NeuroImage 52 (2010) 1059 1069 Contents lsts avalable at ScenceDrect NeuroImage journal homepage wwwelsevercom/locate/ynmg Complex network measures of bran connectvty Uses and nterpretatons Mkal Rubnov a,b,c, Olaf Sporns d, a Black Dog Insttute and School of sychatry, Unversty of New South Wales, Sydney, Australa b Mental Health Research Dvson, Queensland Insttute of Medcal Research, Brsbane, Australa c CSIRO Informaton and Communcaton Technologes Centre, Sydney, Australa d Department of sychologcal and Bran Scences, Indana Unversty, Bloomngton, IN 47405, USA artcle nfo abstract Artcle hstory Receved 1 August 2009 Revsed 1 October 2009 Accepted 2 October 2009 Avalable onlne 9 October 2009 Bran connectvty datasets comprse networks of bran regons connected by anatomcal tracts or by functonal assocatons Complex network analyss a new multdscplnary approach to the study of complex systems ams to characterze these bran networks wth a small number of neurobologcally meanngful and easly computable measures In ths artcle, we dscuss constructon of bran networks from connectvty data and descrbe the most commonly used network measures of structural and functonal connectvty We descrbe measures that varously detect functonal ntegraton and segregaton, quantfy centralty of ndvdual bran regons or pathways, characterze patterns of local anatomcal crcutry, and test reslence of networks to nsult We dscuss the ssues surroundng comparson of structural and functonal network connectvty, as well as comparson of networks across subjects Fnally, we descrbe a Matlab toolbox (http//wwwbran-connectvty-toolboxnet) accompanyng ths artcle and contanng a collecton of complex network measures and large-scale neuroanatomcal connectvty datasets 2009 Elsever Inc All rghts reserved Introducton Modern bran mappng technques such as dffuson MRI, functonal MRI, EEG, and MEG produce ncreasngly large datasets of anatomcal or functonal connecton patterns Concurrent technologcal advancements are generatng smlarly large connecton datasets n bologcal, technologcal, socal, and other scentfc felds Attempts to characterze these datasets have, over the last decade, led to the emergence of a new, multdscplnary approach to the study of complex systems (Strogatz, 2001 Newman, 2003 Boccalett et al, 2006) Ths approach, known as complex network analyss, descrbes mportant propertes of complex systems by quantfyng topologes of ther respectve network representatons Complex network analyss has ts orgns n the mathematcal study of networks, known as graph theory However, unlke classcal graph theory, the analyss prmarly deals wth real-lfe networks that are large and complex nether unformly random nor ordered Bran connectvty datasets comprse networks of bran regons connected by anatomcal tracts or by functonal assocatons Bran networks are nvarably complex, share a number of common features wth networks from other bologcal and physcal systems, and may hence be characterzed usng complex network methods Network characterzaton of structural and functonal connectvty data s ncreasng (Bassett and Bullmore, 2006, 2009 Stam and Correspondng author E-mal address osporns@ndanaedu (O Sporns) Rejneveld, 2007 Bullmore and Sporns, 2009) and rests on several mportant motvatons Frst, complex network analyss promses to relably (Deuker et al, 2009) quantfy bran networks wth a small number of neurobologcally meanngful and easly computable measures (Sporns and Zw, 2004 Achard et al, 2006 Bassett et al, 2006 He et al, 2007 Hagmann et al, 2008) Second, by explctly defnng anatomcal and functonal connectons on the same map of bran regons, network analyss may be a useful settng for explorng structural functonal connectvty relatonshps (Zhou et al, 2006 Honey et al, 2007, 2009) Thrd, comparsons of structural or functonal network topologes between subject populatons appear to reveal presumed connectvty abnormaltes n neurologcal and psychatrc dsorders (Stam et al, 2007, 2009 Bassett et al, 2008 Lestedt et al, 2009 onten et al, 2009 Wang et al, 2009b) In ths artcle, we provde a non-techncal ntroducton to complex network analyss of bran connectvty and outlne mportant conceptual ssues assocated wth ts use We begn by dscussng the constructon of structural and functonal bran connectvty networks We then descrbe the most commonly used measures of local and global connectvty, as well as ther neurobologcal nterpretatons We focus on recently developed network measures (Boccalett et al, 2006 Costa et al, 2007b) and provde a freely avalable Matlab toolbox, contanng these measures, as well as ther weghted and drected varants (Table A1) Fnally, we dscuss some of the ssues assocated wth comparng structural and functonal connectvty n the same subject and comparng connectvty patterns between subjects 1053-8119/$ see front matter 2009 Elsever Inc All rghts reserved do101016/jneuromage200910003
1060 M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 Note that whle we concentrate on the analyss of large-scale connectvty, our dscusson s equally applcable to smaller scale connectvty, wth approprate redefntons For nstance, small-scale bran networks could consst of neurons lnked by synapses or of cortcal columns lnked by ntercolumnar connectons Constructon of bran networks A network s a mathematcal representaton of a real-world complex system and s defned by a collecton of nodes (vertces) and lnks (edges) between pars of nodes (Fg 1) Nodes n large-scale bran networks usually represent bran regons, whle lnks represent anatomcal, functonal, or effectve connectons (Frston, 1994), dependng on the dataset Anatomcal connectons typcally correspond to whte matter tracts between pars of bran regons Functonal connectons correspond to magntudes of temporal correlatons n actvty and may occur between pars of anatomcally unconnected regons Dependng on the measure, functonal connectvty may reflect lnear or nonlnear nteractons, as well as nteractons at dfferent tme scales (Zhou et al, 2009) Effectve connectons represent drect or ndrect causal nfluences of one regon on another and may be estmated from observed perturbatons (Frston et al, 2003) Fg 2 shows llustratve anatomcal, functonal, and effectve connectvty networks, adapted from the study by Honey et al (2007) The anatomcal network represents large-scale connecton pathways between cortcal regons n the macaque, as collated from hstologcal tract tracng studes Functonal and effectve connectvty networks were constructed from tme seres of bran dynamcs smulated on ths anatomcal network The functonal network represents patterns of cross-correlatons between BOLD sgnals estmated from these dynamcs The effectve network represents patterns of causal nteractons, as computed wth transfer entropy, a measure of drected nformaton flow All networks are represented by ther connectvty (adjacency) matrces Rows and columns n these matrces denote nodes, whle matrx entres denote lnks The order of nodes n connectvty matrces has no effect on computaton of network measures but s mportant for network vsualzaton (Fg 2A) The nature of nodes The nature of nodes and lnks n ndvdual bran networks s determned by combnatons of bran mappng methods, anatomcal parcellaton schemes, and measures of connectvty Many combnatons occur n varous expermental settngs (Horwtz, 2003) The choce of a gven combnaton must be carefully motvated, as the nature of nodes and lnks largely determnes the neurobologcal nterpretaton of network topology (Butts, 2009) Nodes should deally represent bran regons wth coherent patterns of extrnsc anatomcal or functonal connectons arcellaton schemes that lump heterogeneously connected bran regons nto sngle nodes may be less meanngful In addton, a parcellaton scheme should completely cover the surface of the cortex, or of the entre bran, and ndvdual nodes should not spatally overlap The use of MEG and EEG sensors may be problematc n ths regard, gven that sensors may detect spatally overlappng sgnals and are generally not algned wth boundares of coherent regons (Ioanndes, 2007) Networks constructed usng dstnct parcellaton schemes may sgnfcantly dffer n ther propertes (Wang et al, 2009a) and cannot, n general, be quanttatvely compared Specfcally, structural and functonal networks may only be meanngfully compared, f these networks share the same parcellaton scheme (Honey et al, 2009) The nature of lnks In addton to the type of connectvty (anatomcal, functonal or effectve) and measure-specfc (eg, tme scale) features of connectvty, lnks are also dfferentated on the bass of ther weght and drectonalty Bnary lnks denote the presence or absence of connectons (Fg 2A), whle weghted lnks also contan nformaton about connecton strengths (Fgs 2B, C) Weghts n anatomcal networks may represent the sze, densty, or coherence of anatomcal tracts, whle weghts n functonal and effectve networks may represent respectve magntudes of correlatonal or causal nteractons Many recent studes dscard lnk weghts, as bnary networks are n most cases smpler to characterze and have a more easly defned null model for statstcal comparson (see below) On the other hand, weghted characterzaton usually focuses on somewhat dfferent and complementary aspects of network organzaton (eg, Saramak et al, 2007) and may be especally useful n flterng the nfluence of weak and potentally non-sgnfcant lnks Weak and non-sgnfcant lnks may represent spurous connectons, partcularly n functonal or effectve networks (Fgs 2B, C top panels) These lnks tend to obscure the topology of strong and sgnfcant connectons and as a result are often dscarded, by applyng an absolute, or a proportonal weght threshold (Fgs 2B, C bottom panels) Threshold values are often arbtrarly determned, and networks should deally be characterzed across a broad range of thresholds Independently, all self-connectons or negatve connectons (such as functonal antcorrelatons) must currently be removed from the networks pror to analyss Future network methods may be able to quantfy the role of negatve weghts n global network organzaton Lnks may also be dfferentated by the presence or absence of drectonalty Thus, anatomcal and effectve connectons may conceptually be represented wth drected lnks Unfortunately, current neuromagng methods are unable to drectly detect anatomcal or causal drectonalty On the other hand, drected anatomcal networks constructed from tract tracng studes (eg, Fg 2A) ndcate the exstence of a large proporton of recprocal connectons n the cortex, whch provdes some valdty for the use of undrected anatomcal networks Drected patterns of effectve connectvty may be nferred from changes n functonal actvty that follow localzed perturbatons Measures of bran networks An ndvdual network measure may characterze one or several aspects of global and local bran connectvty In ths secton, we descrbe measures that varously detect aspects of functonal ntegraton and segregaton, quantfy mportance of ndvdual bran regons, characterze patterns of local anatomcal crcutry, and test reslence of networks to nsult Fg 3 llustrates some basc concepts underlyng these measures, whle Table A1 contans mathematcal defntons of all measures Network measures are often represented n multple ways Thus, measures of ndvdual network elements (such as nodes or lnks) typcally quantfy connectvty profles assocated wth these elements and hence reflect the way n whch these elements are embedded n the network Measurement values of all ndvdual elements comprse a dstrbuton, whch provdes a more global descrpton of the network Ths dstrbuton s most commonly characterzed by ts mean, although other features, such as dstrbuton shape, may be more mportant f the dstrbuton s nonhomogeneous In addton to these dfferent representatons, network measures also have bnary and weghted, drected and undrected varants Weghted and drected varants of measures are typcally generalzatons of bnary undrected varants and
M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 1061 therefore reduce to the latter when computed on bnary undrected networks To llustrate the dfferent representatons and varants of a network measure, we consder a basc and mportant measure known as the degree The degree of an ndvdual node s equal to the number of lnks connected to that node, whch n practce s also equal to the number of neghbors of the node Indvdual values of the degree therefore reflect mportance of nodes n the network, as dscussed below The degrees of all nodes n the network comprse the degree dstrbuton, whch s an mportant marker of network development and reslence The mean network degree s most commonly used as a measure of densty, or the total wrng cost of the network The drected varant of the degree dstngushes the number of nward lnks from the number of outward lnks, whle the weghted varant of the degree, sometmes termed the strength, s defned as the sum of all neghborng lnk weghts It s mportant to note that values of many network measures are greatly nfluenced by basc network characterstcs, such as the number of nodes and lnks, and the degree dstrbuton Consequently, the sgnfcance of network statstcs should be establshed by comparson wth statstcs calculated on null-hypothess networks Null-hypothess networks have smple random or ordered topologes but preserve basc characterstcs of the orgnal network The most commonly used null-hypothess network has a random topology but shares the sze, densty and bnary degree dstrbuton of the orgnal network (Maslov and Sneppen, 2002) Note, however, that ths network may have a dfferent weghted degree dstrbuton, especally f the weght dstrbuton s nonhomogeneous Measures of functonal segregaton Functonal segregaton n the bran s the ablty for specalzed processng to occur wthn densely nterconnected groups of bran regons Measures of segregaton prmarly quantfy the presence of such groups, known as clusters or modules, wthn the network Measures of segregaton have straghtforward nterpretatons n anatomcal and functonal networks The presence of clusters n anatomcal networks suggests the potental for functonal segregaton n these networks, whle the presence of clusters n functonal networks suggests an organzaton of statstcal dependences ndcatve of segregated neural processng Smple measures of segregaton are based on the number of trangles n the network, wth a hgh number of trangles mplyng segregaton (Fg 3) Locally, the fracton of trangles around an ndvdual node s known as the clusterng coeffcent and s equvalent to the fracton of the node's neghbors that are also neghbors of each other (Watts and Strogatz, 1998) The mean clusterng coeffcent for the network hence reflects, on average, the prevalence of clustered connectvty around ndvdual nodes The mean clusterng coeffcent s normalzed ndvdually for each node (Table A1) and may therefore be dsproportonately nfluenced by nodes wth a low degree A classcal varant of the clusterng coeffcent, known as the transtvty, s normalzed collectvely and consequently does not suffer from ths problem (eg, Newman, 2003) Both the clusterng coeffcent and the transtvty have been generalzed for weghted (Onnela et al, 2005) and drected (Fagolo, 2007) networks More sophstcated measures of segregaton not only descrbe the presence of densely nterconnected groups of regons, but also fnd the exact sze and composton of these ndvdual groups Ths composton, known as the network's modular structure (communty structure), s revealed by subdvdng the network nto groups of nodes, wth a maxmally possble number of wthngroup lnks, and a mnmally possble number of between-group lnks (Grvan and Newman, 2002) The degree to whch the network may be subdvded nto such clearly delneated and nonoverlappng groups s quantfed by a sngle statstc, the modularty (Newman, 2004b) Unlke most other network measures, the optmal modular structure for a gven network s typcally estmated wth optmzaton algorthms, rather than computed exactly (Danon et al, 2005) Optmzaton algorthms generally sacrfce some degree of accuracy for computatonal speed One notable algorthm (Newman, 2006) sknowntobequteaccurate and s suffcently fast for smaller networks Another more recently developed algorthm (Blondel et al, 2008) performsmuchfaster for larger networks and s also able to detect a herarchy of modules (the presence of smaller modules nsde larger modules) Both of these algorthms have been generalzed to detect modular structure n weghted (Newman, 2004a) anddrected(lecht and Newman, 2008) networks Other algorthms detect overlappng modular network structure, and hence acknowledge that sngle nodes may smultaneously belong n multple modules (eg, alla et al, 2005) Fgure 2A shows the modular structure of the anatomcal connectvty network of a large porton of the macaque cortex, whle Fgs 2B and C shows functonal and effectve networks reordered by ths modular structure These anatomcal modules correspond to groups of specalzed functonal areas, such as the vsual and somatomotor regons, as prevously determned by physologcal recordngs Anatomcal, functonal, and effectve modules n ths network show extensve overlap Measures of functonal ntegraton Functonal ntegraton n the bran s the ablty to rapdly combne specalzed nformaton from dstrbuted bran regons Measures of ntegraton characterze ths concept by estmatng the ease wth whch bran regons communcate and are commonly based on the concept of a path aths are sequences of dstnct nodes and lnks and n anatomcal networks represent potental routes of nformaton flow between pars of bran regons Lengths of paths consequently estmate the potental for functonal ntegraton between bran regons, wth shorter paths mplyng stronger potental for ntegraton On the other hand, functonal connectvty data, by ts nature, already contan such nformaton for all pars of bran regons (Fg 2B top panel) aths n functonal networks represent sequences of statstcal assocatons and may not correspond to nformaton flow on anatomcal connectons Consequently, network measures based on functonal paths are less straghtforward to nterpret Such measures may be easer to nterpret when nformaton about anatomcal connectons s avalable for the same subjects (eg, Honey et al, 2009) The average shortest path length between all pars of nodes n the network s known as the characterstc path length of the network (eg, Watts and Strogatz, 1998) and s the most commonly used measure of functonal ntegraton The average nverse shortest path length s a related measure known as the global effcency (Latora and Marchor, 2001) Unlke the characterstc path length, the global effcency may be meanngfully computed on dsconnected networks, as paths between dsconnected nodes are defned to have nfnte length, and correspondngly zero effcency More generally, the characterstc path length s prmarly nfluenced by long paths (nfntely long paths are an llustratve extreme), whle the global effcency s prmarly nfluenced by short paths Some authors have argued that ths may make the global effcency a superor measure of ntegraton (Achard and Bullmore, 2007) Note that measures such as the characterstc path length and the global effcency do not ncorporate multple and longer paths, whch may sgnfcantly contrbute to ntegraton n larger and sparser networks (Estrada and Hatano, 2008)
1062 M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 Fg 1 Constructon of bran networks from large scale anatomcal and functonal connectvty datasets Structural networks are commonly extracted from hstologcal (tract tracng) or neuromagng (dffuson MRI) data Functonal networks are commonly extracted from neuromagng (fmri) or neurophysologcal (EEG, MEG) data For computatonal convenence, networks are commonly represented by ther connectvty matrces, wth rows and columns representng nodes and matrx entres representng lnks To smplfy analyss, networks are often reduced to a sparse bnary undrected form, through thresholdng, bnarzng, and symmetrzng aths are easly generalzed for drected and weghted networks (Fg 3) Whle a bnary path length s equal to the number of lnks n the path, a weghted path length s equal to the total sum of ndvdual lnk lengths Lnk lengths are nversely related to lnk weghts, as large weghts typcally represent strong assocatons and close proxmty Connecton lengths are typcally dmensonless and do not represent spatal or metrc dstance The structural, functonal, and effectve connectvty networks n Fg 2 dffer n ther values of the global effcency Structural and effectve networks are smlarly organzed (Honey et al, 2007) and share a hgh global effcency In comparson, functonal networks have weaker connectons between modules, and consequently a lower global effcency Small-world bran connectvty Anatomcal bran connectvty s thought to smultaneously reconcle the opposng demands of functonal ntegraton and segregaton (Tonon et al, 1994) A well-desgned anatomcal network could therefore combne the presence of functonally specalzed (segregated) modules wth a robust number of ntermodular (ntegratng) lnks Such a desgn s commonly termed small-world and ndeed appears to be a ubqutous organzaton of anatomcal connectvty (Bassett and Bullmore, 2006) Furthermore, most studes examnng functonal bran networks also report varous degrees of small-world organzaton It s commonly thought that such an organzaton reflects an
M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 1063 Fg 2 Illustratve anatomcal, functonal, and effectve connectvty networks (A) Large-scale anatomcal connecton network of the macaque cortex Ths network ncludes the ventral and dorsal streams of vsual cortex, as well as groups of somatosensory and somatomotor regons Network modules representng these groups, form largely contguous (color-coded) anatomcal regons Area names and abbrevatons are provded n the supportng nformaton (B, top) Functonal connectvty network, representng cross-correlaton of the regonal BOLD sgnal, as estmated from smulated neural mass model dynamcs Warm colors represent postve correlatons, whle cool colors represent negatve correlatons (B, bottom) The same network, thresholded by removng negatve and self-self correlatons (C, top) Effectve connectvty network, constructed by computng nter-regonal transfer entropy, from the smulated fast tme-scale dynamcs (C, bottom) The same network, thresholded to preserve the strongest connectons, of the same number as n the structural network See Honey et al (2007) for a further exploraton of these networks optmal balance of functonal ntegraton and segregaton (Sporns and Honey, 2006) Whle ths s a plausble hypothess, the somewhat abstract nature of functonal paths (see above) makes nterpretaton of the small-world property less straghtforward n functonal networks A more complete understandng of the relatonshp between bran dynamcs and functonal connectvty wll help to clarfy ths ssue Small-world networks are formally defnedasnetworksthat are sgnfcantly more clustered than random networks, yet have approxmately the same characterstc path length as random networks (Watts and Strogatz, 1998) More generally, small-world networks should be smultaneously hghly segregated and ntegrated Recently, a measure of small-worldness was proposed to capture ths effect n a sngle statstc (Humphres and Gurney, 2008) Ths measure may be useful for snapshot characterzaton of an ensemble of networks, but t may also falsely report a smallworld topology n hghly segregated, but poorly ntegrated networks Consequently, ths measure should not n general be regarded as a substtute for ndvdual assessments of ntegraton and segregaton Anatomcal and effectve networks n Fg 2 are smultaneously hghly segregated and ntegrated, and consequently have small-world topologes In comparson, the functonal network s also hghly segregated but has a lower global effcency, and therefore weaker small-world attrbutes Network motfs Global measures of ntegraton and segregaton bele a rch repertore of underlyng local patterns of connectvty Such local patterns are partcularly dverse n drected networks For nstance, anatomcal trangles may consst of feedforward loops, feedback loops, and bdrectonal loops, wth dstnct frequences of ndvdual loops lkely havng specfc functonal mplcatons These patterns of local connectvty are known as (anatomcal) motfs (Fg 3) The sgnfcance of a motf n the network s determned by ts frequency of occurrence, usually normalzed as the motf z-score by comparson wth ensembles of random null-hypothess networks (Mlo et al, 2002) The frequency of occurrence of dfferent motfs around an ndvdual node s known as the motf fngerprnt of that node and s lkely to reflect the functonal role of the correspondng bran regon (Sporns and Kotter, 2004) The frequency of occurrence of dfferent motfs n the whole network correspondngly represents the characterstc motf profle of the network Functonal actvatons on a local anatomcal crcut may at any gven tme utlze only a porton of that crcut Ths observaton motvated the ntroducton of functonal motfs, defned as possble subsets of connecton patterns embedded wthn anatomcal motfs (Sporns and Kotter, 2004) Functonal motf sgnfcance may be optmally characterzed by weghted
1064 M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 Fg 3 Measures of network topology An llustraton of key complex network measures (n talcs) descrbed n ths artcle These measures are typcally based on basc propertes of network connectvty (n bold type) Thus, measures of ntegraton are based on shortest path lengths (green), whle measures of segregaton are often based on trangle counts (blue) but also nclude more sophstcated decomposton nto modules (ovals) Measures of centralty may be based on node degree (red) or on the length and number of shortest paths between nodes Hub nodes (black) often le on a hgh number of shortest paths and consequently often have hgh betweenness centralty atterns of local connectvty are quantfed by network motfs (yellow) An example three-node and four-lnk anatomcal motf contans sx possble functonal motfs, of whch two are shown one motf contanng dashed lnks, and one motf contanng crossed lnks measures of motf occurrence, known as the motf ntensty, and the motf ntensty z-score (Onnela et al, 2005) Motf ntensty takes nto account weghts of all motf-comprsng lnks and may therefore be more senstve n detectng consstently strong functonal confguratons There s a source of possble confuson n motf termnology Motfs ( structural and functonal ) were ntally consdered only n the context of anatomcal bran networks (Sporns and Kotter, 2004) However, motf measures may also be meanngfully appled to some effectve connectvty networks On the other hand, motfs are generally not used n the analyss of undrected networks, due to the paucty of local undrected connectvty patterns Measures of centralty Important bran regons (hubs) often nteract wth many other regons, facltate functonal ntegraton, and play a key role n network reslence to nsult Measures of node centralty varously assess mportance of ndvdual nodes on the above crtera There are many measures of centralty, and n ths secton, we descrbe the more commonly used measures We also note that motf and reslence measures, dscussed n other sectons, are lkewse sometmes used to detect central bran regons The degree, as dscussed above, s one of the most common measures of centralty The degree has a straghtforward neurobologcal nterpretaton nodes wth a hgh degree are nteractng, structurally or functonally, wth many other nodes n the network The degree may be a senstve measure of centralty n anatomcal networks wth nonhomogeneous degree dstrbutons In modular anatomcal networks, degree-based measures of wthn-module and between-module connectvty may be useful for heurstcally classfyng nodes nto dstnct functonal groups (Gumera and Amaral, 2005) The wthn-module degree z-score s a localzed, wthn-module verson of degree centralty (Table A1) The complementary partcpaton coeffcent assesses the dversty of ntermodular nterconnectons of ndvdual nodes Nodes wth a hgh wthn-module degree but wth a low partcpaton coeffcent(knownasprovncalhubs)arehencelkelytoplay an mportant part n the facltaton of modular segregaton On the other hand, nodes wth a hgh partcpaton coeffcent (known as connector hubs) are lkely to facltate global ntermodular ntegraton Many measures of centralty are based on the dea that central nodes partcpate n many short paths wthn a network, and consequently act as mportant controls of nformaton flow (Freeman, 1978) For nstance, closeness centralty s defned as the nverse of the average shortest path length from one node to all other nodes n the network A related and often more senstve measure s betweenness centralty, defned as the fracton of all shortest paths n the network that pass through a gven node Brdgng nodes that connect dsparate parts of the network often have a hgh betweenness centralty (Fg 3) The noton of betweenness centralty s naturally extended to lnks and could therefore also be used to detect mportant anatomcal or functonal connectons The calculaton of betweenness centralty has been made sgnfcantly more effcent wth the recent development of faster algorthms (Brandes, 2001 Kntal, 2008) Weghted and drected varants of centralty measures are n most cases based on weghted and drected varants of degree and path length (Table A1) Measures of centralty may have dfferent nterpretatons n anatomcal and functonal networks For nstance, anatomcally central nodes often facltate ntegraton, and consequently enable functonal lnks between anatomcally unconnected regons Such lnks n turn make central nodes less promnent and so reduce the senstvty of centralty measures n functonal networks In addton, path-based measures of centralty n functonal networks are subject to the same nterpretatonal caveats as path-based measures of ntegraton (see above) Measures of network reslence Anatomcal bran connectvty nfluences the capacty of neuropathologcal lesons to affect functonal bran actvty For nstance, the extent of functonal deteroraton s heavly determned by the affected anatomcal regon n a stroke, or by the capacty of anatomcal
M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 1065 connectvty to wthstand degeneratve change n Alzhemer's dsease Complex network analyss s able to characterze such network reslence propertes drectly and ndrectly Indrect measures of reslence quantfy anatomcal features that reflect network vulnerablty to nsult One such feature s the degree dstrbuton (Barabas and Albert, 1999) For nstance, complex networks wth power-law degree dstrbutons may be reslent to gradual random deteroraton, but hghly vulnerable to dsruptons of hgh-degree central nodes Most real-lfe networks do not have perfect power law degree dstrbutons On the other hand, many networks have degree dstrbutons that locally behave lke a power law the extent to whch these dstrbutons ft a power law may hence be a useful marker of reslence (Achard et al, 2006) Another useful measure of reslence s the assortatvty coeffcent (Newman, 2002) The assortatvty coeffcent s a correlaton coeffcent between the degrees of all nodes on two opposte ends of a lnk Networks wth a postve assortatvty coeffcent are therefore lkely to have a comparatvely reslent core of mutually nterconnected hgh-degree hubs On the other hand, networks wth a negatve assortatvty coeffcent are lkely to have wdely dstrbuted and consequently vulnerable hgh-degree hubs Related measures of assortatvty computed on ndvdual nodes nclude the average neghbor degree (astor-satorras et al, 2001) and the recently ntroduced local assortatvty coeffcent (raveenan et al, 2008) Indvdual central nodes that score lowly on these measures may therefore compromse global network functon f dsrupted Weghted and drected varants of assortatvty measures are based on the respectve weghted and drected varants of the degree (Barrat et al, 2004 Leung and Chau, 2007) Drect measures of network reslence generally test the network before and after a presumed nsult For nstance, patents wth a progressve neurodegeneratve dsease may be maged over a longtudnal perod Alternatvely, nsults may be computatonally smulated by random or targeted removal of nodes and lnks The effects of such lesons on the network may then be quantfed by characterzng changes n the resultng anatomcal connectvty, or n the emergent smulated functonal connectvty or dynamcal actvty (eg, Alstott et al, 2009) When testng reslence n such a way, t s prudent to use measures that are sutable for the analyss of dsconnected networks For nstance, the global effcency would be preferable to the characterstc path length as a measure of ntegraton Network comparson Complex network analyss may be useful for explorng connectvty relatonshps n ndvdual subjects or between subject groups In ndvdual subjects, comparsons of structural and functonal networks may provde nsghts nto structural functonal connectvty relatonshps (eg, Honey et al, 2007) Across subject populatons, comparsons may detect abnormaltes of network connectvty n varous bran dsorders (Bassett and Bullmore, 2009) The ncreased emphass on structure functon and between-subject comparsons n studes of bran networks wll requre the development of accurate statstcal comparson tools, smlar to those n cellular and molecular bology (Sharan and Ideker, 2006) relmnary steps have already been taken n ths drecton (Costa et al, 2007a) Here we dscuss some general ssues assocated wth network comparson Dfferences n densty between anatomcal and functonal networks make global comparsons between these networks less straghtforward Functonal networks are lkely to be denser than anatomcal networks, as they wll typcally contan numerous connectons between anatomcally unconnected regons (Damoseaux and Grecus, 2009) These dfferences n densty are lkely to become more pronounced n larger, more hghly resolved networks, as anatomcal connectvty n such networks becomes ncreasngly sparse, whle functonal connectvty remans comparatvely dense Notably, comparsons between anatomcal and functonal modular structure (eg, Zhou et al, 2006) reman meanngful despte dfferences n densty Other factors that may affect comparsons of network topology nclude degree and weght dstrbutons The nontrval relatonshp between structural and functonal bran connectvty and the consequent nterpretatonal dffcultes of some functonal measures (see above) make between-subject comparsons of functonal networks dffcult The sgnfcance of such comparsons wll become more obvous wth ncreased knowledge about causal relatonshps between bran regons, as medated by drect anatomcal connectons The development of a detaled anatomcal map of the human bran s an mportant step n ths drecton (Sporns et al, 2005) Bran connectvty analyss software Multple network analyss software packages are freely avalable on the Web These packages nclude command-lne toolboxes n popular languages, such as Matlab (Glech, 2008) and ython (Hagberg et al, 2008), as well as standalone graphcal user nterface software (Batagelj and Mrvar, 2003 NWB-Team, 2006) Some of these packages are especally sutable for the analyss of large networks contanng thousands of nodes, whle others have powerful network vsualzaton capabltes To accompany ths artcle, we developed a freely avalable and open source Matlab toolbox (http//wwwbran-connectvty-toolboxnet) A number of features dstngush our toolbox from most other software packages Our toolbox ncludes many recently developed network measures, whch are dscussed n ths artcle, but are not yet wdely avalable In addton, we provde weghted and drected varants for all our measures many of these varants are lkewse not yet avalable n other software In addton to these features, the toolbox provdes functons for network manpulaton (such as thresholdng) and ncludes algorthms for generatng null-hypothess networks of predetermned (random, ordered and other) topologes Fnally, the toolbox contans datasets for large scale neuroanatomcal networks of the mammalan cortex of several speces The open source nature of our toolbox allows researchers to customze ndvdual functons for ther needs, and to ncorporate functons nto larger analyss protocols Concluson Complex network analyss has emerged as an mportant tool for characterzaton of anatomcal and functonal bran connectvty We descrbed a collecton of measures that quantfy local and global propertes of complex bran networks The accompanyng bran connectvty toolbox allows researchers to start explorng network propertes of complex structural and functonal datasets We hope that the bran mappng communty wll be able to beneft from and contrbute to these tools Acknowledgments We thank Rolf Kötter, atrc Hagmann, Avad Rubnsten, and Chrs Honey for ther contrbutons to the bran connectvty toolbox Jonathan ower and Vassls Tsaras for suggestng valuable mprovements to our toolbox functons and Alan Barrat for defntonal clarfcatons MR s grateful to Mchael Breakspear for hs supervson and support durng ths project MR and OS were supported by the JS McDonnell Foundaton Bran NRG JSMF22002082 MR was supported by CSIRO ICT Centre scholarshp
1066 M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 Appendx A Table A1 Mathematcal defntons of complex network measures (see supplementary nformaton for a self-contaned verson of ths table) Measure Bnary and undrected defntons Weghted and drected defntons Basc concepts and measures Basc concepts and notaton N s the set of all nodes n the network, and n s the number of nodes L s the set of all lnks n the network, and l s number of lnks (, j) s a lnk between nodes and j, (, j N) a j s the connecton status between and j a j = 1 when lnk (, j) exsts (when and j are neghbors) a j = 0 otherwse (a = 0 for all ) We compute the number of lnks as l =, j N a j (to avod ambguty wth drected lnks we count each undrected lnk twce, as a j and as a j ) Lnks (, j) are assocated wth connecton weghts w j Henceforth, we assume that weghts are normalzed, such that 0 w j 1 for all and j l w s the sum of all weghts n the network, computed as l w =, j N w j Drected lnks (, j) are ordered from to j Consequently, n drected networks a j does not necessarly equal a j Degree number of lnks connected to a node Degree of a node, k = X a j jan Weghted degree of, k w = j N w j (Drected) out-degree of, k out = j N a j (Drected) n-degree of, k n = j N a j Shortest path length a bass for measurng ntegraton Number of trangles a bass for measurng segregaton Shortest path length (dstance), between nodes and j, d j = X a uv a uv a g X j where g j s the shortest path (geodesc) between and j Note that d j = for all dsconnected pars, j Number of trangles around a node, t = 1 2 X jhan a j a h a jh Shortest weghted path length between and j, d w j = auv g j w f(w uv ), where f s a map (eg, an nverse) from weght to length and g w j s the shortest weghted path between and j Shortest drected path length from to j, d j = aj g j a j, where g j s the drected shortest path from to j (Weghted) geometrc mean of trangles around, t w = 1 2 jhan w 1 = 3 jw h w jh Number of drected trangles around, t Y = 1 2 jhan a j + a j ð ah + a h Þ a jh + a hj Measures of ntegraton Characterstc path length Characterstc path length of the network (eg, Watts and Strogatz, 1998), L = 1 X L n = 1 X janj n d j an an Weghted characterstc path length, L w = 1 n Drected characterstc path length, L Y = 1 n an an jan j dw j jan j dy j where L s the average dstance between node and all other nodes Global effcency Global effcency of the network (Latora and Marchor, 2001), E = 1 X E n = 1 X n an an janj d 1 j Weghted global effcency, E w = 1 n an Drected global effcency, E Y = 1 n an d w jan j j d Y jan j j 1 1 where E s the effcency of node Measures of segregaton Clusterng coeffcent Clusterng coeffcent of the network (Watts and Strogatz, 1998), C = 1 X C n = 1 X 2t n k an an ðk 1Þ where C s the clusterng coeffcent of node (C = 0 for k b 2) Transtvty Transtvty of the network (eg, Newman, 2003), an T = 2t an k ðk 1Þ Note that transtvty s not defned for ndvdual nodes Local effcency Local effcency of the network (Latora and Marchor, 2001), E loc = 1 X E n loc = 1 X n an an jhanj a ja h k ðk 1Þ h 1 d jh ðn Þ where E loc, s the local effcency of node, and d jh (N ) s the length of the shortest path between j and h, that contans only neghbors of Weghted clusterng coeffcent (Onnela et al, 2005), C w = 1 2t w n an See Saramak et al (2007) for k ðk 1Þ other varants Drected clusterng coeffcent (Fagolo, 2007), C Y = 1 t Y n an ðk out + k n Þðk out + k Þ 2 a ja j jan Weghted transtvty, T w an = 2tw k an ðk 1Þ Drected transtvty, T Y an = ty h an k out + k n k out + k n 1 v2 jan a ja j Weghted local effcency, 1 = 3 E w loc = 1 w jhan j j w h ½d w jhðn ÞŠ 1 2 an k ðk 1Þ Drected local effcency, E Y loc = 1 a jhanj ð j + a j Þða h + a h Þ d Y 1 Y 1 jh ðn Þ + d hj ðn Þ 2n an k out + k n k out + k n 1 2 a jan ja j
M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 1067 Table A1 (contnued) Measure Bnary and undrected defntons Weghted and drected defntons Modularty Modularty of the network (Newman, 2004b), Q = X " e uu X! 2 # e uv uam vam where the network s fully subdvded nto a set of nonoverlappng modules M, and e uv s the proporton of all lnks that connect nodes n module u wth nodes n module v An equvalent alternatve formulaton of the modularty jan a j k k j l δ m m j, (Newman, 2006) s gven by Q = 1 l where m s the module contanng node, and δ m,mj =1fm = m j, and 0 otherwse Weghted modularty (Newman, 2004), Q w = 1 l w jan w j kw k w j l w δ m m j Drected modularty (Lecht andnewman, 2008), Q Y = 1 l jan a j kout k n δ l m m j Measures of centralty Closeness centralty Closeness centralty of node (eg Freeman, 1978), L 1 = d janj j Weghted closeness centralty, L w 1 = jan j dw j Drected closeness centralty, L Y 1 = jan j dy j Betweenness centralty Betweenness centralty of node (eg, Freeman, 1978), 1 b = ðþðn 2Þ h jan h jh j ρ hj ðþ ρ hj Betweenness centralty s computed equvalently on weghted and drected networks, provded that path lengths are computed on respectve weghted or drected paths Wthn-module degree z-score where ρ hj s the number of shortest paths between h and j, and ρ hj () s the number of shortest paths between h and j that pass through Wthn-module degree z-score of node (Gumera and Amaral, 2005), z = k ðm Þ km ð Þ σ km ð Þ where m s the module contanng node, k (m ) s the wthn-module degree of (the number of lnks between and all other nodes n m ), and km ð Þ and σ k(m) are the respectve mean and standard devaton of the wthn-module m degree dstrbuton Weghted wthn-module degree z-score, z w = kw ðm Þ k w ðm Þ Wthn-module out-degree z-score, z out Wthn-module n-degree z-score, z n = kout = kn σ kw ðm Þ ðm Þ k out ðm σ kout ðm Þ m ð Þ σ kn ðm Þ ð Þ k n m Þ artcpaton coeffcent artcpaton coeffcent of node (Gumera and Amaral, 2005), y =1 X k ðmþ 2 k mam where M s the set of modules (see modularty), and k (m) s the number of lnks between and all nodes n module m Weghted partcpaton coeffcent, y w =1 k w ðmþ 2 mam k w Out-degree partcpaton coeffcent, y out =1 k out ðmþ 2 mam k out In-degree partcpaton coeffcent, y n =1 2 k n ðmþ mam k n Network motfs Anatomcal and functonal motfs J h s the number of occurrences of motf h n all subsets of the network (subnetworks) h s an n h node, l h lnk, drected connected pattern h wll occur as an anatomcal motf n an n h node, l h lnk subnetwork, f lnks n the subnetwork match lnks n h (Mlo et al, 2002) h wll occur (possbly more than once) as a functonal motf n an n h node, l h l h lnk subnetwork, f at least one combnaton of l h lnks n the subnetwork matches lnks n h (Sporns and Kotter, 2004) (Weghted) ntensty of h (Onnela et al, 2005), I h = u Π ðjþal w 1 l h h u j where the sum s over all occurrences of h n the network, and L hu s the set of lnks n the uth occurrence of h Note that motfs are drected by defnton Motf z-score z-score of motf h (Mlo et al, 2002), z h = J h h J randh σ J randh where J rand,h and σ J rand,h are the respectve mean and standard devaton for the number of occurrences of h n an ensemble of random networks Motf fngerprnt n h node motf fngerprnt of the network (Sporns and Kotter, 2004), F nh ðhvþ = X an F nh ðhvþ = X J h V an where h s any n h node motf, F nh, (h ) s the n h node motf fngerprnt for node, and J h, s the number of occurrences of motf h around node Intensty z-score of motf h (Onnela et al, 2005), z I h = I h hi randh σ I randh where I rand,h and σ I rand,h are the respectve mean and standard devaton for the ntensty of h n an ensemble of random networks n h node motf ntensty fngerprnt of the network, Fn I h ðhvþ = an FI n h hv = an I h V, where h s any n h node motf, F I n h, (h ) s the n h node motf ntensty fngerprnt for node, and I h, s the ntensty of motf h around node (contnued on next page)
1068 M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 Table A1 (contnued) Measure Bnary and undrected defntons Weghted and drected defntons Measures of reslence Degree dstrbuton Cumulatve degree dstrbuton of the network (Barabas and Albert, 1999), k ð Þ = X ðkvþ k Vzkp Cumulatve weghted degree dstrbuton, k ð w Cumulatve out-degree dstrbuton, k ð out Þ = k Vzk Cumulatve n-degree dstrbuton, k n = k Vzk Þ = k Vzk w pkv, out pkv n pkv where p(k ) s the probablty of a node havng degree k Average neghbor degree Average degree of neghbors of node (astor-satorras et al, 2001), jan k nn = a jk j k Assortatvty coeffcent Assortatvty coeffcent of the network (Newman, 2002), l 1 h ðjþal k k j l 1 ðjþal 1 2 k 2 + k j r = l 1 h ðjþal 1 2 k2 + k 2 j l 1 ðjþal 1 2 k + k 2 j Other concepts Degree dstrbuton preservng network randomzaton Degree-dstrbuton preservng randomzaton s mplemented by teratvely choosng four dstnct nodes 1, j 1, 2, j 2 N at random, such that lnks ( 1, j 1 ), ( 2, j 2 ) L, whle lnks ( 1, j 2 ), ( 2, j 1 ) L The lnks are then rewred such that ( 1, j 2 ), ( 2, j 1 ) L and ( 1, j 1 ), ( 2, j 2 ) L, (Maslov and Sneppen, 2002) Lattczaton (a lattce-lke topology) results f an addtonal constrant s mposed, 1 +j 2 + 2 +j 1 b 1 +j 1 + 2 +j 2 (Sporns and Kotter, 2004) Average weghted neghbor degree (modfed from Barrat et al, 2004), k w nn = w jan jk w j k w Average drected neghbor degree, k Y nn = janð a j + a j Þ k out + k n 2 k out + k n Weghted assortatvty coeffcent (modfed from Leung and Chau, 2007), h r w l 1 ð jþal = w jk w k w j l 1 2 1 ð jþal 2 w jðk w + k w j Þ h 2 l 1 1 ð jþal 2 w j ðk w Þ 2 + ðk w j Þ 2 1 ÞaL 2 w jðk w + k w j Þ l 1 ð j Drected assortatvty coeffcent (Newman, h 2002), 2 r Y = l 1 ð jþal kout l 1 1 k out ðjþal 2 k n j h 2 n 2 + k l 1 1 ð jþal 2 kout j l 1 j + k n j h 2 1 ð ÞaL 2 kout + k n j The algorthm s equvalent for weghted and drected networks In weghted networks, weghts may be swtched together wth lnks n ths case, the weghted degree dstrbuton s not preserved, but may be subsequently approxmated on the topologcally randomzed graph wth a heurstc weght reshufflng scheme Measure `of network small-worldness Network small-worldness (Humphres and Gurney, 2008), S = C = C rand L = L rand where C and C rand are the clusterng coeffcents, and L and L rand are the characterstc path lengths of the respectve tested network and a random network Small-world networks often have S 1 Weghted network small-worldness, S w = Cw = C w rand L w = C w rand Drected network small-worldness, S Y = CY = C Y rand L Y = C Y rand In both cases, small-world networks often have S 1 All bnary and undrected measures are accompaned by ther weghted and drected generalzatons Generalzatons that have not been prevously reported (to the best of our knowledge) are marked wth an astersk ( ) The Bran Connectvty Toolbox contans Matlab functons to compute most measures n ths table Appendx B Supplementary data Supplementary data assocated wth ths artcle can be found, n the onlne verson, at do101016/jneuromage200910003 References Achard, S, Bullmore, E, 2007 Effcency and cost of economcal bran functonal networks LoS Comput Bol 3, e17 Achard, S, Salvador, R, Whtcher, B, Sucklng, J, Bullmore, E, 2006 A reslent, lowfrequency, small-world human bran functonal network wth hghly connected assocaton cortcal hubs J Neurosc 26, 63 72 Alstott, J, Breakspear, M, Hagmann,, Cammoun, L, Sporns, O, 2009 Modelng the mpact of lesons n the human bran LoS Comput Bol 5, e1000408 Barabas, AL, Albert, R, 1999 Emergence of scalng n random networks Scence 286, 509 512 Barrat, A, Barthelemy, M, astor-satorras, R, Vespgnan, A, 2004 The archtecture of complex weghted networks roc Natl Acad Sc U S A 101, 3747 3752 Bassett, DS, Bullmore, E, 2006 Small-world bran networks Neuroscentst 12, 512 523 Bassett, DS, Bullmore, ET, 2009 Human bran networks n health and dsease Curr Opn Neurol 22, 340 347 Bassett, DS, Meyer-Lndenberg, A, Achard, S, Duke, T, Bullmore, E, 2006 Adaptve reconfguraton of fractal small-world human bran functonal networks roc Natl Acad Sc U S A 103, 19518 19523 Bassett, DS, Bullmore, E, Verchnsk, BA,Mattay, VS, Wenberger, DR, Meyer-Lndenberg, A, 2008 Herarchcal organzaton of human cortcal networks n health and schzophrena J Neurosc 28, 9239 9248 Batagelj, V, Mrvar, W, 2003 ajek analyss and vsualzaton of large networks In Jünger, M, Mutzel, (Eds), Graph Drawng Software Sprnger, Berln, pp 77 103 Blondel, VD, Gullaume, J-L, Lambotte, R, Lefebvre, E, 2008 Fast unfoldng of communtes n large networks J Stat Mech 2008, 10008 Boccalett, S, Latora, V, Moreno, Y, Chavez, M, Hwang, DU, 2006 Complex networks Structure and dynamcs hys Rep 424, 175 308 Brandes, U, 2001 A faster algorthm for betweenness centralty J Math Socol 25, 163 177 Bullmore, E, Sporns, O, 2009 Complex bran networks graph theoretcal analyss of structural and functonal systems Nat Rev, Neurosc 10, 186 198 Butts, CT, 2009 Revstng the foundatons of network analyss Scence 325, 414 416 Costa, LdF, Kaser, M, Hlgetag, C, 2007a redctng the connectvty of prmate cortcal networks from topologcal and spatal node propertes BMC Syst Bol 1, 16 Costa, LDF, Rodrgues, FA, Traveso, G, Boas, RV, 2007b Characterzaton of complex networks a survey of measurements Adv hys 56, 167 242 Damoseaux, JS, Grecus, MD, 2009 Greater than the sum of ts parts A revew of studes combnng structural connectvty and restng-state functonal connectvty Bran Struct Funct 213, 525 533 (Epub 2009 Jun 30) Danon, L, Daz-Gulera, A, Duch, J, Arenas, A, 2005 Comparng communty structure dentfcaton J Stat Mech 2005, 09008 Deuker, L, Bullmore, ET, Smth, M, Chrstensen, S, Nathan, J, Rockstroh, B, Bassett, DS, 2009 Reproducblty of graph metrcs of human bran functonal networks NeuroImage 47, 1460 1468 Estrada, E, Hatano, N, 2008 Communcablty n complex networks hys Rev, E Stat Nonlnear Soft Matter hys 77, 036111 Fagolo, G, 2007 Clusterng n complex drected networks hys Rev, E Stat Nonlnear Soft Matter hys 76, 026107 Freeman, LC, 1978 Centralty n socal networks conceptual clarfcaton Soc Netw 1, 215 239
M Rubnov, O Sporns / NeuroImage 52 (2010) 1059 1069 1069 Frston, KJ, 1994 Functonal and effectve connectvty n neuromagng a synthess Hum Bran Mapp 2, 56 78 Frston, KJ, Harrson, L, enny, W, 2003 Dynamc causal modellng NeuroImage 19, 1273 1302 Grvan, M, Newman, MEJ, 2002 Communty structure n socal and bologcal networks roc Natl Acad Sc U S A 99, 7821 7826 Glech, D, 2008 Matlabbgl (verson 40) Gumera, R, Amaral, LAN, 2005 Cartography of complex networks modules and unversal roles J Stat Mech 2005, 02001 Hagberg, AA, Schult, DA, Swart, J, 2008 Explorng network structure, dynamcs, and functon usng networkx In Varoquaux, G, Vaught, T, Mllman, J (Eds), roceedngs of the 7th ython n Scence Conference (Scy2008) asadena, CA USA, pp 11 15 Hagmann,, Cammoun, L, Ggandet, X, Meul, R, Honey, CJ, Wedeen, VJ, Sporns, O, 2008 Mappng the structural core of human cerebral cortex LoS Bol 6, e159 He, Y, Chen, ZJ, Evans, AC, 2007 Small-world anatomcal networks n the human bran revealed by cortcal thckness from MRI Cereb Cortex 17, 2407 2419 Honey, CJ, Kotter, R, Breakspear, M, Sporns, O, 2007 Network structure of cerebral cortex shapes functonal connectvty on multple tme scales roc Natl Acad Sc U S A 104, 10240 10245 Honey, CJ, Sporns, O, Cammoun, L, Ggandet, X, Thran, J, Meul, R, Hagmann,, 2009 redctng human restng-state functonal connectvty from structural connectvty roc Natl Acad Sc U S A 106, 2035 2040 Horwtz, B, 2003 The elusve concept of bran connectvty NeuroImage 19, 466 470 Humphres, MD, Gurney, K, 2008 Network small-world-ness a quanttatve method for determnng canoncal network equvalence LoS ONE 3, e0002051 Ioanndes, AA, 2007 Dynamc functonalconnectvty Curr Opn Neurobol 17, 161 170 Kntal, S, 2008 Betweenness centralty algorthms and lower bounds arxv, 08091906v0802 Latora, V, Marchor, M, 2001 Effcent behavor of small-world networks hys Rev Lett 87, 198701 Lecht, EA, Newman, ME, 2008 Communty structure n drected networks hys Rev Lett 100, 118703 Lestedt, SJ, Coumans, N, Dumont, M, Lanquart, J, Stam, CJ, Lnkowsk,, 2009 Altered sleep bran functonal connectvty n acutely depressed patents Hum Bran Mapp 30, 2207 2219 Leung, CC, Chau, HF, 2007 Weghted assortatve and dsassortatve networks model hysca, A 378, 591 602 Maslov, S, Sneppen, K, 2002 Specfcty and stablty n topology of proten networks Scence 296, 910 913 Mlo, R, Shen-Orr, S, Itzkovtz, S, Kashtan, N, Chklovsk, D, Alon, U, 2002 Network motfs smple buldng blocks of complex networks Scence 298, 824 827 Newman, MEJ, 2002 Assortatve mxng n networks hys Rev Lett 89, 2087011 2087014 Newman, MEJ, 2003 The structure and functon of complex networks SIAM Rev 45, 167 256 Newman, MEJ, 2004a Analyss of weghted networks hys Rev, E Stat Nonlnear Soft Matter hys 70, 056131 Newman, MEJ, 2004b Fast algorthm for detectng communty structure n networks hys Rev, E 69, 066133 Newman, MEJ, 2006 Modularty and communty structure n networks roc Natl Acad Sc U S A 103, 8577 8582 NWB-Team, 2006 Network workbench tool Indana unversty, Northeastern Unversty, and Unversty of Mchgan Onnela, J, Saramak, J, Kertesz, J, Kask, K, 2005 Intensty and coherence of motfs n weghted complex networks hys Rev, E Stat Nonlnear Soft Matter hys 71, 065103 alla, G, Dereny, I, Farkas, I, Vcsek, T, 2005 Uncoverng the overlappng communty structure of complex networks n nature and socety Nature 435, 814 818 astor-satorras, R, Vazquez, A, Vespgnan, A, 2001 Dynamcal and correlaton propertes of the nternet hys Rev Lett 87, 258701 raveenan, M, rokopenko, M, Zomaya, AY, 2008 Local assortatveness n scale-free networks Europhys Lett 84, 28002 onten, SC, Douw, L, Bartolome, F, Rejneveld, JC, Stam, CJ, 2009 Indcatons for network regularzaton durng absence sezures weghted and unweghted graph theoretcal analyses Exp Neurol 217, 197 204 Saramak, J, Kvela, M, Onnela, J, Kask, K, Kertesz, J, 2007 Generalzatons of the clusterng coeffcent to weghted complex networks hys Rev, E Stat Nonlnear Soft Matter hys 75, 027105 Sharan, R, Ideker, T, 2006 Modelng cellular machnery through bologcal network comparson Nat Botechnol 24, 427 433 Sporns, O, Kotter, R, 2004 Motfs n bran networks LoS Bol 2, e369 Sporns, O, Zw, JD, 2004 The small world of the cerebral cortex Neuronformatcs 2, 145 162 Sporns, O, Honey, CJ, 2006 Small worlds nsde bg brans roc Natl Acad Sc U S A 103, 19219 19220 Sporns, O, Tonon, G, Kotter, R, 2005 The human connectome a structural descrpton of the human bran LoS Comput Bol 1, e42 Stam, CJ, Rejneveld, JC, 2007 Graph theoretcal analyss of complex networks n the bran Nonlnear Bomed hys 1, 3 Stam, CJ, Jones, BF, Nolte, G, Breakspear, M, Scheltens,, 2007 Small-world networks and functonal connectvty n Alzhemer's dsease Cereb Cortex 17, 92 99 Stam, CJ, de Haan, W, Daffertshofer, A, Jones, BF, Manshanden, I, van Cappellen van Walsum, AM, Montez, T, Verbunt, J, de Munck, JC, van Djk, BW, Berendse, HW, Scheltens,, 2009 Graph theoretcal analyss of magnetoencephalographc functonal connectvty n Alzhemer's dsease Bran 132, 213 224 Strogatz, SH, 2001 Explorng complex networks Nature 410, 268 276 Tonon, G, Sporns, O, Edelman, GM, 1994 A measure for bran complexty relatng functonal segregaton and ntegraton n the nervous system roc Natl Acad Sc U S A 91, 5033 5037 Wang, J, Wang, L, Zang, Y, Yang, H, Tang, H, Gong, Q, Chen, Z, Zhu, C, He, Y, 2009a arcellaton-dependent small-world bran functonal networks a restng-state fmri study Hum Bran Mapp 30, 1511 1523 Wang, L, Zhu, C, He, Y, Zang, Y, Cao, Q, Zhang, H, Zhong, Q, Wang, Y, 2009b Altered small-world bran functonal networks n chldren wth attenton-defct/hyperactvty dsorder Hum Bran Mapp 30, 638 649 Watts, DJ, Strogatz, SH, 1998 Collectve dynamcs of small-world networks Nature 393, 440 442 Zhou, C, Zemanova, L, Zamora, G, Hlgetag, CC, Kurths, J, 2006 Herarchcal organzaton unveled by functonal connectvty n complex bran networks hys Rev Lett 97, 238103 Zhou, D, Thompson, WK, Segle, G, 2009 Matlab toolbox for functonal connectvty Neuromage 2009 Oct 147(4)1590-1607 Epub 2009 Jun 8