brain graphs: graphical models of the human brain ......cp07ch05-bullmore ari 9 march 2011 21:34...

CP07CH05-Bullmore ARI 9 March 2011 21:34

Brain Graphs: GraphicalModels of the HumanBrain ConnectomeEdward T. Bullmore1 and Danielle S. Bassett2

1Behavioural & Clinical Neuroscience Institute, Department of Psychiatry, University ofCambridge, Cambridge Biomedical Campus, Cambridge CB2 0SZ, United Kingdom;email: [email protected] of Physics, University of California, Santa Barbara, Santa Barbara,California 93106

Annu. Rev. Clin. Psychol. 2011. 7:113–40

First published online as a Review in Advance onDecember 3, 2010

The Annual Review of Clinical Psychology is onlineat clinpsy.annualreviews.org

This article’s doi:10.1146/annurev-clinpsy-040510-143934

Copyright c© 2011 by Annual Reviews.All rights reserved

548-5943/11/0427-0113$20.00

Keywords

network, systems, topological, connectome, connectivity,neuroimaging

Abstract

Brain graphs provide a relatively simple and increasingly popular wayof modeling the human brain connectome, using graph theory to ab-stractly define a nervous system as a set of nodes (denoting anatomi-cal regions or recording electrodes) and interconnecting edges (denot-ing structural or functional connections). Topological and geometricalproperties of these graphs can be measured and compared to randomgraphs and to graphs derived from other neuroscience data or other(nonneural) complex systems. Both structural and functional humanbrain graphs have consistently demonstrated key topological propertiessuch as small-worldness, modularity, and heterogeneous degree dis-tributions. Brain graphs are also physically embedded so as to nearlyminimize wiring cost, a key geometric property. Here we offer a con-ceptual review and methodological guide to graphical analysis of humanneuroimaging data, with an emphasis on some of the key assumptions,issues, and trade-offs facing the investigator.

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Graph: a model of acomplex systemcompletely defined bya set of nodes orvertices and the edgesor lines drawn betweenthem; mathematicaltheory of randomgraphs originated withEuler and later Erdosand Renyi in the1950s. Analysis ofnonrandom graphs hasgrown rapidly as anaspect of complexityscience in the past20 years

Contents

WHAT IS A BRAIN GRAPH?. . . . . . . . 114WHY BOTHER WITH BRAIN

GRAPHS AS MODELS OF THEHUMAN BRAINCONNECTOME? . . . . . . . . . . . . . . . . 114Generalizability . . . . . . . . . . . . . . . . . . . . 114Interpretability . . . . . . . . . . . . . . . . . . . . 116Clinical Relevance . . . . . . . . . . . . . . . . . 117Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

HOW TO CONSTRUCT ABRAIN GRAPH . . . . . . . . . . . . . . . . . . . 118What Is a Node? . . . . . . . . . . . . . . . . . . . 118What Is an Edge? . . . . . . . . . . . . . . . . . . 120

MEASURES ON GRAPHS. . . . . . . . . . . 127Topological Measures . . . . . . . . . . . . . . 127Geometric Measures . . . . . . . . . . . . . . . 129

COMPARING AND VISUALIZINGBRAIN GRAPHS . . . . . . . . . . . . . . . . . 130Comparing Graphs . . . . . . . . . . . . . . . . 130Visualizing Graphs . . . . . . . . . . . . . . . . . 131

BEYOND THE SIMPLESTGRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . 131Directional Connections

and Causal Relationships . . . . . . . . 131Weighted Network Analysis . . . . . . . . 132

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . 133

WHAT IS A BRAIN GRAPH?

A brain graph is a model of a nervous systemas a number of nodes interconnected by a setof edges. For example, the edges can repre-sent functional or structural connections be-tween cortical and subcortical regional nodesbased on analysis of human neuroimaging data.Once a brain graph has been constructed bydefining the nodes and edges, its topologicalproperties can be measured by a rich array ofmetrics that has been developed recently in thefield of statistical physics of complex networks(Albert & Barabasi 2002) and historically builton the concepts of graph theory (Erdos &Renyi 1959). Since the nodes of a brain graphcan be spatially localized, or physically em-

bedded, its geometrical properties can also beestimated and potentially related to networktopology.

To date, most such human brain graphshave specified binary connectivity—the edgesbetween nodes are undirected and unweighted;see Figure 1. The construction of such binarybrain graphs is the focus of this article, althoughwe also briefly describe the construction of di-rected and/or weighted brain graphs.

WHY BOTHER WITH BRAINGRAPHS AS MODELS OF THEHUMAN BRAIN CONNECTOME?

Brain graphs are simple models of the real un-derlying connectome (Sporns et al. 2005). Theyare properly based on a number of more-or-less explicit, and more-or-less realistic, tech-nical assumptions. For example, we will usu-ally assume that the nodes are independent andinternally coherent, and we may also assumethat all the edges signify the same strength ofconnection between nodes. Such assumptionsinevitably entail some loss of information inthe resulting graphs compared to the multivari-ate datasets from which they were constructed.However, the technical constraints and robustsimplifications of graph theoretical analysis, ap-plied to human neuroimaging data, are worthit—arguably—for two strategic reasons: gener-alizability and interpretability.

Generalizability

Graph theoretical analysis is potentially ap-plicable to any scale, modality, or volumeof neuroscientific data (Bassett & Bullmore2006, Bullmore & Sporns 2009, Sporns,2010). We can say this with some confidencebecause graph theory has already proven to beapplicable to a considerable diversity of com-plex systems, including markets, ecosystems,computer circuits, and gene-gene interactomes(Barabasi 2009). Some of these systems havebeen graphically modeled on a much biggerscale than has so far been attempted for ner-vous systems. For example, graph models have

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been constructed for the World Wide Webcomprising up to 200 million nodes (Websites)and 1.5 billion edges (links) (Barabasi & Albert1999, Broder et al. 2000); and for the brain genetranscriptome, comprising up to 20,000 nodes,each representing expression of geneticallyspecific mRNA, with each edge representingsignificant coexpression of mRNA related toa pair of genes (Oldham et al. 2006). Thus,graph theory potentially provides a commonlanguage for the analysis of complex systemsin general, and we expect to be able to use it todescribe some of the key topological propertiesof nervous systems from the cellular scale of theneuronal connectome, exemplified by that ofthe nematode worm, Caenorhabditis elegans, tothe whole-brain scale of human neuroimagingdata.

Within the domain of human brain map-ping, graph models have now been reportedfor all major modalities of magnetic resonanceimaging (MRI) and neurophysiological data.Functional brain graphs have been constructedfrom functional MRI (fMRI) (Achard &Bullmore 2007, Achard et al. 2006, Eguıluzet al. 2005, Liu et al. 2008, Salvador et al. 2005a,van den Heuvel et al. 2008), electroencephalog-raphy (EEG) (Micheloyannis et al. 2006, Stamet al. 2007a), and magnetoencephalography(MEG) data (Bassett et al. 2006, Deuker et al.2009, Stam 2004). Structural brain graphshave been constructed from diffusion tensorimaging (DTI) or diffusion spectrum imaging(DSI) (Gong et al. 2008, Hagmann et al. 2008)and conventional MRI data (Bassett et al.2008, He et al. 2007). This degree of gener-alizability immediately supports comparison oftopological parameters between structural andfunctional networks. For example, it has beendiscovered that fMRI and DTI brain graphsconsistently demonstrate some common globaltopological properties (Honey et al. 2009, Parket al. 2008, Skudlarski et al. 2008, Zalesky &Fornito 2009; see Figure 1), including:

� small-worldness—indicating a balancebetween network segregation andintegration,

Topology: spatialproperties that areinvariant undercontinuousdeformation aretopological aspects of asystem. In braingraphs, topologicalanalysis considers theconnectivity betweennodes regardless oftheir physical oranatomical locations

Connectivity: ameasure of associationbetween neurons orbrain regions. Inhuman neuroimaging,functional connectivitymeans that two regionsdemonstrate similardynamics over time,whereas effectiveconnectivity meansthat one region has acausal effect ondynamics in anotherregion

Degree: the degree ofa node is the numberof edges connecting itto the rest of thenetwork; thedistribution of degreesover all nodes in thenetwork can bedescribed as a degree(probability)distribution. Braingraphs typically have abroad-scale degreedistribution, implyingthat at least a few“hub” nodes will havehigh degree

� modularity—indicating a decomposabil-ity of the system into smaller subsystems,and

� heterogeneous degree distributions—broad-scale or fat-tailed probability dis-tributions of degree, indicating the likelypresence of network hubs or highly con-nected nodes.

It is conceptually easier to link the braingraphs derived from these different data typesto each other than it would be if each imag-ing dataset were described in terms of somemodality-specific measure of association be-tween regions, e.g., tractographic connectionprobabilities from DSI or correlations be-tween regional fMRI time series. Facilitat-ing between-modality translation of resultscan be important for methodological cross-validation and, more fundamentally, for in-forming our understanding of how functionalnetworks might interact with the substrate of arelatively static structural network (Honey et al.2009).

The generalizability of graph theory alsoallows us to compare the topological propertiesof large-scale or macro networks representedby neuroimaging to those of small-scale orcellular networks measured by microscopy ormicroelectrode recording. This is potentiallyimportant in defining organizational propertiesof nervous systems that are conserved acrossscales of space and time, and across differentspecies. More radically, the generalizability ofgraph theory encourages the translation of newideas from analysis of nonneural complex sys-tems, such as principles of high-performancemicroprocessor design, to quantification of thehuman brain connectome (Bassett et al. 2010).We can thus begin to address questions such as,what is special about the human connectomecompared to a variety of other complex,information-processing systems? Do humanbrain networks represent a singular pinnacleof organizational complexity or are they oneof a universality class of superficially diversenetworks that share important topologicalproperties in common? These questions aboutconservation of topological principles across

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Small-worldness:originallydefined as thecombination of highclustering and shortcharacteristic pathlength; subsequentlyalso definedas the combination ofhigh global and localefficiency of informa-tion transfer betweennodes of a network

Modularity: modularnetworks are thosethat are composed oftopological modules orcommunities. Amodule is defined as agroup of nodes thathave manyconnections to othernodes within themodule but fewconnections to nodesoutside the module

Cost: a topologicaldefinition of the cost ofa system is the numberof edges in proportionto the total number ofpossible edges; aphysical definitionweights the cost by thegeometric distancebetween nodes. Brainnetworks economize(but do not minimize)both topological andphysical costs

Efficiency: a wordwith many meaningsbut usually understoodas a measure ofinformation transferbetween nodes; anetwork with highglobal efficiency willhave a shortcharacteristic pathlength. Cost-efficiencyrelates the efficiency ofthe graph to its cost orconnection density

information-processing systems are potentiallyimportant because topological conservationsuggests that diverse systems have conver-gently evolved to satisfy universal optimizationcriteria. The identity and interdependence ofsuch putative network selection criteria are notyet fully established, but plausible candidatesinclude minimization of wiring cost (a geo-metric measure of physical distance betweenconnected nodes), maximization of efficiencyof information transfer (a topological measureinversely related to path length between nodes),hierarchical modularity, and high dimensionalinterconnect topology (Bassett et al. 2010,Robinson et al. 2009). Through the prism ofgraph theory, we can begin to test ideas aboutevolution and development of human brainnetworks that are informed by what we knowabout the selection of many other, perhapsexperimentally more tractable, networks.

Interpretability

Biological and behavioral interpretability isalways an issue in human neuroimaging. Theanatomical and physiological significance ofstructural and functional MRI signals hasbeen extensively debated but not yet entirelyresolved (Lee et al. 2010, Lerch et al. 2006).Imaging studies of brain systems or networksinevitably depend on some measure of signalassociation or covariation between regions,but the neurobiological basis is not yet settledfor either interregional correlations in brainstructure estimated over subjects in MRI orinterregional correlations in brain functionestimated over time in fMRI. Lacking aclear structural or physiological substrate forchanges in anatomical or functional connectiv-ity, measured by statistical association betweenregions at the systems level of neuroimaging,it is difficult to predict how changes in suchdescriptive statistics should be related to cog-nitive or behavioral performance of the system.For example, is it cognitively “good” or “bad”to have a greater-than-average correlationbetween a pair of fMRI time series represent-

ing, say, left middle frontal gyrus and righthippocampus? Similarly, it is difficult to explainwhy specific neuropsychiatric disorders, e.g.,schizophrenia, have often been associated witha profile of both abnormally increased anddecreased magnitude of connectivity acrossdifferent brain regions (Rubinov et al. 2009,Whitfield-Gabrieli et al. 2009). Is less connec-tivity always a sign of the disease process, andmore connectivity in a patient group always asign of a compensatory process, or can excessconnectivity be directly pathological?

The translation of modality-specific con-nectivity statistics to topological measureson brain graphs may help us to find moresecure cognitive and clinical interpretationsof neuroimaging systems. This proposition isfar from proven yet, although there are someencouraging early signs in its favor. For exam-ple, three recent studies—using fMRI (van denHeuvel et al. 2009), MEG (Bassett et al. 2009),and DTI (Li et al. 2009)—have indepen-dently reported associations between generalintelligence or executive task performanceand topological measures of brain networkefficiency or cost-efficiency. In general, highercognitive performance has been associated withbrain graphs globally configured for greaterefficiency—speed and fidelity—of parallelinformation transfer between regional nodes.This observation is compatible with neuropsy-chological theories that higher-order cognitivefunctions depend on distributed processing(Fodor 1983) across a large, integrated “neu-ronal workspace” (Dehaene & Naccache 2001):A network with higher global efficiency willbe more optimized as a cognitive workspace.There have also been early reports that disease-related changes in topological properties ofbrain graphs can be related to other aspects ofthe disorder in question. For example, reduc-tions in network efficiency have been associatedwith greater white matter lesion load in pa-tients with multiple sclerosis (He et al. 2009a),and reductions in nodal degree (the number ofedges connecting a regional node to the restof the brain graph) have been associated with


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greater severity of local amyloid depositionin patients with Alzheimer’s disease (Buckneret al. 2009). It is also notable that moderatelevels of heritability have been reported forbrain graph topology measured in a twin studyusing EEG (Smit et al. 2008), suggesting thatthere may be important genetic effects onvariation in brain graph metrics. Collectively,these and other early results suggest that it willbe interesting to apply graph theory more ex-tensively to the cognitive, clinical, and geneticinterpretation of neuroimaging systems.

Clinical Relevance

The organization of brain graphs is modulatedby an array of factors that varies throughout thehealthy population, including behavioral vari-ability (Bassett et al. 2009), cognitive ability (Liet al. 2009, van den Heuvel et al. 2009), sharedgenetic factors (Smit et al. 2008), genetic infor-mation (Schmitt et al. 2008), age (Meunier et al.2008, Micheloyannis et al. 2009), and gender(Gong et al. 2009). The architecture of an indi-vidual’s connectivity is inherently dynamic, be-ing altered by experimental tasks (Bassett et al.2006, de Vico Fallani et al. 2008a) and drugtreatment (Achard & Bullmore 2007, Schwarzet al. 2009).

Complex network theory is particularlyappealing when applied to the study of clinicalneuroscience, where many cognitive andemotional disorders have been characterized asdysconnectivity syndromes (Catani & ffytche2005), as indicated by abnormal phenotypicprofiles of anatomical and/or functional con-nectivity between brain regions. For example,in schizophrenia, a profound disconnectionbetween frontal and temporal cortices has beensuggested to characterize the brain (Friston &Frith 1995); in contrast, people with autism maydisplay a complex pattern of hyperconnectivitywithin frontal cortices but hypoconnectivity be-tween the frontal cortex and the rest of the brain(Courchesne & Pierce 2005). In fact, a wealthof clinical and disease states have recently beenshown to manifest themselves by abnormal

cortical graph organization: schizophrenia(Bassett et al. 2008, 2009; Liu et al. 2008;Lynall et al. 2010; Micheloyannis et al. 2006;Rubinov 2009), Alzheimer’s disease (He et al.2008, 2009a; Stam 2010; Stam et al. 2007a;Supekar et al. 2008), epilepsy (Horstmann et al.2010, Raj et al. 2010, van Dellen et al. 2009),multiple sclerosis (He et al. 2009b), acute de-pression (Leistedt et al. 2009), absence seizures(Ponten et al. 2009), medial temporal lobeseizures (Ponten et al. 2007), attention deficithyperactivity disorder (Wang et al. 2010),stroke (de Vico Fallani et al. 2009, Wang et al.2009), spinal cord injury (de Vico Fallani et al.2008b), fronto-temporal lobar degeneration(de Haan et al. 2009), and early blindness (Shuet al. 2009). Together, these studies highlightthe extended clinical relevance of graphicalanalysis of human neuroimaging data.

Caveats

So far we have emphasized the attractive sim-plicity, generalizability, interpretability, andclinical relevance of brain graphs. Now we re-turn to our major caveat: graph analysis of neu-roimaging data is not “plug and play.” It is amodel building exercise, entailing arbitrary as-sumptions and decisions, which can have influ-ential effects on the results of the analysis. Itis also relativistic: Many of the results from abrain graph will need to be calibrated by com-parison to the extreme bounds of random andregular graphs (see Figure 1), e.g., to quantifysmall-worldness of the brain networks, or wemay wish to compare one group of brain graphsto another. How best to compare topologicalmetrics between graphs is not a trivial ques-tion. In addition to these relatively specialistquestions about construction and comparisonof brain graphs, any approach to systems anal-ysis of neuroimaging data also raises a numberof issues about data acquisition and preprocess-ing, statistical hypothesis testing and multiplecomparisons, visualization, etc. In what follows,we attempt to address some of these caveats ingreater detail.


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HOW TO CONSTRUCT ABRAIN GRAPH

The two key questions to address in construct-ing a graphical model of a brain network are,(a) What is a node, and (b) What is an edge?

What Is a Node?

Before we describe the previously used defini-tions of nodes in neuroscience and neuroimag-ing, it is important to describe what propertiesa node should have in general (Butts 2009). Ingraphical models, a node is a portion of thesystem that is separable from the other por-tions of the system in some way; i.e., nodes aremeant to be inherently independent or distinctin the system under study. To put it anotherway, the interactions between nodes will be-come continuously less meaningful the moresimilar the nodes are to each other. In additionto being independent, the nodes should be in-ternally coherent or homogeneous, i.e., nodesshould be encapsulated informational compo-nents that have internal integrity and externalindependence (Butts 2008, 2009; Rubinov &Sporns 2010).

Cellular systems. A cellular nervous systemhas an immediately obvious graphical decon-struction: each neuron can be considered anindependent and homogeneous node, and thesynapses between neurons can be consideredas edges. This is the intuitively straightforwardbasis for the graphical analysis of the cellularconnectome of the nematode worm, C. elegans(Bassett et al. 2010, Watts & Strogatz 1998),which comprises about 300 cellular nodes andabout 7,600 synaptic edges. It has also been thebasis for graphical modeling of small-worldcellular networks in vertebrate brainstem(Humphries et al. 2006). However, we notethat C. elegans is currently the only organismfor which the cellular connectome has beencompletely described; graphical modeling ofcellular connectomes in other species will bechallenged by their greater size and variability,and even in this most tractable case, graphical

analysis involves making choices about whetherall types of neurons should have equal statusas nodes [more than 100 classes of neurons arerecognized in C. elegans (White et al. 1986)].

Electrophysiological data. For graphicalanalysis of neurophysiological data on elec-tromagnetic fields, including microelectroderecordings from cortical tissue (Yu et al. 2008)as well as scalp electrode recordings in EEG orsurface sensors in MEG, it may be reasonableto define each electrode or signal sensor as anode. Defining nodes as MEG or EEG sensorswill preserve the native covariance structure ofthe data, but without appropriate preprocess-ing (Stam et al. 2007b), this will include strongcorrelations between neighboring sensors dueto volume conduction of electrical activity froma single source in the brain to multiple nearbyelectrodes or sensors on the scalp surface. Ingraphical terms, untreated volume conductionwill be represented by a regular lattice-likestructure of highly clustered connectionsbetween spatially neighboring sensors, whichcould clearly confound analysis of brainnetwork properties such as small-worldness.

An alternative approach is to reconstruct thesensor data in terms of anatomically locatedsources and define each source as a node (Palvaet al. 2010a,b). This allows the operator somediscretion in how many nodal sources shouldbe reconstructed from a fixed number of sen-sors, and it deals with the issue of volume con-duction. However, it is also important to bearin mind that some source reconstruction algo-rithms, such as the synthetic aperture magne-tometry beamformer, solve the inverse problemby diagonalizing the sensor covariance matrix torender sources as statistically independent fromeach other (see, e.g., Cheyne et al. 2006). Thisis obviously unlikely to represent an optimalstarting point for an analysis of functional con-nectivity between sources. Other reconstruc-tion algorithms assume stationarity (constantmean and variance over time) of the time series,which is unlikely to be realistic over the dura-tion of a typical experimental recording (from


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tens to thousands of seconds) (Kowalik & Elbert1994).

Thus we can see that a preprocessing step(source reconstruction) that might seem attrac-tive for defining the nodes of an MEG networkcould have severe effects on the covariance be-tween nodes that will later be used to definethe edges of the network (Vrba & Robinson2001). Most graphical studies of neurophysio-logical data so far have used sensors as nodes(Bassett et al. 2006, 2009; Stam 2004, 2010),implicitly sacrificing anatomical resolution andindependence of nodes for greater data fi-delity of edges representing covariation be-tween nodes. This trade-off has not been ex-tensively studied in terms of its impact onstatistical properties of graph metrics derivedfrom neurophysiological data. It seems clearfrom preliminary work that headline prop-erties of human brain functional networks—such as small-worldness and broad-scale de-gree distributions—are qualitatively conservedwhether network nodes are defined as sources(Palva et al. 2010b) or sensors (Deuker et al.2009), but this is an area where there is likely tobe further significant methodological develop-ment in the future.

Tract-tracing and MRI data. For graphi-cal analysis of tract-tracing data on large-scaleaxonal projections between regions of mam-malian cortex, nodes have usually been de-fined cytoarchitectonically as Brodmann areas(Hilgetag et al. 2000; Scannell et al. 1995, 1999;Young 1992, 1993). In some of the early graph-ical studies of human neuroimaging data, anapproximately equivalent convention was fol-lowed to define nodes (Achard et al. 2006,Bassett et al. 2008, Wang et al. 2008). Each in-dividual’s brain image was coregistered with ananatomically parcellated template image, andthe mean signal over all voxels in each region ofthe template image was taken as the nodal valueat that anatomical location. Several similar butnot identical template images are available (seeRelated Resources) and have been used for thispurpose. It has been shown that use of differentanatomical templates for analysis of the same

imaging data can lead to subtly different graph-ical results (Wang et al. 2010). It has also beensuggested by analysis of simulated fMRI datathat specification of functionally, rather thananatomically, defined nodes may improve fi-delity of functional network modeling (Smithet al. 2010).

The main advantage of using an anatomi-cally defined template for nodal parcellation ofneuroimaging data is that it can support directcomparison of results to prior neuroimaging orprimate neuroanatomy studies using the sameor a similar template. The main disadvantage,which is probably more significant, is thatthe size of different template regions can varyconsiderably. For fMRI analysis, for examplein the Automated Anatomical Labeling (AAL)template (Tzourio-Mazoyer et al. 2002)(Figure 2), the middle frontal gyrus regioncomprises many more voxels than the hip-pocampus region. Nodal values obtained byaveraging across many voxels in larger regionswill be less noisy than nodal values estimated byaveraging across smaller regions, leading to abias in favor of stronger statistical associationsbetween larger regions of the template image(Salvador et al. 2008). One way of dealingwith this bias is to randomly sample the samenumber of voxels in estimation of each nodalvalue, but this is likely to mean that largernodes are less internally coherent.

An increasingly preferred approach to defi-nition of nodes in fMRI data has been to stip-ulate that each node should comprise an equalnumber of voxels and that nodes should collec-tively cover the brain without prior anatomicalinformation being used to guide their size orlocation (Zalesky et al. 2010b). If this approachis adopted, the key parameter to choose is thespatial size of each node, which can range froma minimum of 1 voxel to a maximum in the or-der of 104 voxels. We can see immediately thateach spatial scale represents a trade-off betweenthe defining nodal properties of independenceand coherence: Single voxel nodes will be morecoherent but less independent of each other (tothe extent that the image is spatially smooth),whereas larger nodes, located regardless of


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approximate cytoarchitectonic fields, will beless coherent but more independent.

There has been limited systematic explo-ration of the effect of nodal size on overallnetwork properties. A few fMRI studiesadopting the minimum nodal size of one voxelhave reported scale-free or power law degreedistributions of the resulting networks (Eguıluzet al. 2005, van den Heuvel et al. 2008), whereasthe majority of fMRI and MRI studies usinganatomical templates with larger nodes havereported exponentially truncated power lawdegree distributions (e.g., Achard et al. 2006).This indicates that some network properties,such as the form of the degree distribution,may be qualitatively affected by the size of thenodes—in this case, voxel nodes leading to net-works with a larger probability of very highlyconnected hubs. It is not yet clear whether thisapparent difference in degree distributionalproperties as a function of the spatial scale ofnetwork nodes reflects a biological differencein network organization at different spatialscales or the reduced independence of voxelnodes that are small compared to the spatialsmoothness of the image. However, a studyof anatomical networks derived from the sameset of DTI and DSI data as nodal size wascontinuously increased from small nodes,comprising a local cube of {3 × 3 × 3} voxels,to large nodes, comprising {100 × 100 ×100} voxels, found that the form of the degreedistribution and other topological propertieswere quite consistently expressed across scales,although the quantitative values of topologicalparameters were scale dependent (Zalesky et al.2010b).

Another factor to take into considerationwhen deciding the size of nodes is that smallernodes will naturally be more numerous, and hy-pothesis testing of nodal statistics, e.g., to pro-duce a cortical surface map of between-groupdifferences in nodal degree, will require a largernumber of statistical tests and therefore morestringent multiple comparisons corrections. Inother words, the greater spatial resolution oflocal network properties conferred by smallernodes comes at the cost of more conservative

significance thresholds to control type I (falsepositive) error in the context of multiple com-parisons. This is a familiar trade-off in the con-text of mass univariate statistical testing for clas-sical brain activation mapping.

What Is an Edge?

We have seen that there is no absolutely corrector straightforward answer to the question ofdefining nodes. The question of how to defineedges in a brain graph is even more open toa variety of legitimate choices. As before, thecase of C. elegans can provide a deceptivelysimple template: The edges in this cellularconnectome are usually defined as the synapsesbetween neurons, which are highly reliablebetween different worms and have been exactlydescribed (White et al. 1986). But shouldelectrical and chemical synapses be given equalweight as edges? Should the directionality ofchemical synapses be respected by specificationof directed edges in the corresponding graph?In practice, most graphical analyses of C. eleganshave adopted the simplest possible choice—allsynaptic connections are represented as undi-rected and unweighted (or equally weighted)edges—but this is clearly not the only possiblechoice. When we turn to the graphical analysisof neuroimaging, neurophysiological, andother data where we lack a “ground truth”knowledge of the physical connections be-tween nodes, a reasonable specification ofedges becomes more challenging.

To explore the methodological issuesin this context, we consider the illustrativeexample of building a graph model for thehuman functional connectome based on asingle fMRI dataset (see Figure 2A). Severalquestions immediately arise; e.g., under whatexperimental conditions should the data berecorded, and how should they be preprocessedprior to graph analysis? To date, most graphmodels of fMRI data have been based ondata recorded with participants lying quietlyin the scanner, at rest. However, there is noreason in principle why graphs could not beconstructed from data recorded while subjects


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perform experimentally controlled cognitivetasks; indeed, this strategy is likely to becomemore popular in the future, as attention focuseson the question of how graphical parametersof functional network organization can berelated to cognitive performance (see FutureIssues and Palva et al. 2010a for an example oftask-related network analysis using MEG).

For resting-state data analysis, one com-mon question is how long a functional MRItime series should be to support investigationof functional associations and networks. Froma strictly technical point of view, a larger num-ber of time points will generally improve theprecision of time series model parameter esti-mates. The frequency bandwidth of a digitaltime series process is limited by the Nyquist fre-quency ( fN = fS/2, where fS is the samplingfrequency) at the upper end and by the length ofthe time series at the lower end ( fmin = 1/T ,where T is the number of time points in theseries). Within these limits, higher-frequencycomponents will generally be estimated withgreater precision than low-frequency compo-nents (Achard et al. 2008). In fMRI, inter-est is often focused on a low-frequency range<0.1 Hz, which is considered to be more purelyrepresentative of neuronal (noncardiorespira-tory) sources of (co)variation, and the samplinginterval is typically on the order of seconds.These considerations indicate that fMRI timeseries will ideally be recorded over at least 5 to10 minutes (Van Dijk et al. 2010), and longertime series (about 30 minutes) have been re-ported (Achard et al. 2006). However, verylong periods of fMRI recording may be diffi-cult for subjects to tolerate without excessivehead movement or changes in brain state such asfalling asleep. In general, basing functional con-nectivity or network analysis on very long peri-ods of time series, although technically prefer-able, incurs the assumption that brain func-tional systems are in the same (stationary) stateover the period of observation, which becomesincreasingly implausible as the length of obser-vation increases. The assumption of stationar-ity also arises if several short segments of fMRItime series, e.g., the resting-state epochs of a

classical blocked periodic activation paradigm,are concatenated to form a composite time se-ries for connectivity and network analysis (Fairet al. 2007). This procedure assumes that thebrain is in approximately the same functionalstate before and after performance of cogni-tive tasks, although there is some evidence thatendogenous brain dynamics recorded immedi-ately after effortful task performance may bedifferent from the dynamics preceding task per-formance (Barnes et al. 2009).

In addition to the question of time serieslength, there are multiple preprocessing stepscommonly applied in fMRI analysis that willlikely have as yet incompletely characterizedeffects on functional brain graphs. For ex-ample, nodal time series are often correctedfor head movement and for other “nuisancecovariates,” such as global brain or whitematter mean fluctuations, before estimationof association metrics (Poldrack et al. 2008).Such preprocessing steps can significantly alterthe specificity, strength, and localization ofmeasured functional associations and thereforemay substantially alter the topology of braingraphs derived from the association matrices(Murphy et al. 2009, Van Dijk et al. 2010,Weissenbacher et al. 2009).

However, many of these issues concerningdata acquisition and preprocessing are genericto fMRI studies of brain functional connectiv-ity and are not uniquely problematic from theperspective of graph analysis. To focus morespecifically on the issues arising in graph analy-sis, we use a representative single fMRI dataset(results shown in Figure 2A), acquired overthe course of about 37 minutes (equivalent to2,048 time points at each voxel) from a healthyvolunteer in the resting state. The data werepreprocessed to correct voxel times series forhead movement and slice timing offsets, thenregionally parcellated using the AAL template(Tzourio-Mazoyer et al. 2002), which defines90 cortical and subcortical regions; method-ological details are described in Achard et al.(2006). Thus, the dataset available for graphanalysis comprised a {N × T} or {90 × 2048}multivariate time series (see Figure 2A).


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Association matrix: amatrix of connectivitymeasures betweeneach possible pair ofthe nodes in thesystem; applying aglobal threshold to anassociation matrix togenerate a binaryadjacency matrix is thesimplest and mostfrequently usedapproach to graphanalysis of humanneuroimaging data

Functional association. Given a multivariatetime series like an fMRI dataset (or an MEGor EEG or MEA recording), the first signif-icant choice we have to make is how to de-scribe the statistical association between eachnodal time series. In general, we estimate thepair-wise association ai, j between the ith andjth nodes, i �= j = 1, 2, 3, . . . , N and compilethese statistics for all possible pairs in a {N ×N} interregional association matrix, A (seeFigure 2A,B). Many measures of associationcould be used for this purpose, and they can becategorized by various criteria. Following Fris-ton (1994), we can distinguish measures of func-tional connectivity, such as correlation coeffi-cients, from measures of effective connectivity,such as path coefficients. A functional connec-tivity statistic measures the extent to which twoprocesses behave similarly over time; an effec-tive connectivity statistic measures the extent towhich one process can be predicted or explainedby the other. Thus, the association matrix gen-erated by estimating the functional connectivitybetween each pair of nodes will be symmetric,whereas the association matrix generated by aneffective connectivity analysis need not be. Todate, almost all graphical analyses of fMRI datahave been based on a symmetric association ma-trix generated by some measure of functionalconnectivity between nodes; however, there isno reason in principle why methods of effectiveconnectivity analysis, such as path analysis ordynamic causal modeling, could not be used asthe statistical basis for a functional brain graph.

Moreover, this distinction between func-tional and effective connectivity, althoughinfluential, does not exhaust the ways in whichvarious association measures can be distin-guished (David et al. 2004). Some measures offunctional connectivity, such as the correlationcoefficient, will only capture linear interactionsbetween time series, whereas other measures,such as the mutual information (Bassett et al.2009), phase synchronization (Kitzbichleret al. 2009, Palva et al. 2010a), or synchroniza-tion likelihood (Stam 2004), are sensitive toboth linear and nonlinear interactions. Somemeasures are sensitive to association between

nodal time series subtended by a specific fre-quency interval, such as the wavelet correlation(Bullmore et al. 2004) or coherence in thefrequency domain (Salvador et al. 2005b).Some measures, such as the partial correlation(Salvador et al. 2005a) or partial coherence(Salvador et al. 2005b), are particularly sen-sitive to the specific association between eachpair of nodes and will discount third-partyeffects, such as shared inputs from a third nodeor global mode of covariation.

We cannot offer definitive guidance aboutwhich of these or other possible associationstatistics is “best” for the purposes of graphanalysis [though see David et al. (2004) andSmith et al. (2010) for evaluation of multiplepossible connectivity metrics in the contextof simulated data]. But we can offer somegeneral suggestions about how to deal withthis moment of choice. First, it is wise to bearin mind the nature and limitations of the data,and the hypothetical question of interest. Inour illustrative example, we have a resting statefMRI data matrix where the number of timepoints (2,048) is about one order of magnitudegreater than the number of nodes (90). Underthese circumstances, it is probably advisable tostart with a simple measure of stationary associ-ation, especially if focusing on a low-frequencyinterval, e.g., ≤0.1 Hz, to mitigate nonneuronalcontributions to covariation between nodes. Ifthe time series were longer, it might be moreattractive to look at dynamic or nonstationaryaspects of association; if the number of nodeswas smaller relative to the number of timepoints, more sophisticated measures of associ-ation such as dynamic causal modeling mightbe computationally tractable; if the data hadbeen acquired during performance of multiplediscrete cognitive trials, rather than at rest, itmight be interesting to measure the correlationof event-related response amplitude betweennodes (Yoon et al. 2008).

Whichever measure of association is cho-sen, our second piece of general advice wouldbe to look at the association matrix carefullybefore proceeding to graphical analysis (Figure3A,B). Several simple exploratory analyses may


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be informative and will build a preliminaryunderstanding of the data before they are moreabstractly represented by a graphical model.For example, it may be useful to calculate thegrand mean of the association matrix and, foreach node, its mean association with the restof the matrix: we refer to these as measures ofthe strength of association. The mean strengthof association for the ith node is simplyai,. = 1

N

∑j ai, j . It may also be useful to con-

sider the between-node variation in strengthof association and the within-node variation ofassociation to all other nodes in the system: Wecan refer to these as measures of the diversityof association. It can also be informative to useprincipal component (PC) analysis to identifymajor modes of covariation between multiplenodes; for example, the ratio of the firsteigenvalue, λ1, to the sum of other eigenvaluescan provide a measure of the global integrationof the system I = λ1∑N

2 λ j(Lynall et al. 2010,

Tononi et al. 1994). Many other exploratorymultivariate techniques could be used at thisstage.

The basic idea is to get acquainted with theassociation matrix in relatively simple termsbefore it is transformed to an adjacency matrixand described topologically. Such an incre-mental approach will generally prevent us fromthinking about the results of graphical analysisas the output of a “black box” procedure andwill likely be predictive of some of the key topo-logical results. For example, the most highlyconnected nodes or hubs of a brain graphwill typically be those nodes with the highestmean strength of connectivity (Figure 4).The value of preliminary exploratory analysisis particularly clear when we are dealingwith several individual association matricesrepresenting brain systems in subjects drawnfrom different patient groups or studied underdifferent experimental conditions (see below).

Structural association. Conceptually iden-tical approaches can be taken to the graphicalanalysis of structural networks derived frommeasures of anatomical connectivity between

nodal regions. Anatomical connectivity can bedefined in different ways, based on differentkinds of MRI data. For diffusion tensor orspectrum imaging, it is possible to assign aprobability of axonal connection between anypair of gray matter regions on the basis oftractographic analysis of an individual dataset.For conventional (e.g., T1-weighted) MRIdata, anatomical connectivity has been inferredby thresholding a matrix of interregionalcovariation in cortical thickness or volume ofmultiple regions (Bassett et al. 2008, He et al.2007) (see Figure 2C). Strong between-subjectcovariation in local gray matter measurementshas been interpreted as indicative of axonalconnectivity between covarying regions, onthe grounds that connectivity has mutuallytrophic effects on the growth of connectedneurons or regions, leading to correlations inthe size of the regions when they are measuredafter a period of development (Mechelli et al.2005, Wright et al. 1999). This is not the onlypossible interpretation—as noted, the cellularsubstrates for many neuroimaging phenomenaare unresolved—but there is some empiricalevidence that regions with high gray mattercovariation measured in conventional MRI alsohave a high probability of being connected byaxonal projections inferred from tractographicanalysis of DTI or DSI data (Lerch et al. 2006).

Both MRI and DT/SI-based anatomicalnetworks have their limitations. The maindrawback of MRI-based anatomical connectiv-ity analysis, besides the question of its cellularsubstrate, is that it depends on covariation overindividual subjects. Precise estimation there-fore demands MRI data, ideally on hundredsof subjects, to yield a single association matrix,and thus a single network, for the whole group.This precludes analysis of how individualdifferences in anatomical network organizationmight be associated with other individualdifferences, such as variability in performanceof a cognitive task. In contrast, the main advan-tage of DT/SI-based anatomical connectivityanalysis, in addition to its more clearly definedcellular substrate, is that an association matrixcan be estimated for each subject, where ai, j


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denotes the connection probability betweenregions i and j estimated by some tractographyalgorithm (Hagmann et al. 2008, Iturria-Medina et al. 2007). This allows investigatorsto explore associations between measures onindividual anatomical networks and other indi-vidually variable measures, such as functionalnetworks derived from fMRI, or cognitive vari-ables such as IQ scores or executive function,or clinical variables such as white matter lesionload in patients with multiple sclerosis.

The main current disadvantage of DT/SI-based networks is that tractography onavailable data seems generally to underestimatethe probability of connections between regionsthat are a long distance apart in the brain.This is because long-distance projections aremore likely to intersect with other, differentlyorientated projections, and it is more difficultto trace the course of a single tract in the con-text of crossing fibers. More recent acquisitionsequences, such as the HARDI sequence forDSI, have been shown to generate networkswith a higher probability of long-distance con-nections than older, classical DTI sequences(Zalesky et al. 2010b). Foreseeable improve-ments in diffusion data acquisition and analysistools can be expected to produce high-qualityanatomical networks in single subjects.

Thresholding and connection density.Having thus estimated an association matrixfrom the data, the next crucial question is, whatkind of graph do we wish to construct from theassociation matrix? We will continue to illus-trate possible solutions in relation to a func-tional association matrix based on our illustra-tive fMRI dataset, but many of the issues applydirectly to graphical analysis of other associa-tion matrices. As already noted, most investiga-tors using fMRI for this purpose to date haveestimated a measure of functional connectivitybetween nodes, such as the correlation coeffi-cient, and then thresholded the resulting asso-ciation matrix A to create a binary adjacencymatrixA (Figure 5). We can describe this moreformally by saying that a threshold τ is applied

to each element ai, j of the association matrix,and if ai, j ≥ τ , the corresponding element ofthe adjacency matrix αi, j is set to unity; other-wise, if ai, j < τ , αi, j = 0. If there is a nonzeroelement in the adjacency matrix, this is equiva-lent to saying there is an unweighted and undi-rected edge between the corresponding nodesof the network. In other words, the threshold-ing operation on the association matrix will de-fine the edges in the adjacency matrix and willtherefore have a strong influence on the topol-ogy of the network. So a key subsidiary questionthat arises is, what should be the value of thethreshold τ?

Let’s consider the limiting cases first, ini-tially assuming for the sake of simplicity thatthe association between each pair of nodal timeseries has been described in terms of the ab-solute correlation coefficient 0 ≤ |ai, j | ≤ 1. Ifτ = 0, then all elements of the association ma-trix will pass the threshold, all elements of theadjacency matrix will be nonzero, all possibleedges in the graph will exist, and the connec-tion density of the graph κ will be maximized.We can generally define the connection densityor topological cost of the graph as

0 ≤ κ ≤ 1 = Eτ

N (N −1)2

, (1)

where Eτ is the number of edges generated bythresholding at some value of τ , and N (N −1)

2 isthe maximum number of edges that could existin a network of N nodes. Clearly, when τ = 0,κ = 1. It should also be clear that if τ = 1in this case, then no elements of the associa-tion matrix will pass the threshold, all elementsof the adjacency matrix will have the value ofzero, there will be no edges in the graph, andits connection density will be zero. The im-portant generalization is that if the thresholdis set below the minimum value of the asso-ciation matrix, the adjacency matrix will havemaximum connection density, whereas if thethreshold is set above the maximum value ofthe association matrix, the adjacency matrix willhave minimum connection density, and as thethreshold is gradually increased from minimum


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to maximum values, this will result in monoton-ically, but not necessarily linearly, decreasingconnection density of the graph (Figure 5).Thus, we can describe any threshold on anyassociation measure in terms of the connec-tion density or topological cost of the result-ing graph, and this will always fall in the range0 ≤ κ ≤ 1 (Stam et al. 2007a). We will seethat this translation from threshold value τ toconnection density κ is useful, but it does notyet answer the question of which threshold(s)to apply.

There are two broad approaches to thresh-olding: We can search for a single, in some senseoptimal, threshold to apply to each associationto decide if it should be an edge and describethe topological properties of the resulting net-work only at that threshold (Achard et al. 2006,He et al. 2007), or we can threshold the as-sociation matrix at many different values anddescribe the resulting network properties as afunction of changing threshold or connectiondensity (Achard & Bullmore 2007, Bassett et al.2008).

One way to define a single threshold isby controlling the probability of type I (falsepositive) error on multiple hypothesis testingof each element in the association matrix.For example, in our illustrative example,we have 4,050 unique elements of the as-sociation matrix. We could set τ such thatProb(ai, j > τ ) ≤ 0.05, which will result inabout 200 bidirectional nonzero elements inthe adjacency matrix under the null hypothesis.More conservatively, we could set τ so as tocontrol the expected number of false positives,or the false discovery rate (Genovese et al.2002, He et al. 2007). These are both examplesof mass bivariate hypothesis testing, where astatistical hypothesis is tested independentlyfor each of a large number of bivariate measuresof association between nodes. An alternativeapproach may be to apply a preliminary statis-tical threshold to each edge and then test thenull hypothesis at the level of suprathresholdclusters of interconnected edges (Zalesky et al.2010a). This is akin to the use of cluster-level

Clustering:a measure of thecliquishness ofconnections betweennodes in a topologicalneighborhood of thegraph; the nearestneighbors of a highlyclustered node willalso be the nearestneighbors of eachother. Related to localefficiency and fault-tolerance of thenetwork

statistics in classical fMRI activation mapping(Bullmore et al. 1999), although in this caseclusters of edges are defined topologicallyrather than clusters of voxels being defined byspatial proximity, as in the classical case.

Our preference has been to explore networkproperties as a function of changing threshold.As the threshold is gradually relaxed, moreedges are added to the network, so it becomesincreasingly dense or less sparsely connected(see Figure 4). The complex or nonrandomtopology of brain graphs is typically clearestin relatively low-cost networks, i.e., those withconnection densities less than about 0.5. In thisfirst half of the possible cost range, increasingconnection density is associated with a dispro-portionate increase in global and local efficiencyof network topology, and the small-world prop-erties of the system are most clearly demon-strated by comparison to random networks(Figure 6). The greater-than-linear increase inefficiency as a function of cost means that thecost-efficiency difference is typically positiveand has a maximum value at connection densityabout 0.3 (Achard & Bullmore 2007, Bassettet al. 2009) (see Figure 6). At lower thresholds,associated with connection densities greaterthan 0.5, the law of diminishing returns seemsto apply. Addition of extra edges, or increasingconnection cost, is associated with relativelymodest increments of network efficiency; small-world properties are less clearly delineated;and the networks become indistinguishablefrom random graphs at highest connectioncosts.

This behavior allows us to define a small-world cost regime as the range of connectiondensities over which the network exhibits thesmall-world characteristic: a clustering coeffi-cient greater than the clustering coefficient ofa random network and a path length about thesame as the path length of a random network(Humphries et al. 2006, Watts & Strogatz1998); for mathematical definitions, see the fol-lowing section on Measures on Graphs: Topo-logical Measures. This definition of small-worldness clearly requires that the experimental


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network be compared to random networksand regular lattices (Bassett et al. 2010, Kaiser& Hilgetag 2006, Watts & Strogatz 1998).Each of these benchmark graphs has a differentdegree distribution; that is, a node in each ofthese networks will have a different probabilityof being a highly connected hub. This degreedistributional variation could predispose thenetwork to have higher or lower values ofa given graph metric, such as the clusteringcoefficient. It is therefore often useful to assesswhether the topological structure present in abrain network under study is due to the degreedistribution alone or to additional architecturalconstraints. Thus, it is common practice tocompare brain network properties to both (a) apure random network and (b) a random networkthat has been constructed to retain the identicaldegree distribution of the brain network understudy (Maslov & Sneppen 2002).

When random graphs are constructed overa range of connection densities, it is found thatthe network becomes fragmented, into a gi-ant connected cluster and a number of smallerislands, at a characteristic critical connectiondensity. Thus, for a random graph to be en-tirely connected, the mean degree of the Nnodes needs to be at least 2× log(N ), or equiv-alently, the connection density κ needs to beat least (2 log(N ))N

N (N −1)/2 . We can say that there isa percolation threshold, defined as the low-est connection density at which the graphbecomes entirely connected and informationcan percolate freely throughout the wholesystem.

When brain graphs are constructed byglobal thresholding over a range of connec-tion densities, they also tend to become frag-mented at lower connection densities (see, e.g.,Figure 4). The percolation threshold is variablebetween individuals and data types but is typi-cally in the range 0.1 < κ < 0.5. The fragmen-tation or percolation properties of networks canbe topologically informative in their own rightand are also closely related to measures of a net-work’s robustness to random error or targetedattack (Achard et al. 2006, Honey & Sporns2008, Lynall et al. 2010). But this behavior also

alerts us to the fact that sparsely connected net-works, which tend to be more complex or lessrandom topologically, will often comprise dif-ferent numbers of fully connected nodes, andthis will need to be controlled when it comesto comparing any topological metric betweennetworks. One simple way of controlling thenumber of connected nodes is to measure graphmetrics only over the range of costs for whichall individual graphs are entirely connected(Bassett et al. 2008, Lynall et al. 2010). How-ever, if some of the individual networks havehigh percolation thresholds, this may forcecomparison of metrics over a less sparsely con-nected cost range, where differences from ran-domness will tend to be less salient.

An alternative to global thresholding forgraph construction may help to circumventsome of these issues. For example, a graph canbe constructed from the minimum spanningtree (MST) (Hagmann et al. 2008). The MSTfully connects N nodes with N − 1 edges,with low connection density in the order ofN/N2. This means that the MSTs of any twoconnectivity matrices will be guaranteed to beentirely connected at sparse connection densi-ties and therefore could support comparison oftopological metrics controlled for number ofedges and number of connected nodes over amore interesting cost range than global thresh-olding. However, MSTs are by definitionacyclic and will not demonstrate biologicallyplausible clustering of connections betweentopologically local nodes. Once the MST hasbeen defined as the “skeleton” of the brainnetwork, it must therefore be grown somehow,e.g., by addition of extra edges that pass a globalthreshold value, such that graph metrics can beestimated over a full cost range of connectiondensities, without any change in the numberof connected network nodes (Alexander-Blochet al. 2010). Minimum spanning trees andrelated methods of graph construction mayprove to be advantageous as a technical basisfor comparing networks, matched for numberof nodes and edges, between different groupsor experimental conditions. However, thereare likely to be significant future developments


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concerning the optimal methods for filtering anassociation matrix to construct a brain graph.

MEASURES ON GRAPHS

Brain graphs are models of physically embed-ded information-processing networks. Eachnode has an anatomical address in physicalspace and a network role in topological space.The brain as a whole is a high-performancenetwork—with many global topological prop-erties in common with high-performancecomputer chips, economic markets, and othercomplex systems—physically embedded as asulco-gyrally convoluted sheet of processingelements interconnected by a core of whitematter cabling.

There are therefore two main classes ofmeasure on brain graphs: topological and ge-ometric. Topological metrics capture the rela-tions between nodes regardless of their physicallocation—an edge can count identically as anedge whether it connects two locally neighbor-ing nodes or a pair of nodes located far apartfrom each other; a hub node with many edges,or high degree, will count as a hub whereverit is located. Physical metrics capture the re-lations between nodes in Euclidean space andwill have continuous values in SI units. Thereis obvious potential interest in understandingthe interaction between brain network topol-ogy and geometry. There are many methodsby which topological and geometric measurescould be combined; for example, we can weighttopological edges by physical distance betweennodes for weighted network analysis. We con-sider that this topo-physical mapping of brainnetworks is likely to become of considerable in-terest in the future, although it has not yet beenmuch developed.

Topological Measures

Because network analysis is based on the math-ematical field of graph theory, there is a wealthof previously defined metrics that can be used tocharacterize the topological architecture of thebrain’s anatomical or functional connectivity.

These are parameters of global network organi-zation, and many of them can also be estimatedat the single-node or edge level of the graph(see Rubinov & Sporns 2010 for review).

Degree and degree distribution. Perhapsthe simplest topological measure is the degreeof a node, k, which is defined as the numberof edges emanating from that node. Degree,sometimes called degree centrality, has beenused to discriminate between nodes in thesystem that are well connected, i.e., so-calledhubs, and nodes that are less well connected, ornonhubs (see Figure 1). Due to their relativelyincreased connectivity, high-degree nodes arelikely to play an important role in the system’sdynamics. The probability distribution fornodal degree is the degree distribution ofthe network. Brain graphs generally haveheterogeneous or broad-scale degree distribu-tions, meaning that the probability of a highlyconnected hub is higher than in a comparablerandom network (see Figure 6C). Most studieshave found that an exponentially truncatedpower law is the best form of degree distri-bution to fit to networks based on functionaland structural MRI data. Some studies havereported that the degree distribution of fMRInetworks follows a power law, which impliesthat the probability of a highly connected hubis higher than it would be if the degree distri-bution were exponentially truncated. Differentdegree distributions imply differences in net-work growth rules. For example, a power lawdegree distribution is compatible with growthby a simple preferential attachment rule,whereby each new node is more likely to formconnections with existing hubs, and there is nolimit to the number of connections a hub cansustain; whereas an exponentially truncated dis-tribution implies that there may also be physicalconstraints on the number of connections thatany single node can sustain. It will be interestingto explore computational models of networkgrowth more extensively in relation to emerg-ing empirical data on brain network degreedistributions and other topological properties.


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Although degree provides a simple measureof a node’s hubness, it is not the only measureof a node’s significance for the flow of informa-tion through the network. The complementarymetric of betweenness centrality describes howmany shortest paths between any two nodesin the system must pass through the node inquestion (Freeman 1977). A brain regional nodewith high centrality is therefore potentially aninformation bottleneck because it will be in-volved in many of the shortest paths betweenother regions of the whole brain network. Al-though betweenness centrality is the oldest andperhaps most used centrality measure, closenesscentrality, eigenvector centrality, and edge cen-trality all provide similar but not identical infor-mation regarding nodal importance (Lohmannet al. 2010, Sporns et al. 2007). Centrality mea-sures can also be estimated for each edge, as wellas for each node, in the network.

Small-worldness and efficiency. The twometrics originally used to characterize the C. el-egans neuronal system were the clustering coef-ficient, which is a measurement of the efficiencyof local connectivity, and path length, whichis a simplified measurement of the global effi-ciency of information transfer on the network(Watts & Strogatz 1998) (see Figure 3C). Asdescribed in a previous section, these two met-rics enable us to define the small-world prop-erty, in which the network exhibits a clusteringcoefficient, C, greater than the clustering coef-ficient of a random network, Cr ; a path length,L, about the same as the path length of a ran-dom network, Lr ; and therefore a small-worldscalar σ > 1 (Humphries et al. 2006, Watts &Strogatz 1998), where

σ =CCrLLr

. (2)

It is important to note here that reporting asmall-world scalar σ > 1 is not enough toprove small-worldness; both γ = C/Cr > 1and λ = L/Lr ∼ 1 are also required. Thereason that the relation σ > 1 is not enoughto prove small-worldness is that it is possibleto have regular network structures with large

clustering coefficients (for example, giving γ =3) but also long path lengths (for example givingλ = 2), which together provide a σ value greaterthan 1. In other words, regular-lattice-like net-works may have small-world scalars σ > 1, andso to prove small-worldness, both the γ and λ

values need to be reported.The small-world scalar is dependent on

the calculation of a path length, which can betroublesome for networks that contain one ormore disconnected nodes. The path length of adisconnected node is infinity (it cannot transferinformation to any other node on the network),and so the average path length of a networkthat contains a disconnected node (such asmany fMRI networks at sparse threshold;Achard et al. 2006) will also be equal to infinity.In a complementary formalism, Latora &Marchiori (2001) introduced the global ef-ficiency as an alternative metric of globalintegration that is inversely proportional tothe characteristic path length of the network,thus allowing computation of a finite value forgraphs with disconnected nodes (Achard &Bullmore 2007). In addition to the global effi-ciency, Latora & Marchiori (2001) also definedthe local efficiency of each node, which is similarbut not equivalent to its clustering coefficient orfault tolerance (see Figure 3D). Subsequently,Achard & Bullmore (2007) defined the nodalefficiency as inversely proportional to the pathlength of connections between a single nodeand every other node in the network. It can beseen that there are many metrics that includethe word efficiency in their name, and care isrequired to avoid terminological confusion.

As connection density increases, there willbe an increase in global efficiency for anygraph: More edges make it easier to get fromone node to another, but each extra edge adds amarginal cost to the overall topological cost ofthe network. In brain graphs, efficiency tendsto increase faster than linearly as connectiondensity is increased from zero to about 20%,but at progressively higher costs, the extraadvantage in efficiency is less than the incre-mental cost. This means that the differencebetween global efficiency and topological cost,


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so-called cost-efficiency (Achard & Bullmore2007, Bassett et al. 2009, Deuker et al. 2009), isgenerally positive and has a maximum value at acritical connection density, which is somewhatvariable between subjects but typically aboutκ ∼ 0.3. The topological cost-efficiency ofbrain networks is consistent with their classi-fication as economical small-world networks(Latora & Marchiori 2001, 2003).

Modularity. A brain graph can generallybe subdivided or partitioned into subsets ormodules of nodes (Chen et al. 2008, Meunieret al. 2008). Finding the mathematicallyoptimal modular decomposition for a brainnetwork is not trivial, and several alternativealgorithms have been proposed. In general, theaim is to find the partition that maximizes theratio of intramodular to intermodular edges.Thus the nodes in any module will be moredensely connected to each other than to nodesin other modules (Blondel et al. 2008, Leicht& Newman 2008, Newman 2006). The in-tramodular degree is a measure of the numberof connections a node makes with other nodesin the same module. The participation coeffi-cient is a measure of the ratio of intramodularconnectivity to intermodular connectivity foreach node (Guimera et al. 2007). These andrelated metrics can be used to define nodes as“connectors” (with high intermodular connec-tions) or “provincials” (with low intermodularconnections). We can see that resolving themodular or community structure of a braingraph is likely to add important informationabout which anatomical regions or tracts havethe most critical topological roles in trans-mission of information across brain networks.We also know from recent work that brainnetworks are not only modular at a global scale,they also demonstrate a hierarchical modularity(Meunier et al. 2009) or nested arrangement ofmodules within modules, which may be advan-tageous in terms of stability, evolvability, andefficiency of physical embedding of complexnetworks in general (Bassett et al. 2010, Simon1962).

Other topological metrics. There is noshortage of topological metrics that could beapplied to brain graphs. Here we have summa-rized only a few of the parameters that havebeen most extensively investigated in the neu-roimaging literature to date. The literature onstatistical physics of complex networks is theprimary source for many other metrics, tools,and concepts for brain graph analysis (Albert& Barabasi 2002, Strogatz 2001). There is alsoa rapidly growing literature on specialist ap-plications to human neuroimaging and otherneurophysiological data (for recent reviews,see Bassett & Bullmore 2006, 2009; Bullmoreet al. 2009; Bullmore & Sporns 2009; Hagmannet al. 2010; and Wang et al. 2009). In gen-eral, it is worth remembering that many topo-logical metrics will be strongly correlated witheach other and with more elementary statisticalproperties of the data. Different metrics oftenprovide convergent angles on the same aspectsof network organization: For example, a morehub-dominated degree distribution was asso-ciated with greater clustering of connectionsin fMRI data (Lynall et al. 2010). It is corre-spondingly unlikely that any single metric willturn out to be uniquely important in capturingthe complexity of human brain networks (Costaet al. 2007).

Geometric Measures

The main geometric measure on a graph isthe distance between connected nodes. Thisis sometimes described for convenience asthe wiring length of a connection. In humanbrain networks based on DTI or fMRI data,most connections have short wiring length, butthe probability distribution is heavy-tailed andthere is a significant minority of long-distanceedges (see Figure 6D). Further analysis hasshown that the human brain is economically butnot minimally wired, meaning the total wiringcost of the network (the sum of all physical dis-tances between connected nodes) is less thanit would be if the same nodes were wired upat random but more than if the nodes wererewired to minimize wiring cost (Chen et al.


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2006, Kaiser & Hilgetag 2006). This observa-tion implies that conservation of wiring cost hasbeen positively if not uniquely selected in evo-lution of large-scale brain networks, and it iscompatible with much prior data and analysisindicating that many aspects of brain anatomi-cal organization can be approximated by wiringminimization principles (Attwell & Laughlin2001, Chen et al. 2006, Kaiser & Hilgetag 2006,Niven & Laughlin 2008).

The distance is usually estimated as theEuclidean distance between regional centroidcoordinates in stereotactic space. It might bepossible to substitute curvilinear estimatesof distance from tractographic modeling ofDTI/DSI data in the future. The distancebetween regions can be used, for example, toweight the edges of the network under study.In this case, we could perform what is called aweighted network analysis, in which the weightsof the edges are used in the computation of allgraph metrics calculated on the network.

Although for the most part topological andgeometric measures have remained separatein network analysis, the interactions betweentopology and space specifically for spatially em-bedded systems such as the brain will likelyprove interesting in future. One interesting av-enue will be to assess the spatial distributionof graph metrics throughout the physical spaceof the system. This has been done in the con-text of other more generally spatially embed-ded systems: The spatial distributions of cen-trality measures have been used, for example, tostudy urban street architectures (Crucitti et al.2006). In addition to studying the spatial dis-tributions of graph metrics, we may be able toapply metrics that inherently blend topologi-cal and geometric properties, such as the Rent’sexponent. Based on work in the computer sci-ence literature and recently applied to humanbrains, the topo-physical metric known as Ren-tian scaling has been used to quantify the cost-efficiency of physical connectivity in terms ofboth wiring and connection complexity (Bassettet al. 2010). It is likely that additional metricsand analysis methods will provide added insightinto the complex interaction between network

topology and physical placement in the humanbrain, both functionally and structurally.

COMPARING AND VISUALIZINGBRAIN GRAPHS

Comparing Graphs

Brain graph analysis inevitably means makingcomparisons between networks. For instance,we often want to know if some aspect of brainnetwork organization is nonrandom. This willmean comparing the brain network derivedfrom neuroimaging data to a random networkgenerated by computer simulation of an Erdos-Renyı random graph (Erdos & Renyi 1959).For a given number of nodes, we can generatea large number of random graphs with some-what variable topology and use the distributionof topological metrics in the random graphs as apoint of reference to judge the nonrandomnessof the same metrics measured in the neuroimag-ing data. We may also want to compare braingraphs to minimally wired versions of the samenetwork, or to compare brain graphs betweengroups of subjects, such as a patients and healthyvolunteers. In all these examples of compar-ing brain graphs, it is important to observe twogolden rules (Bollobas 1985): The graphs to becompared must have (a) the same number ofnodes and (b) the same number of edges.

This is because the quantitative values oftopological metrics will depend on both the sizeand connection density of the graphs, and in or-der to identify topological differences betweengraphs that specifically point to the differencebetween groups, it is important to control thesegeneral effects before making any quantitativecomparisons. To test statistical hypothesesabout differences between groups of networks,we would also recommend the adoption of non-parametric techniques, e.g., permutation testsfor the difference in topological metrics be-tween two groups of brain graphs. The distribu-tional properties of topological metrics are notwell known, and the sample sizes in most neu-roimaging experiments are not large, present-ing challenges for asymptotic theory. More pos-itively, a computationally intensive approach


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to inference by Monte Carlo simulations ordata resampling offers substantial advantagesin terms of flexibility, precision, and validity oftesting that are only discounted somewhat bythe costs entailed in greater processing time.

In addition to comparing topologicalstructures of entire graphs, we could focuson subnetworks of graphs that are differentbetween groups. Recent progress in appli-cations of machine learning can be used touncover unique subgraphs that discriminatebetween two weighted (sparse or complete)graphs (Richiardi et al. 2010), and permutationtesting on topological clusters of connectionshas proven to be statistically powerful inidentification of abnormal subnetworks infMRI data on people with schizophrenia(Zalesky et al. 2010a). Techniques like thesemay provide a new avenue for the applicationof complex network analysis to smaller subsetsof whole-brain connectivity profiles.

Visualizing Graphs

Brain graphs can be visualized in real space, asseen in Figure 6B, to show their actual physical

structure. In a complementary visualization,brain graphs can be plotted in topologicalspace. For example, nodes can be visualizedas close to each other if they are part of thesame topologically defined cluster or moduleand far apart from each other if they are sep-arated by a long path length (see Figure 6A).Also, topological properties taken from thesebrain graphs can be visualized in anatomicalspace—so-called topophysical mapping. Forexample, we can produce cortical surface mapsof the degree of cortical nodes, highlightingthe anatomical distribution of network hubs(see Figure 4). See Table 1 for availablesoftware that can be used to visualize braingraphs.

BEYOND THE SIMPLEST GRAPHS

Directional Connectionsand Causal Relationships

Network edges may be directed (drawn as ar-rows) or undirected (drawn as lines). A directededge makes a claim about the causal relationsbetween nodes, whereas an undirected edge is

Table 1 Available tools for network analysis of brains

Human brain atlases Software SiteAAL WFU PickAtlas http://www.fmri.wfubmc.edu/cms/Brodmann MRICRO http://www.cabiatl.com/mricro/Freesurfer Freesurfer http://surfer.nmr.mgh.harvard.edu/Harvard-Oxford FSL http://www.fmrib.ox.ac.uk/fsl/LPBA40 LONI http://www.loni.ucla.edu/Atlases/Reference networks Laboratory SiteC. elegans (N = 131,277) Kaiser http://www.biological-networks.orgMacaque (N = 95) Kaiser http://www.biological-networks.orgMacaque (N = 71,47) Sporns http://www.indiana.edu/cortex/Macaque Visual (N = 30,32) Sporns http://www.indiana.edu/cortex/Cat (N = 95,52) Sporns http://www.indiana.edu/cortex/Network Toolboxes Language SiteMatlab BGL MatlabBrain Connectivity Toolbox Matlab http://www.indiana.edu/cortex/Brainwaver RNetwork visualization Description Sitegplot Matlab http://www.mathworks.com/matlabcentral/fileexchangePajek Closed source http://pajek.imfm.si/doku.phpCaret Van Essen http://brainvis.wustl.edu/wiki/index.php


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agnostic about causality—it is simply a claimabout association. The directionality or causal-ity of neuronal interactions is not always eas-ily estimated from data. In the neural systemof C. elegans, many synaptic connections be-tween cells are electrical (not chemical), andthis has usually been modeled as a single undi-rected edge between neuronal nodes (Kaiser &Hilgetag 2006, Watts & Strogatz 1998). Sim-ilarly, most interregional axonal connectionsin mammalian cortex are reciprocal, and mostgraphical models of macaque and cat cortexhave had undirected edges (Kaiser & Hilgetag2006, Sporns et al. 2004). In human neuroimag-ing data, it is currently more difficult to assigndirectionality to associations between regions,whether measured by structural MRI, DTI,DSI, or fMRI. Most graphs constructed fromfMRI data have therefore measured symmet-ric measures of association or functional con-nectivity, like simple correlation, partial corre-lation, mutual information, or synchronizationlikelihood, and constructed undirected graphson this basis. However, it is possible to modeldirectional edges (Chen & Herskovits 2007)by using, for example, partial directed coher-ence (Baccala & Sameshima 2001), dynamiccausal modeling (Friston et al. 2003), structuralequation modeling (Buchel & Friston 1997), orGranger causality (Bernasconi & Konig 1999)as measures of asymmetric association or ef-fective connectivity. To date, however, it isunclear whether it is possible to scale thesemethods to systems with more than a handfulof nodes; for the possibility of such a large-scale analysis using structural equation mod-eling, see Kenny et al. (2009). Computationalmodels have shown that more complex and bi-ologically plausible dynamics can be generatedfrom directed graphs of brain anatomy based ontract tracing data on primates than can be gen-erated from undirected graphs based on diffu-sion spectrum imaging data on humans (Knocket al. 2009). It seems likely that further devel-opment of data and methods for analysis of di-rected brain graphs will be a priority for futuretechnical innovation.

Weighted Network Analysis

Although the majority of network studies ofneuroimaging data have been performed on bi-nary or unweighted networks, in which eachedge has a weight of one and each nonedge hasa weight of zero, there has been a growing in-terest in the use of weighted networks, in whichedges may have continuously variable weights(see Rubinov & Sporns 2010 for a recentreview and http://sites.google.com/a/brain-connectivity-toolbox.net/bct for the BrainConnectivity Toolbox software library, whichprovides code for many of the metrics). Onesimple approach to weighted network analysisis to start with a binary network constructedat some cost and then assign a weight wi, j toeach edge. The weights could be the physi-cal distance between nodes or the strength offunctional connectivity between nodes. Mostof the topological metrics available for analysisof binary networks have been generalized forweighted network analysis, so weighting a bi-nary adjacency matrix does not restrict the op-tions for topological analysis, and it does retainmore physical information in the graph model.The weighting of brain networks by physi-cal connection distance or wiring cost seemslikely to be of interest given the prior evidencethat wiring costs are highly economized, if notstrictly minimized, in animal nervous systems atmany spatial scales. It is likely that future studieswill increasingly use DT/SI-based measures ofanatomical connection distance between nodesas the weights on anatomical and functionalnetworks.

A more radical approach, sometimes alsocalled weighted network analysis, is to estimatemetrics that are analogous to topological met-rics on a thresholded graph without applyingany threshold to the association matrix. For ex-ample, unsupervised learning methods, such asthe spin-glass algorithm, can be used to decom-pose an association matrix into clusters that cor-respond closely to the modules identified bya topological analysis of the adjacency matrixgenerated by thresholding the association ma-trix (Alexander-Bloch et al. 2010). It is likely


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that there will be further consideration of ap-proaches to network analysis that do not de-pend on a thresholding step, but of course oneof the benefits of global thresholding is that itwill remove a proportion of the noisy or ran-dom edges from the graph. Metrics estimateddirectly on the unthresholded association ma-trix will generally have lower signal-to-noise ra-tios than metrics estimated on sparsely thresh-olded graphs.

CONCLUSIONS

Graph analysis is rapidly growing in popularityas an approach to modeling the complexity ofthe human brain connectome. The fundamen-

tal motivations for graph theory as a methodof brain network analysis are its relative sim-plicity and high degrees of generalizability andinterpretability. However, like all other model-ing endeavors, the results of brain graph anal-ysis are underpinned by basic assumptions orchoices, many of which will represent a trade-off between competitive criteria. We have triedto elucidate some of the conceptual issues andrecent methodological advances that will be rel-evant to an investigator wanting to use thesetechniques to analyze neuroimaging or otherneuroscientific data. Our hope has been to stim-ulate informed use and further methodologicaldevelopment of graph theoretical networks asmodels of the human brain connectome.

SUMMARY POINTS

1. Brain graphs are apparently simple but powerful models of the brain’s structural orfunctional connectome; graphical analysis is applicable to many scales and types of neu-roscience data and is interpretable in relation to general principles of complex systemorganization.

2. Key questions to address in any graphical analysis are, what is a node and what is anedge? In neuroimaging or neurophysiological data, a node will typically be a region ofthe image, or a sensor or electrode of the recording array, and edges will be defined bythresholding a measure of statistical association between nodes. Many methodologicalissues attend the specification of both nodes and edges.

3. It is recommended to explore network properties over a range of connection densities ortopological costs. Sparsely thresholded networks (with cost less than 20%) demonstratenonrandom properties such as small-worldness and modularity more saliently and aremore likely to be fragmented or not entirely connected.

4. Once a brain graph has been constructed, many topological and geometrical propertiescan be estimated—early work has focused on broad-scale degree distributions, econom-ical small-worldness and cost-efficiency, and modularity. These and other nonrandomproperties of network organization have been found consistently across many differenttypes of neuroscientific data, suggesting that they represent highly conserved generalprinciples of connectome organization.

5. Network analysis is relativistic—brain graphs need to be compared to each other andto benchmark networks. In making comparisons, it is generally advisable to ensure thatthe connection density and number of entirely connected nodes is equivalent betweengraphs.

6. Weighted network analysis allows incorporation of more physical data, such as the con-nection distance between nodes, in the topological analysis of network properties.


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FUTURE ISSUES

1. Scale. Early brain graph studies have typically used a few images. With the fund-ing of the National Institutes of Health Human Connectome Project (http://www.humanconnectomeproject.org) and the related growth of interest in graphicalanalysis, we can expect publication of brain graphs based on hundreds or thousands ofsubjects, allowing more powerful studies of genetic and other causes of variation in braingraph parameters.

2. Causality. It will be necessary to optimize techniques to capture directional interac-tions between neuronal populations and to model large causal systems as directed braingraphs. Most human brain graph studies to date have assumed undirected edges, whichis technically simpler but neglects this biologically important aspect of neural systems.

3. Cognition. Aspects of brain graph topology have been linked to cognitive and behavioralperformance, but the relationships between psychological and topological properties ofbrain networks are likely to be investigated more extensively in the future (for example,by greater consideration of task-related functional neuroimaging data).

4. Biomarkers. The potential utility of graphical metrics as diagnostic markers of neuropsy-chiatric syndromes will be explored on the basis of larger clinical samples and in relationto various dysconnectivity models of schizophrenia and other disorders likely to resultfrom developmentally perturbed formation of the brain connectome.

5. Network movies. Most graphical analyses have provided a static view of brain func-tional network organization over several seconds or minutes of observation; however,rapid functional network configuration—or nonstationarity—is theoretically importantfor adaptivity of cognitive function and may be studied by construction of network moviesrepresenting time-resolved change in network architecture.

DISCLOSURE STATEMENT

ETB is employed half-time by GlaxoSmithKline and half-time by the University of Cambridgeand is a stockholder in GSK.

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←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Figure 2From data to association matrix. (A) An anatomical template image (left) is used to parcellate the voxel-level fMRI data. Regional meantime series (middle) are estimated for each of the N = 90 regions in the parcellation template. The pair-wise association ai, j isestimated between the ith and jth nodes, i �= j = 1, 2, 3, . . . , N and compiled for all possible pairs to form a {N × N} interregionalassociation matrix, A (right). If we choose the measure of association to be the absolute Pearson’s correlation between two time series,then this value ranges between 0 and 1. (B) For EEG or MEG data, neurophysiological time series are measured by an array of sensors(left), each of which provides a nodal time series (middle) at frequencies generally higher than those measured by fMRI. Due to thisincrease in temporal resolution, a wide variety of association metrics can be applied, including mutual information, synchronization,and phase coherence (right), to construct an association matrix. (C) Several morphometric variables can be computed on a regional basis(left) from individual structural MRI images, including gray matter volume, cortical thickness, surface area, and curvature (middle). Thecorrelation or partial correlation of these regional morphometric variables over subjects provides an association matrix (right), whichcan be used as a measure of anatomical connectivity (Lerch et al. 2006).


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_________ _________ _________

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SubcorticalOccipitalFrontal

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Medial TemporalSubcortical

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TemporalParietal

Correlation

10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

70

80

90

10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

70

80

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10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

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0.2

0.3

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A

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____

_______________________

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________ _________

________

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←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Figure 5From association matrix to adjacency matrix. (A) The association matrix represents the absolute value of all pair-wise waveletcorrelations estimated for a single fMRI dataset (see Achard et al. 2006). The 90 nodal regions of the Automated Anatomical Labeling(AAL) template are ordered into left and right hemispheres, and then into six anatomical clusters (medial temporal, subcortical,occipital, frontal, temporal, parietal) as defined by Salvador et al. (2005a). (B) The three adjacency matrices shown were obtained bythresholding the association matrix in A at costs of κ = 0.15 (left), κ = 0.30 (middle), and κ = 0.45 (right). White elements in theadjacency matrices indicate the existence of an edge, and black elements indicate the absence of an edge. Note that the density ofconnections or topological cost of the matrix increases with decreasing threshold. (C) Plot of the connection density or cost (x-axis) as afunction of the threshold applied to the association matrix ( y-axis) to construct the adjacency matrix; association matrices thresholded athigher values will have fewer edges than those thresholded at lower values. Colored lines in the plot indicate the costs at which thethree adjacency matrices shown in B were calculated.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

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0.3

0.35

0.4

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Topological Space Physical Space

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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Loc

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30


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←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Figure 6Topological and geometrical properties of functional brain graphs. (A) For a single fMRI dataset, we calculated the adjacency matrix ata cost of κ = 0.05, and the edges present in this adjacency matrix were plotted here in topological space so that the distance betweennodes is larger if those nodes are separated by a longer path length and smaller if they are separated by a shorter topological pathlength. The size of nodes indicates nodal degree whereas the color of the node indicates its lobar identity. (B) Brain graph plotted inphysical space where the distance between nodes is the Euclidean distance between regional centroids in the anatomical space of thereal brain. The size of nodes indicates nodal degree whereas the color of the node indicates its lobar identity. (C) Plot of the degreedistribution (red ) and distribution fit (black) of the brain graph, showing the predominance of low-degree nodes and the presence of afew high-degree hubs. (D) Plot of the distribution of physical distance of connections of the brain graph (red ) in comparison to thedistance distribution in a minimally rewired network ( black). (E) Plot of global efficiency versus cost for the brain network (red ) and adistribution of 100 comparable random networks (black). (F) Plot of local efficiency versus cost for the brain network (red ) anda distribution of 100 comparable random networks (black). (G) Plot of the cost-efficiency versus cost for the brain network (red ) and adistribution of 100 comparable random networks (black). (H) Plot of small-worldness versus cost for the brain network (red ). Thesmall-worldness scalar σ = 1 in a random graph.


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CP07-FrontMatter ARI 8 March 2011 4:13

Annual Review ofClinical Psychology

Volume 7, 2011 Contents

The Origins and Current Status of Behavioral Activation Treatmentsfor DepressionSona Dimidjian, Manuel Barrera Jr., Christopher Martell, Ricardo F. Munoz,

and Peter M. Lewinsohn � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Animal Models of Neuropsychiatric DisordersA.B.P. Fernando and T.W. Robbins � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �39

Diffusion Imaging, White Matter, and PsychopathologyMoriah E. Thomason and Paul M. Thompson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �63

Outcome Measures for PracticeJason L. Whipple and Michael J. Lambert � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �87

Brain Graphs: Graphical Models of the Human Brain ConnectomeEdward T. Bullmore and Danielle S. Bassett � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 113

Open, Aware, and Active: Contextual Approaches as an EmergingTrend in the Behavioral and Cognitive TherapiesSteven C. Hayes, Matthieu Villatte, Michael Levin, and Mikaela Hildebrandt � � � � � � � � 141

The Economic Analysis of Prevention in Mental Health ProgramsCathrine Mihalopoulos, Theo Vos, Jane Pirkis, and Rob Carter � � � � � � � � � � � � � � � � � � � � � � � � � 169

The Nature and Significance of Memory Disturbance in PosttraumaticStress DisorderChris R. Brewin � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 203

Treatment of Obsessive Compulsive DisorderMartin E. Franklin and Edna B. Foa � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 229

Acute Stress Disorder RevisitedEtzel Cardena and Eve Carlson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 245

Personality and Depression: Explanatory Models and Reviewof the EvidenceDaniel N. Klein, Roman Kotov, and Sara J. Bufferd � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 269

vi

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CP07-FrontMatter ARI 8 March 2011 4:13

Sleep and Circadian Functioning: Critical Mechanismsin the Mood Disorders?Allison G. Harvey � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 297

Personality Disorders in Later Life: Questions About theMeasurement, Course, and Impact of DisordersThomas F. Oltmanns and Steve Balsis � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 321

Efficacy Studies to Large-Scale Transport: The Development andValidation of Multisystemic Therapy ProgramsScott W. Henggeler � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 351

Gene-Environment Interaction in Psychological Traits and DisordersDanielle M. Dick � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 383

Psychological Treatment of Chronic PainRobert D. Kerns, John Sellinger, and Burel R. Goodin � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 411

Understanding and Treating InsomniaRichard R. Bootzin and Dana R. Epstein � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 435

Psychologists and Detainee Interrogations: Key Decisions,Opportunities Lost, and Lessons LearnedKenneth S. Pope � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 459

Disordered Gambling: Etiology, Trajectory,and Clinical ConsiderationsHoward J. Shaffer and Ryan Martin � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 483

Resilience to Loss and Potential TraumaGeorge A. Bonanno, Maren Westphal, and Anthony D. Mancini � � � � � � � � � � � � � � � � � � � � � � � 511

Indexes

Cumulative Index of Contributing Authors, Volumes 1–7 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 537

Cumulative Index of Chapter Titles, Volumes 1–7 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 540

Errata

An online log of corrections to Annual Review of Clinical Psychology articles may befound at http://clinpsy.annualreviews.org

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brain graphs: graphical models of the human brain ......cp07ch05-bullmore ari 9 march 2011 21:34...

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