games are more frequent between members of the same conferences. (iv) Polblogs network \\cite{Lada2005}. A directed network of hyperlinks between weblogs on US politics around the time of the 2004 presidential election. We convert this network to an undirected network by treating all directed links as undirected edges and reserve the largest component. Each weblog has a political leaning: blue (left or liberal) or red (right or conservative), and thus this network contains two communities. (v) Polbooks network \\cite{Krebs0000,Newman0000}. In this network, nodes represent books about US politics sold in Amazon.com, and edges connecting frequently co-purchasing book pairs \\cite{Lu2012}. Each book is assigned a political tag (liberal, neutral or conservative), and thus the network is consisted of three communities. (vi) Highschool network \\cite{Moody2001}. A directed network represents the friendship choices made by students from different grades. We also convert this network to an undirected network. Fundamental structural statistics of these six networks are shown in Table 1.\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig2_NMI_groundtruth_rev7.eps} \n \\caption{ The NMI values of the three indices on the six real networks with known ground truth. }\n\\label{fig_Illustration}\n\\end{figure}\n\n\n\nLFR networks \\cite{Lancichinetti2008} are classical benchmark networks generated by computer. In an LFR network, degrees are distributed in a power law with exponent $2<\\gamma<3$, and the sizes of communities also follow a power-law distribution with exponent $1<\\beta< 2$. Besides, the community size $s$ and node degree $k$ satisfy the constraints $s_{min}>k_{min}$ and $s_{max}>k_{max}$. An important mixing parameter $\\mu$ represents the ratio between the external degree of an arbitrary node with respect to its community and the total degree of this node. With the increase of $\\mu $, the community structure of the network becomes ambiguous. \n\n\n\n\\section{Results}\n\n\\begin{comment}\nIn this section, we evaluate the performance of LSI and the greedy algorithm with the LSI on the ground truth networks and benchmark networks by the NMI \\cite{Kuncheva2004}. Given a confusion matrix $\\textbf{N}$, whose element $N_{ij}$ on the $i$th row and the $j$th column represents the number of nodes contained in both the $i$th standard community and the $j$th community detected by the algorithm, the NMI is defined as\n\n\\begin{equation}\nNMI = \\frac{{ - 2\\sum\\limits_{i = 1}^{{C_r}} {\\sum\\limits_{j = 1}^{{C_f}} {{N_{ij}}N\/{N_{i.}}{N_{.j}}} } }}{{\\sum\\limits_{i = 1}^{{C_r}} {{N_{i.}}\\log ({N_{i.}}\/N) + \\sum\\limits_{j = 1}^{{C_f}} {{N_{.j}}\\log ({N_{.j}}\/N)} } }},\n\\end{equation}\nwhere $N$ is the number of nodes in a network. $C_r$ is the number of the standard communities. $C_f$ denotes the number of communities detected by the algorithm to be evaluated. $N_{i.}$ and $N_{.j}$ stand for the sum of all elements in the $i$th row and the $j$th column of $\\textbf{N}$ respectively. The value of the NMI is between 0 and 1. The higher its value, the more accurate identified communities. \n\\end{comment}\n\n\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig3_NMI_LFR_rev3.eps} \n \\caption{How NMI changes with varying parameters in LFR networks. The default values of the four parameters are $\\mu=0.4$, $\\gamma=2.5$, $\\beta=1.5$ and $\\langle k\\rangle=20$. In each plot, only one parameter is varying. The results are averaged over 10 networks with a fixed number of nodes $N=1000$. \n } \n\\label{fig_Illustration}\n\\end{figure}\n\nWe evaluate algorithms' performance by NMI \\cite{Kuncheva2004}: the higher the NMI is, the better the algorithm is. Figure 2 shows that LSI can obtain better community structure than LMD and GMD for the real networks except the polbooks network. Figure 3 compares the performance of the three indices on LFR networks. It is shown that LSI remarkably outperforms GMD and LMD. \n\nFigure 4 shows the relationship between nodes' LSI values and degrees for a real network and an artificially generated network. The difference between LSI and degree is remarkable, namely LSI can identify some low-degree nodes as central nodes while reject some high-degree nodes. If one would like to correctly detect a real community, it is natural to expect that one or more nodes in this community should be first identified as central nodes. If none of nodes in this community are central nodes, the probability that this community could be perfectly detected is very tiny. We set the number of central nodes identified by GMD being equal to the number of central nodes identified by LSI, and then compare how many real communities having central nodes by the three indices. As shown in Tables 2 and 3, more real communities have at least one central node by LSI than by LMD or GMD. In particular, as shown in Table 3, for LFR benchmark networks, most communities have at least one central node according to LSI, however, more than a half communities do not have any central nodes by GMD and LMD. Indeed, LMD only identify very few central nodes and most communities do not have any central nodes accordingly. This may be the reason why LSI can outperform LMD and GMD in community detection.\n\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig4_LCI_degree_20220104.eps} \n \\caption{LSI versus degree. Each data point represents one node. Nodes with $LSI\\geq0$ are colored by yellow, while nodes with $LSI<0$ are colored by red. (a) The result for the polblogs network. (b) The result for the LFR network with parameters $N=1000$, $\\langle k\\rangle=20$, $k_{max}=100$, $\\gamma=2.5$, $\\beta=1.5$ and $\\mu=0.2$, where $k_{max}$ is the maximal degree. } \n\\label{fig_Illustration}\n\\end{figure}\n\n\n\\begin{table}\n\\centering \n\\caption{The number of real communities $m$ and the number of central nodes identified by the three indices on the six real networks. Each number in bracket is the number of real communities containing at least one central node. The number of central nodes for GMD is set to be equal to the number of central nodes identified by LSI. }\n\\begin{indented}\n\\item[]\\begin{tabular}\n{>{\\columncolor{white}}ccccccc} \n\\hline\n\\rowcolor[gray]{0.9} indicators &karate &dolphins &football &polblogs &polbooks &highshool\\\\ \n\\hline\n$m$ &2 &4 &12 &2 &3 &8 \\\\ \nLSI &5(2) &19(4) &65(12) &123(2) &24(3) &80(7) \\\\ \nLMD &2(2) &5(3) &32(11) &6(2) &3(3) &12(6) \\\\ \nGMD &5(2) &19(3) &65(12) &123(2) &24(3) &80(7) \\\\ \n\\hline\n\\end{tabular}\n\\end{indented} \n\\end{table} \n\n\\begin{table} \n\\centering \n\\caption{The number of real communities $m$ and the number of central nodes identified by the three indices on the LFR networks. Each number in bracket is the number of real communities containing at least one central node. The number of central nodes for GMD is set to be equal to the number of central nodes identified by LSI.}\n\\begin{indented}\n\\item[]\\begin{tabular} \n{>{\\columncolor{white}}ccccccccc} \n\\hline\n\\rowcolor[gray]{0.9} indicators &$\\mu=0.1$ &$\\mu=0.2$ &$\\mu=0.3$ &$\\mu=0.4$ &$\\mu=0.5$ \\\\ \n\\hline\n$m$ &31 &31 &30 &35 &32 \\\\ \nLSI &242(29) &200(29) &187(25) &176(27) &166(23) \\\\ \nLMD &9(8) &6(4) &5(3) &5(4) &4(4) \\\\ \nGMD &242(13) &200(15) &187(15) &176(15) &166(15) \\\\ \n\n\\hline\n\\end{tabular} \n\\end{indented}\n\\end{table} \n\n\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig5_0211.eps} \n \\caption{The NMI values of the proposed greedy algorithm based on LSI and the Newman fast algorithm for the six real networks. }\n\\label{fig_IIlustration}\n\\end{figure}\n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig6_0211.eps} \n \\caption{The NMI values of the proposed greedy algorithm based on LSI and the Newman fast algorithm for the LFR networks. The default values of the four parameters are $\\mu=0.4$, $\\gamma=2.5$, $\\beta=1.5$ and $\\langle k\\rangle=20$. In each plot, only one parameter is varying. The results are averaged over 10 networks with a fixed number of nodes $N=1000$. }\n\\label{fig_IIlustration}\n\\end{figure}\n\n\nThe Newman fast algorithm is the most well-known greedy algorithm. It starts with every node being a single community, and then join a pair of provisional communities at each step to maximize the modularity $Q$, until all nodes agglomerated together to form one big community. The progress can be represented as a dendrogram, namely a tree that shows the order of all joins. The best community partition is obtained by cutting the dendrogram at the layer corresponding to the maximal $Q$. We compare the Newman fast algorithm with the proposed greedy algorithm based on LSI. As shown in figure 5, the proposed algorithm performs better than the Newman fast algorithm on real networks except the polbooks network. Figure 6 reports the comparison on LFR networks, indicating that the proposed algorithm outperforms the Newman fast algorithm and the superiority is robust.\n \n\n\\section{Conclusion and Discussion}\n\nIn this paper, we proposed a novel index LSI and a variant of greedy expanding algorithm that ensures all provisional communities are weak communities (i.e., internal degree is larger than external degree). The LSI performs remarkably better than the GMD and LMD in detecting communities if we apply those indices to identify central nodes for community expanding. In addition, the proposed greedy algorithm based on LSI outperforms the well-known Newman fast algorithm. \n\n\\begin{figure}[H]\n\\setlength{\\abovecaptionskip}{0.cm}\n\\setlength{\\belowcaptionskip}{-0.cm}\n\\centering\n\t\\includegraphics[width=0.8\\textwidth]{fig7_b.eps} \n \\caption{Illustration of the community structure of the polbooks network. Red, blue and black nodes belong to conservative, liberal, and neutral communities, respectively.} \n\\label{fig_Illustration}\n\\end{figure}\n\nPolbooks is the only network where the proposed algorithm is slightly worse than benchmark algorithms. Figure 7 illustrates the community structure of the polbooks network, from which we can observe that the neutral community is not a weak community, with external edges far more than internal edges. As our algorithm avoids weak communities, it may not perform well if some ground-truth communities themselves are not weak communities. It is hard to say whether this is a disadvantage or an advantage, since if a ground-truth community is not a weak community, it usually contains some domain-specific reasons and thus may not be a typical community from the purely topological viewpoint. \n\nThe proposed method can be extended to handle directed networks and weighted networks by slightly modifying the definition of LSI. If we allow non-central nodes to be merged to more than one community, the proposed algorithm can deal with the overlapping community structure. We can also add a free parameter $\\alpha$ to the quality function as $F(C_i) = {d_{in}(C_i)}\/{[d_{in}(C_i) + d_{out}(C_i)]^\\alpha}$, and then uncover the hierarchical community structure with different resolutions via tuning the parameter $\\alpha$.\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\\addcontentsline{toc}{section}{Acknowledgements}\nThis work was supported by the National Natural Science Foundation of China under Grant Nos. 71874172, 11975071, and the Thousand Talents Program of Sichuan Province under Grant No. 17QR003.\n\n\n\n\n\n\n\\section*{References}\n\\addcontentsline{toc}{section}{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}