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Graphtheoryasatooltounderstandbraindisorders
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Graphtheoryapproachestofunctionalnetworkorganizationinbraindisorders:
Acritiqueforabravenewsmall-world
MichaelN.Hallquist1,2
and
FrankG.Hillary1,2,3
1DepartmentofPsychology,PennsylvaniaStateUniversity,UniversityPark,PA
2SocialLifeandEngineeringSciencesImagingCenter,UniversityPark,PA
3DepartmentofNeurology,HersheyMedicalCenter,Hershey,PA
Corresponding author: Frank G. Hillary, Department of Psychology, Penn State University, 140 Moore Building, University Park, PA 16802. Email: [email protected].
This research was supported by a grant from the National Institute of Mental Health to MNH (K01 MH097091).
Acknowledgments.WethankZachCeneviva,AllenCsuk,RichardGarcia,andRiddhiPatelfortheirworkcollecting,organizing,andcodingreferencesfortheliteraturereviewandmanuscript.Keywords:graphtheory,braindisorders,networkneuroscience,proportionalthresholding
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Abstract
Overthepasttwodecades,resting-statefunctionalconnectivity(RSFC)methods
haveprovidednewinsightsintothenetworkorganizationofthehumanbrain.Studiesof
braindisorderssuchasAlzheimer’sdiseaseordepressionhaveincreasinglyadaptedtools
fromgraphtheorytocharacterizedifferencesbetweenhealthyandpatientpopulations.
Here,weconductedareviewofclinicalnetworkneuroscience,summarizing
methodologicaldetailsfrom106RSFCstudies.Althoughthisapproachisprevalentand
promising,ourreviewidentifiedfourkeychallenges.First,thecompositionofnetworks
variedremarkablyintermsofregionparcellationandedgedefinition,whichare
fundamentaltographanalyses.Second,manystudiesequatedthenumberofconnections
acrossgraphs,butthisisconceptuallyproblematicinclinicalpopulationsandmayinduce
spuriousgroupdifferences.Third,fewgraphmetricswerereportedincommonacross
studies,precludingmeta-analyses.Fourth,somestudiestestedhypothesesatonelevelof
thegraphwithoutaclearneurobiologicalrationaleorconsideringhowfindingsatonelevel
(e.g.,globaltopology)arecontextualizedbyanother(e.g.,modularstructure).Basedon
thesethemes,weconductednetworksimulationstodemonstratetheimpactofspecific
methodologicaldecisionsoncase-controlcomparisons.Finally,weoffersuggestionsfor
promotingconvergenceacrossclinicalstudiesinordertofacilitateprogressinthis
importantfield.
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Introduction
Effortstocharacterizea“humanconnectome”havebroughtsweepingchangesto
functionalneuroimagingresearch,withmanyinvestigatorstransitioningfromindicesof
localbrainactivitytomeasuresofinter-regionalcommunication(Friston,2011).Thebroad
goalofthisconceptualrevolutionistounderstandthebrainasafunctionalnetworkwhose
coordinationisresponsibleforcomplexbehaviors(Biswaletal.,2010).Theprevailing
approachtostudyingfunctionalconnectomesinvolvesquantifyingcouplingoftheintrinsic
brainactivityamongregions.Inparticular,resting-statefunctionalconnectivity(RSFC)
methods(Biswal,Yetkin,Haughton,&Hyde,1995)focusoninter-regionalcorrespondence
inlow-frequencyoscillationsoftheBOLDsignal(approximately0.01-0.12Hz).
WorkoverthepasttwodecadeshasdemonstratedthevalueofRSFCapproachesfor
mappingfunctionalnetworkorganization,includingtheidentificationofseparablebrain
subnetworks(Biswaletal.,2010;Lairdetal.,2009;Poweretal.,2011;Smithetal.,2009;
vandenHeuvel&HulshoffPol,2010).BecauseRSFCmethodsdonotrequirethestudy-
specificdesignsandcognitiveburdenassociatedwithtask-basedfMRIstudies,RSFCdata
aresimpletoacquireandhavebeenusedinhundredsofstudiesofhumanbrainfunction.
Nevertheless,therearenumerousmethodologicalchallenges,includingconcernsaboutthe
qualityofRSFCdata(Poweretal.,2014)andtheeffectofdataprocessingonsubstantive
conclusions(Ciricetal.,2017;Hallquist,Hwang,&Luna,2013;Shirer,Jiang,Price,Ng,&
Greicius,2015).
RSFCstudiesofbraininjuryordiseasetypicallyexaminedifferencesinthe
functionalconnectomesofaclinicalgroup(e.g.,Parkinson’sdisease)comparedtoa
matchedcontrolgroup.Therearespecificmethodologicalandsubstantiveconsiderations
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Graphtheoryasatooltounderstandbraindisorders
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thatapplytoRSFCstudiesofbraindisorders.Forexample,differencesintheoveralllevel
offunctionalconnectivitybetweenpatientandcontrolgroupscouldleadtodifferencesin
thenumberofspuriousconnectionsinnetworkanalyses,potentiallyobscuringmeaningful
groupcomparisons(vandenHeuveletal.,2017).Likewise,thereisincreasingconcernin
theclinicalneurosciencesthatanunacceptablysmallpercentageoffindingsarereplicable
(Mülleretal.,2017).Suchconcernsechothegrowingemphasisonopen,reproducible
practicesinneuroimagingmoregenerally(Poldracketal.,2017).
Inthispaper,wereviewthecurrentstateofgraphtheoryapproachestoRSFCinthe
clinicalneurosciences.Basedonkeythemesinthisliterature,weconductedtwonetwork
simulationstodemonstratethepitfallsofspecificanalysisdecisionsthathaveparticular
relevancetocase-controlstudies.Finally,weproviderecommendationsforbestpractices
topromotecomparabilityacrossstudies.
Ourreviewdoesnotdirectlyaddressmanyimportantmethodologicalissuesthat
areactiveareasofinvestigation.Forexample,detectingandcorrectingmotion-related
artifactsremainsacentralchallengeinRSFCstudies(Ciricetal.,2017;Dosenbachetal.,
2017)thatisespeciallyproblematicinclinicalanddevelopmentalsamples(Greene,Black,
&Schlaggar,2016;VanDijk,Sabuncu,&Buckner,2012).Similarly,brainparcellation—
definingthenumberandformofbrainregions—isoneofthemostimportantinfluences
onthecompositionofRSFCnetworks.Therearenumerousparcellationapproaches,
includinganatomicalatlases,functionalboundarymapping,anddata-drivenalgorithms
(Goñietal.,2014;Honnoratetal.,2015).InordertofocusonRSFCgraphtheoryresearch
intheclinicalliterature,whererelevant,wereferreaderstomorefocusedtreatmentsof
importantissuesthatarebeyondthescopeofthispaper.
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ALiteratureReviewofClinicalNetworkNeuroscienceStudies
Graphtheoryisabranchofdiscretemathematicsthathasbeenappliedinnumerous
studiesofbrainnetworks,bothstructuralandfunctional.Agraphisacollectionofobjects,
calledverticesornodes;thepairwiserelationshipsamongnodesarecallededgesorlinks
(Newman,2010).GraphscomposedofRSFCestimatesamongregionsprovideawindow
intotheintrinsicconnectivitypatternsinthehumanbrain.Figure1providesasimple
schematicofthemostcommongraphtheoryconstructsandmetricsreportedinRSFC
studies.Formorecomprehensivereviewsofgraphtheoryapplicationsinnetwork
neuroscience,seeBullmoreandSporns(2009,2012),orFornito,Zalesky,andBullmore
(2016).
Figure1.Atoygraphandrelatedgraphtheoryterminology.ReprintedwithpermissionfromHillary&Grafman,2017.
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Thegoalofourliteraturesearchwastoobtainarepresentativecross-sectionof
graph-theoreticRSFCstudiesspanningneurologicalandmentaldisorders.Wefocusedon
functionalconnectivity(FC)asopposedtostructuralconnectivity,whereadistinctsetof
methodologicalcritiquesarelikelyrelevant.Also,wereviewedfMRIstudiesonly,
excludingelectroencephalography(EEG)andmagnetoencephalography(MEG).Although
thereareimportantadvantagesofEEG/MEGinsomerespects(Papanicolaouetal.,2017),
wefocusedonfMRIinpartbecausethevastmajorityofclinicalRSFCstudieshaveusedthis
modality.Inaddition,therearefMRI-specificconsiderationsfornetworkdefinitionand
spatialparcellationinRSFCstudies.WeconductedtworelatedsearchesofthePubMed
database(http://www.ncbi.nlm.nih.gov/pubmed)toidentifyarticlesfocusingongraph-
theoreticapproachestoRSFCinmentalandneurologicaldisorders(fordetailsonthe
queries,seeMethods).
ThesesearcheswereruninApril2016andresultedin626potentialpapersfor
review(281fromneurologicalquery,345frommentaldisorderquery).Studieswere
excludediftheywerereviews,casestudies,animalstudies,methodologicalpapers,used
electrophysiologicalmethods(e.g.,EEGorMEG),reportedonlystructuralimaging,ordid
notfocusonbraindisorders(e.g.,healthybrainfunctioning,normalaging).After
exclusions,thesetwosearchesyieldeddistinct106studiesincludedinthereview(see
Table1).AfulllistingofallstudiesreviewedisprovidedintheSupplementaryMaterials.
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Table1.Clinicaldisordersrepresentedinthereviewof106clinicalnetworkneurosciencestudies
Note.ADHD:attentiondeficithyperactivitydisorder,MCI:mildcognitiveimpairment
Below,wesummarizeimportantgeneralthemesfromtheliteraturereview,
includingtheheterogeneityofdataanalyticapproachesacrossgraphtheoreticalstudies.
Wethenturnourfocustotwocriticalissueswithimportantimplicationsforinterpreting
networkanalysesincase-controlstudies:1)networkthresholdingand2)matchingthe
hypothesistothelevelofinquiryinthegraph.Foreachoftheseissues,weoffernetwork
simulationstoillustratetheimportanceoftheseissuesforcase-controlcomparisons.
CreatingComparableNetworksinClinicalSamples
Definingnodesinfunctionalbrainnetworks.GraphtheoryanalysesofRSFCdata
fundamentallydependonthedefinitionofnodes(i.e.,brainregions)andedges(i.e.,the
quantificationoffunctionalconnectivity).Fornetworkanalysestorevealnewinsightsinto
ClinicalPhenotype n(frequency)Alzheimer’sDisease/MCI 19(17.9%)Epilepsy/Seizuredisorder 13(12.3%)Depression/Affective 12(11.3%)Schizophrenia 11(10.4%)Alcohol/SubstanceAbuse 7(6.6%)Parkinson’sDisease/Subcortical 6(5.7%)TraumaticBrainInjury 6(5.7%)Anxietydisorders 5(4.7%)ADHD 5(4.7%)Stroke 4(2.8%)Cancer 3(2.8%)MultipleSclerosis 2(1.9%)AutismSpectrumDisorder 2(1.9%)Disordersofconsciousness 2(1.9%)Somatizationdisorder 2(1.9%)DualDiagnosis 2(1.9%)OtherNeurologicaldisorder 3(2.8%)OtherPsychiatricdisorder 2(1.9%)
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clinicalphenomena,investigatorsmustchooseaparcellationschemethatrobustlysamples
theregionsandnetworksofinterest.Ourliteraturereviewrevealedthat76%ofstudies
definedgraphsbasedoncomprehensiveparcellations(i.e.,samplingmostorallofthe
brain),whereas24%analyzedconnectivityintargetedsub-networks(e.g.,motorregions
only;Table2a).Furthermore,wefoundsubstantialheterogeneityincomprehensive
parcellations,rangingfrom10to67632nodes(Mode=90;M=1129.2;SD=7035.9).In
fact,whereas25%ofstudieshad90nodes(mostoftheseusedtheAALatlas;Tzourio-
Mazoyeretal.,2002),thefrequencyofallotherparcellationsfellbelow5%,resultinginat
least50distinctparcellationsin106studies.
Table2.Networkconstructionandedgedefinition
a.Networkconstruction n(frequency)Comprehensiveregionsampling 80(75.5%)Targetedregionsampling 26(24.5%)
b.Edgedefinition n(frequency)Weightednetwork 48(45.3%)Binarynetwork 42(39.6%)Both 13(12.3%)Unknown/unclear 3(2.8%)
c.EdgeFCstatistic n(frequency)Correlation(typically,Pearson’sr) 82(77.3%)Partialcorrelation 11(10.4%)Waveletcorrelation 6(5.7%)Causalmodeling(e.g.,SEM) 3(2.8%)Other/unclear 4(3.7%)
d.TreatmentofNegativeedges n(frequency)Notreported/unclear 61(57.5%)Discardnegativevalues 23(21.7%)Absolutevalue 10(9.4%)Maintained/analyzed 9(8.5%)Othertransformation 3(2.8%)Note.FC=functionalconnectivity;SEM=structuralequationmodeling
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Althoughitmayseemself-evident,itbearsmentioningthatseveralpopular
parcellationschemesprovideabroad,butnotcomplete,samplingoffunctionalbrain
regions.Forexample,recentparcellationsbasedonthecorticalsurfaceofthebrain(e.g.,
Glasseretal.,2016;Gordonetal.,2016)haveprovidedanewlevelofdetailonfunctional
boundariesinthecortex.Yetifaresearcherisinterestedcortical-subcorticalconnectivity,
itiscrucialthattheparcellationbeextendedtoincludeallrelevantregions.
Thereareadvantagesandchallengestoeveryparcellationapproach(e.g.,Honnorat
etal.,2015;Poweretal.,2011);here,wefocusontwospecificconcerns.First,agoalof
mostclinicalnetworkneurosciencestudiesistodescribegroupdifferencesinwhole-brain
connectivitypatternsthatarereasonablyrobusttothegraphdefinition.Thus,investigators
maywishtouseatleasttwoparcellationsinthesamedatasettodetermineifthefindings
areparcellation-dependent.Becauseofthefundamentaldifficultyincomparingunequal
networks(vanWijk,Stam,&Daffertshofer,2010),onewouldnotexpectidenticalfindings.
Inparticular,globaltopologicalfeaturessuchasefficiencyorcharacteristicpathlength
mayvarybyparcellation,butotherfeaturessuchasmodularityandhubarchitectureare
likelytobemorerobust.Applyingmultipleparcellationstothesamedatasetincreasesthe
numberofanalysesmultiplicatively,aswellastheneedtoreconcileinconsistentfindings.
Nevertheless,webelievethatthisdirectionholdspromiseforsupportingreproducible
findingsinclinicalRSFCstudies(Craddock,James,Holtzheimer,Hu,&Mayberg,2012;Roy
etal.,2017;Schaeferetal.,inpress).
Second,differencesinparcellationfundamentallylimittheabilitytocompare
studies,bothdescriptivelyandquantitatively.Asnotedabove,ourreviewrevealed
substantialheterogeneityinparcellationschemesacrossstudiesofbraindisorders.
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Extendingourconcernaboutparcellationdependence,suchheterogeneitymakesit
impossibletoascertainwhetherdifferencesbetweentwostudiesofthesameclinical
populationareanartifactofthegraphdefinitionorameaningfulfinding.Moreover,
whereasmeta-analysesofstructuralMRIandtask-basedfMRIstudieshavebecome
increasinglypopular(e.g.,Goodkindetal.,2015;Mülleretal.,2017),suchanalysesarenot
currentlypossibleingraph-theoreticstudiesinpartbecauseofdifferencesinparcellation.
Toresolvethisissue,weencouragescientiststoreportresultsforafield-standard
parcellation,whilealsoallowingforadditionalparcellationsthatmayhighlightspecific
findings.
Definingedgesinfunctionalbrainnetworks.Parcellationdefinesthenodes
comprisingagraph,butanequallyimportantdecisionishowtodefinefunctional
connectionsamongnodes(i.e.,theedges).Thevastmajority(77%)ofthestudiesreviewed
usedbivariatecorrelation,especiallyPearsonorSpearman,asthemeasureofFC.InCritical
Issue1below,weconsiderhowFCestimatesarethresholdedinordertodefinesedgesas
presentorabsentinbinarygraphs.
ThestatisticalmeasureofFChasimportantimplicationsfornetworkdensityand
theinterpretationofrelationshipsamongbrainregions.Theprevailingapplicationof
bivariatecorrelationdoesnotseparatethe(statistically)directconnectivitybetweentwo
regionsfromindirecteffectsattributabletoadditionalregions.Bycontrast,partial
correlationmethodsrelyoninvertingthecovariancematrixamongallregions,thereby
removingcommonvarianceanddefiningedgesbasedonuniqueconnectivitybetween
regions(Smithetal.,2011).Theadvantagesoffullversuspartialcorrelationmetricsfor
definingFCisatopicofactiveinquirythatisbeyondthescopeofthisreview(e.g.,Cassidy,
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Rae,&Solo,2015;Varoquaux&Craddock,2013).Nevertheless,wewishtohighlightthatas
theaveragefullcorrelationincreaseswithinanetwork,partialcorrelationvalues,on
average,mustdecrease.Incasesofneuropathologywhereonemightexpectdistinctgains
orlossesoffunctionalconnections,theuseofpartialcorrelationshouldbeinterpreted
basedupontherelativeedgedensityandmeanFC.Ofthestudiesreviewedhere,10%used
partialcorrelation,butvirtuallynostudyaccountedforpossibledifferencesinedgedensity
(seeTable2c).
Inaddition,fullcorrelationsofRSFCdatatypicallyyieldanFCdistributioninwhich
mostedgesarepositive,butanappreciablefractionarenegative.Thereremainslittle
consensusforhandlingorinterpretingnegativeedgeweightsinRSFCgraphanalyses(cf.
Murphy&Fox,2017).Inourreview,57%ofthestudiesreportedinsufficientorno
informationabouthownegativeedgeswerehandledingraphanalyses(Table2d).Twenty-
onepercentofstudiesdeletednegativeedgespriortoanalysis,and9%includedthe
negativeweightsaspositiveweights(i.e.,usingtheabsolutevalueofFC).Asdetailed
elsewhere,somegraphmetricsareeithernotdefinedorneedtobeadaptedwhennegative
edgesarepresent(Rubinov&Sporns,2011).
Importantly,themeanofthefullrankRSFCdistributiondependsonwhetherglobal
signalregression(GSR)isincludedinthepreprocessingpipeline(Murphy,Birn,
Handwerker,Jones,&Bandettini,2009).WhenGSRisincluded,thereisoftenabalance
betweenpositiveandnegativecorrelations.IfGSRisincludedasanuisanceregressor,a
largefractionofFCestimatesmaysimplybediscardedasirrelevanttocase-control
comparisons,whichisamajor,untestedassumption.ThemeaningofnegativeFC,however,
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remainsunclear,withseveralinvestigatorsattributingnegativecorrelationstostatistical
artifactsandGSR(Murphyetal.,2009;Murphy&Fox,2017;Saadetal.,2012).
GiventhatnegativecorrelationsareobservableintheabsenceofGSR,however,
othershaveexaminedwhethernegativeweightscontributedifferentiallytoinformation
processingwithinthenetwork(Parenteetal.,2017).Negativecorrelationsmayalsoreflect
NMDAactionincorticalinhibition(Anticevicetal.,2012).Theseconnectionsbear
considerationgiventhatbrainnetworkscomposedofonlynegativeconnectionsdonot
retainasmall-worldtopology,butdohavepropertiesdistinctfromrandomnetworks
(Parenteetal.,2017;Schwarz&McGonigle,2011).Altogether,theomissionof
methodologicaldetailsaboutnegativeFCinempiricalreportsseverelyhampersthe
resolutionofthisimportantchoicepointindefininggraphs.
Degreedistributionasafundamentalgraphmetric.Afterresolvingthe
questionsofnodeandedgedefinition,wealsobelieveitiscrucialforstudiestoreport
informationaboutglobalnetworkmetricssuchascharacteristicpathlength,clustering
coefficient(akatransitivity),anddegreedistribution.AsnotedinTable3,localandglobal
efficiencywerecommonlyreported(71%and74%,respectively)andtypicallyacross
multipleFCthresholds.However,ourreviewrevealedthatonly27%ofstudiesprovided
cleardescriptivestatisticsformeandegree,and16%plottedthedegreedistribution.In
binarygraphs,thedegreedistributiondescribestherelativefrequencyofedgesforeach
nodeinthenetwork.Asimilarpropertycanbequantifiedbyexaminingthestrength(aka
cost)distributioninweightedgraphs.Wearguethatformalpresentationofthedegree
(strength)distributionisvitaltounderstandinganyRSFCnetworkforafewreasons.First,
itprovidesasanitycheckonthedata.Becausehumanneuralsystemshavebeenwiredto
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maximizecommunicationwhileminimizingcost(Bassettetal.,2009;Betzeletal.,2017;
Chen,Wang,Hilgetag,&Zhou,2013;Tomasi,Wang,&Volkow,2013),themostcommon
degreedistributioninhealthyandclinicalfunctionalconnectomesispowerlaw(Achard,
Salvador,Whitcher,Suckling,&Bullmore,2006).Second,reportingdetailsofthedegree
distributionfacilitatescomparisonsofgraphtopologyacrossstudies,aswellastheimpact
ofanalyticdecisionssuchasFCthresholding.Finally,viewingthedegreedistributionmay
offerotherwiseunavailableinformationaboutthenetworktopologyinhealthyandclinical
samples.Forexample,inpriorwork,weexaminedthepowerlaw“tail”ofthedegree
distributiontounderstandtheimpactofthemosthighlyconnected,andrare,nodesonthe
network(Hillaryetal.,2014).
Table3.GraphmetricscommonlyreportedinclinicalRSFCstudies
Graphmetrics n(frequency)DegreeDistribution(plotted) 17(16.0%)MeanDegree(weightedorbinary) 29(27.4%)Clustering/Localefficiency 76(71.7%)PathLength/Globalefficiency 79(74.5%)Smallworldness 33(31.1%)Modularity(e.g.,Q-value) 21(19.8%)
Note.Totalfrequencyisgreaterthan100%becausesomestudiesreportedmorethanoneofthesemetrics.
CriticalIssue1:EdgeThresholdingandComparingUnequalNetworks
Wenowfocusonedgethresholding—thatis,howtotransformacontinuous
measureofFCintoanedgeinthegraph.Inourreview,39%ofstudiesbinarizedFCvalues
suchthatedgeswereeitherpresentorabsentinthegraph,whereas45%ofstudies
retainedFCasedgeweights(seeTable2b).Regardlessofwhetherinvestigatorsanalyze
binaryorweightednetworks,therearefundamentalchallengestocomparingnetworks
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thatdiffereitherintermsofaveragedegree(k)orthenumberofnodes(N)(Fornitoetal.,
2016;vanWijketal.,2010).Inparticular,comparinggroupsongraphmetricssuchaspath
lengthandclusteringcoefficientcanbeambiguousbecausethesemetricshave
mathematicaldependenciesonbothkandN.
Mostbrainparcellationapproachesdefinegraphswithanequivalentnumberof
nodes(N)ineachgroup.Ontheotherhand,connectiondensityisoftenavariableof
interestinclinicalstudieswherethepathologymayalternotonlyconnectionstrength,but
alsothenumberofconnections.IfNisconstant,variationinkbetweengroupsconstrains
theboundariesoflocalandglobalefficiency.Iftwogroupsdiffersystematicallyinedge
density,thisalmostguaranteesbetween-groupdifferencesinmetricssuchasclustering
coefficientandpathlength.Determiningwheretointerveneinthiscircularproblemhas
greatimportanceinclinicalnetworkneuroscience,wherehypothesesoftenfocusonthe
numberandstrengthofnetworkconnections.
Toaddressthisissue,severalinvestigatorshaverecommendedproportional
thresholding(PT)inwhichtheedgedensityisequatedacrossnetworks(Achard&
Bullmore,2007;Bassettetal.,2009;Power,Fair,Schlaggar,&Petersen,2010).
Furthermore,toreducethepossibilitythatfindingsarenotspecifictothechosendensity
threshold,29%ofstudieshavetestedforgroupdifferencesacrossarange(e.g.,5-25%;
Table4).However,wearguethatdefiningedgesbasedonPTmaynotbeidealforclinical
studies,wherethereareoftenregionaldifferencesinfunctionalcouplingorpathology-
inducedalterationsinthenumberoffunctionalconnections(Hillary&Grafman,2017).For
example,indepression,thedorsomedialprefrontalcortexexhibitsenhancedconnectivity
withdefaultmode,cognitivecontrol,andaffectivenetworks(Sheline,Price,Yan,&Mintun,
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Graphtheoryasatooltounderstandbraindisorders
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2010).Asdemonstratedbelow,whenFCdiffersinselectedregionsbetweengroups,PTis
vulnerabletoidentifyingspuriousdifferencesinnodalstatistics(e.g.,degree).The
concernsexpressedhereextendfromvandenHeuvelandcolleagues(2017),who
demonstratedthatPTincreasesthelikelihoodofincludingspuriousconnectionsinthe
networkwhengroupsdifferinmeanFC.
Table4.Thresholdingmethodfordefiningedgesingraphs
Natureofthresholding n(frequency)Allconnectionsretained 6(5.7%)Singlebyvalue 25(23.6%)Multiplebyvalue 28(26.4%)Singlebydensity 5(4.7%)Multiplebydensity 31(29.2%)Bothbyvalueanddensity 7(6.6%)Other(connectionslost) 3(2.8%)Unknown/unclear 1(0.9%)
Note.Value:thresholdingdeterminedbyFCstrength;Single:analysesreportedatasinglethresholdvalue;Multiple:networkexaminedacrossmultiplethresholds
SimulationtodemonstrateaproblemwithPTingroupcomparisons:Whack-
a-node.Inthefirstsimulation—‘whack-a-node’—weexaminedtheconsequencesofPT
onregionalinferenceswhengroupsdifferinFCforselectedregions.Unlikeempirical
resting-statefMRIdata,wheretheunderlyingcausalprocessesremainrelativelyunknown,
simulationsallowonetotesttheeffectofbiologicallyplausiblealterations(e.g.,
hyperconnectivityofcertainnodes)onnetworkanalayses.Simulationscanalsoclarifythe
effectsofalternativeanalyticdecisionsonsubstantiveconclusionsinempiricalstudies.For
detailsonoursimulationapproach,seeMethods.Briefly,weusedanetworkbootstrapping
approachtosimulateresting-statenetworkdataforacase-controlstudywith50patients
and50controls.Werepeatedthissimulation100times,increasingconnectivityfor
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Graphtheoryasatooltounderstandbraindisorders
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patientsinthreerandomlytargetednodes(hereaftercalled‘Positive’)anddecreasing
connectivityslightly,butnonsignficantly,inthreerandomnodes(‘Negative’).Wecompared
changesinPositiveandNegativenodestothreeComparatornodesthatdidnotdiffer
betweengroups.
Resultsofwhack-a-nodesimulation.
Proportionalthresholding.Inamultilevelregressionofgroupdifferencet-statistics
(patient–control)ondensitythreshold,wefoundthatPTwassensitiveto
hyperconnectivityofPositivenodes,reliablydetectinggroupdifferences,averaget=12.4
(SD=1.15),averagep<.0001(Figure2a).Importantly,however,degreewassignificantly
lowerinpatientsthancontrolsforNegativenodes,averaget=-6.52(SD=1.15),averagep
<.001.WedidnotfindanysystematicdifferencebetweengroupsinComparatornodes,
averaget=-.22(SD=.65),averagep=.47.
Thesegroupdifferenceswerequalifiedbyasignificantdensityxnodetype
interaction(p<.0001)suchthatgroupdifferencesforPositivenodeswerelargerathigher
densities(Figure3,toppanel),B=11.32(95%CI=10.59–12.04),t=30.72,p<.0001.
Conversely,wefoundequal,butopposite,shiftsinNegativenodes(Figure3,middlepanel):
groupdifferencesbecameincreasinglynegativeathigherdensities,B=-11.93,(95%CI=-
12.65–-11.2),t=-32.74,p<.0001.However,wedidnotobserveanassociationbetween
densityandgroupdifferencesinComparatornodes,B=.31,p=.40(Figure3,bottom
panel).
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Figure2.Theeffectofthresholdingmethodongroupdifferencesindegreecentrality.Thecentralbarofeachrectangledenotesthemediantstatistic(patient–control)across100replicationdatasets(patientn=50,controln=50),whereastheupperandlowerboundariesdenotethe90thand10thpercentiles,respectively.Thedarklineatt=0reflectsnomeandifferencebetweengroups,whereasthelightgraylinesatt=-1.99and1.99reflectgroupdifferencesthatwouldbesignificantatp=.05.a)Graphsbinarizedat5%,15%,and25%density.b)Graphsbinarizedatr={.2,.3,.4}.c)Graphsbinarizedatr={.2,.3,.4},withdensityincludedasabetween-subjectscovariate.d)Strengthcentrality(weightedgraphs).
−505
1015
0.05 0.15 0.25Density
Mea
n t
Density thresholda)
−505
1015
0.4 0.3 0.2Pearson r
FC Thresholdb)
−505
1015
0.4 0.3 0.2Pearson r
Mea
n t
FC threshold, covary densityc)
−505
1015
Node TypePositiveNegativeComparator
Weightedd)
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Figure3.Theeffectofdensitythresholdongroupdifferencesindegreecentrality.Dotsdenotethemeantstatisticatagivendensity,whereasverticallinesdenotethe95%confidenceinterval.Allstatisticsreflectgroupdifferencesindegreecentralitycomputedongraphsbinarizedatdifferentdensities.
Functionalconnectivity(FC)thresholding.Ingraphsthresholdedatdifferinglevelsof
FC(rsrangingbetween.2and.5),wefoundreliableincreasesinPositivenodesinpatients,
averaget=6.37,averagep<.0001(Figure2b).Negativenodes,however,werenot
significantlydifferentbetweengroups,averaget=-1.75,averagep=.23.Neitherdid
Comparatornodesdifferbygroup,averaget=.05,p=.50.UnlikePT,forFC-thresholded
graphs,wedidnotfindasignificantthresholdxnodetypeinteraction,𝜒2(16)=13.38,p=
.65.
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Comparator
Negative
Positive
0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25
11
12
13
−7
−6
−5
−0.4
−0.2
Density threshold
Aver
age
grou
p di
ffere
nce t
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Graphtheoryasatooltounderstandbraindisorders
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Functionalconnectivitythresholdingwithdensityascovariate.Bydefinition,FC
thresholdingcannothandletheproblemofgroupdifferencesinmeanFC.Thus,theriskof
FCthresholdingaloneisthatnodalstatisticsmayreflectgroupdifferencesinmeanFCthat
affectinterpretationoftopologicalmetrics(e.g.,smallworldness).Tomitigatethisconcern,
onecouldthresholdatatargetFCvalue,thenincludegraphdensityforeachsubjectasa
covariate(assuggestedbyvandenHeuveletal.,2017).AsdepictedinFigure2c,however,
althoughthisstatisticallycontrolsfordensity,italsoreintroducestheconstraintthatthe
groupsmustbeequalinaveragedegree.Asaresult,thepatternofeffectsmirrorsthePT
graphs(Figures2a,3).Morespecifically,groupdifferencesinPositivenodeswerereliably
detected,averaget=12.39,averagep<.0001.Butpatientsappearedtobesignificantly
morehypoconnectedinNegativenodescomparedtocontrols,averaget=-6.59,averagep
<.001.
Weightedanalysis.Inanalysesofstrengthcentrality,weobservedsignificantly
greaterdegreeinPositivenodes,averaget=5.5,averagep=.0003(Figure2d).Asexpected
givenoursimulationdesign,NegativeandComparatornodeswerenotsignificantly
differentbetweengroups,averageps=.19and.49,respectively.
Discussionofwhack-a-nodesimulation.Inour‘whackanode’simulation,three
nodeswererobustlyhyperconnectedin‘patients’,whilethreenodeswereweakly
hypoconnected.Theprincipalfindingofthissimulationwasthatenforcingequalaverage
degreeusingPTcanspuriouslymagnifychangesingroupcomparisonsofnodalstatistics.
Whenthegroupswereotherwiseequal,hyperconnectivityinselectednodeswas
accuratelydetectedusingPTacrossdifferentgraphdensities.However,nodesthatwere
weaklyhypoconnectedtendedtobeidentifiedasstatisticallysignificant.Theinclusionof
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Graphtheoryasatooltounderstandbraindisorders
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lowmagnitude,spuriousconnectionsintothenetworkinheresfromthewayinwhichPT
handlesthetailsoftheFCdistribution.ByretainingonlyedgesatthehighendoftheFC
distribution,edgesthatareonthecuspofthatcriterionaremostvulnerabletobeing
removed.Forexample,atadensityof25%,smallvariationinFCstrengthnearthe75th
percentilecouldleadtoinclusionoromissionofanedge.Asaresult,ifFCforedges
incidenttoagivennodetendstobeweakerinonegroupthantheother,thenbinarygraphs
generatedusingPTwillmagnifythestatisticalsignificanceofdifferencesindegree
centrality.TotheextentthatnodaldifferencesinFCstrengthrepresentashiftinthe
centraltendencyofthedistribution,thisproblemisnotsolvedbyapplyingmultipledensity
thresholds(Figure3).WeobservedthesameproblemifthedirectionofFCchangeswas
flippedinthesimulation:underPT,groupcomparisonsweresignificantforweakly
hyperconnectednodesifsomenodeswererobustlyhypoconnected(detailsavailablefrom
thefirstauthoruponrequest).
WeexaminedFCthresholding(here,usingPearsonrasthemetric)andweighted
analysesascomparisonstoPT.Thesemethodsdonotsufferfromthespuriousdetectionof
nodaldifferencesevidentunderPT.Rather,FCthresholdingaccuratelydetected
hyperconnectednodesacrossdifferentthresholdswhilenotmagnifyingsignificanceofthe
weaklyhypoconnectednodes.However,asnotedelsewhere(vandenHeuveletal.,2017;
vanWijketal.,2010),iftwogroupsdifferinmeanFC,thresholdingatagivenlevel(e.g.,r=
.3)inbothgroupswillleadtodifferencesingraphdensity.Thiscouldmanifestas
widespreaddifferencesinnodalstatisticsduetoglobaldifferencesinthenumberofedges.
Weconsideredwhetherincludingper-subjectgraphdensityasacovariateingroup
differenceanalysescouldretainthedesirableaspectsofFCthresholdingwhileaccounting
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Graphtheoryasatooltounderstandbraindisorders
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forthepossibilityofglobaldifferencesinFC.Wefound,however,thatstatistically
covaryingfordensitywasqualitativelysimilarinitseffectstoPTbecauseitconstrainsthe
sumofdegreechangesacrossthenetworktobezerobetweengroups(i.e.,equalaverage
degree).
ThegoalofoursimulationwastoprovideaproofofconceptthatPTmaynegatively
affectnodalstatisticsincase-controlgraphstudiesbyenforcingequalaveragedegree.We
didnot,however,testarangeofparameterstoidentifytheconditionsunderwhichthis
concernholdstrue.Thesimulationfocusedspecificallyondegreecentralityinthebinary
caseandstrengthintheweightedcase.Whileuntested,weanticipatethattheseeffects
likelygeneralizetoothernodalmeasuressuchaseigenvectorcentrality.Importantly,the
problemswithPThighlightedaboveoccurregardlessofedgedensity(Figure3),sotheuse
ofmultipleedgedensitiesdoesnotadequatelyaddressthe“whack-a-node”issue.
CriticalIssue2:Matchingtheorytoscale:Telescopinglevelsofanalysisingraphs
Thesecondmajormethodologicalthemefromourliteraturereviewconcernsthe
alignmentbetweenneurobiologicalhypothesesandgraphanalyses.Werefertothisas
theory-to-scalematching.RSFCgraphsoffertelescopinglevelsofinformationabout
intrinsicconnectivitypatterns,fromglobalinformationsuchasaveragepathlengthto
detailssuchasconnectivitydifferencesinaspecificedge.Forexample,inmajordepression,
resting-statestudieshavefocusedanalysesonspecificsubnetworkscomposedofthe
dorsalmedialprefrontalcortex,anteriorcingulatecortex(ACC),amygdala,andmedial
thalamus(forareview,seeWang,Hermens,Hickie,&Lagopoulos,2012).
Weproposethatgraphanalysesshouldbeconceptualizedandreportedintermsof
telescopinglevelsofanalysisfromglobaltospecific:topology,modularstructure,nodal
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Graphtheoryasatooltounderstandbraindisorders
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effects,edgeeffects.Ourreviewoftheclinicalnetworkneuroscienceliteraturerevealed
thatthemajorityofstudies(69%;seeTable5)testedwhethergroupsdifferedonglobal
metricssuchassmall-worldness.Importantly,however,manystudiesprovidedlimited
theoreticaljustificationforwhythepathophysiologyofagivenbraindisordershouldalter
theglobaltopologyofthenetwork.Inthefollowing,weprovidemorespecificcommentson
pathlengthandsmall-worldness,andhowinvestigatorsmightideallymatchhypothesesto
levelsofanalysis.
Table5.Levelofgroupdifferencehypothesesingraphanalyses(i.e.,telescoping)
Hypothesislevel n(frequency)Global 73(68.9%)Module(subnetwork) 41(38.7%)Nodal(Region) 52(49.1%)Edge 1(0.9%)Note.Global:examiningwhole-graphnetworkfeatures(e.g.,small-worldness),Module:examiningsubnetworks/modules(e.g.,defaultmodenetwork).Totalfrequencyisgreaterthan100%becausesomestudiestestedhypothesesatmultiplelevels.
Theclinicalmeaningfulnessofsmall-worldness.Inadefiningstudyfornetwork
neuroscience,WattsandStrogatz(1998)demonstratedthattheorganizationofthecentral
nervoussysteminC.elegansreflecteda“small-world”topology(cf.Muldoon,Bridgeford,&
Bassett,2016).Theimpactofthisfindingcontinuestoresonateinthenetwork
neuroscienceliterature20yearslater(Hilgetag&Kaiser,2004;Sporns&Zwi,2004),with
manystudiesfocusingon“disconnection”andthelossofnetworkefficiencyasquantified
bysmall-worldness(31%ofthestudiesreviewedhere).Althoughsmall-worldtopologies
havebeenobservedinmoststudiesofbrainfunction(Bassett&Bullmore,2016),the
relevanceofthisorganizationforfacilitatinghumaninformationprocessingremains
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Graphtheoryasatooltounderstandbraindisorders
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unclear.Otherfeaturesofhumanneuralnetworks,suchasmodularity,mayhavemore
importantimplicationsfornetworkfunctioning(Hilgetag&Goulas,2015).Highernetwork
modularityreflectsagraphinwhichtheconnectionsamongnodestendtoformmore
denselyconnectedcommunities.Asoriginallyobservedinscale-freenetworks,graphswith
highermodularitytendtoberobusttorandomnetworkdisruption(Albert,Barabási,Carle,
&Dougherty,1998).
Itremainsuncertainthat,asageneralrule,brainpathologyshouldbereflectedina
measureofsmall-worldness.Forexample,asmall-worldtopologyispreservedevenin
experimentsthatdramaticallyreducesensoryprocessingviaanesthesiainprimates
(Vincentetal.,2007)andindisordersofconsciousnessinhumans(Croneetal.,2014).
Recognizingthatconventionalmeasuresofsmall-worldness(e.g.,Humphries&Gurney,
2008)dependondensityanddonothandlevariationinconnectionstrength,recent
researchhasrecastthisconceptanditsquantificationintermsofthe“smallworld
propensity”ofanetwork(Muldoonetal.,2016).
Althoughwedonotdisputethevalueofglobalgraphmetricssuchassmall-
worldness,bydefinition,theyprovideinformationatonlythemostmacroscopiclevel.
Groupdifferencesinglobalmetricsmaylargelyreflectmorespecificeffectsatfinerlevels
ofthegraph.Forexample,removingconnectionsinfunctionalhubregionsselectivelytends
toreduceglobalefficiencyandclustering(Hwang,Hallquist,&Luna,2013).Likewise,
failingtoidentifygroupdifferencesinglobalstructuredoesnotimplyequivalenceatother
levelsofthegraph(e.g.,nodesormodules).Todemonstratethepointthatsubstantial
groupdifferencesinfinerlevelsofthegraphmaynotbeevidentinglobalmetrics,we
conductedasimulationinwhichthegroupsdifferedsubstantiallyinmodularstructure.
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Graphtheoryasatooltounderstandbraindisorders
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Simulationtodemonstratetheimportanceofunderstandinggraphsat
multiplelevels:globalinsensitivitytomodulareffects.Extendingthebasicapproachof
ourwhack-a-nodesimulation,weuseda13-moduleparcellationofthe264-node
groundtruthgraphinasimulationof50‘controls’and50‘patients’(modularstructure
fromPoweretal.,2011).Morespecifically,weincreasedFCinthefronto-parietalnetwork
(FPN)anddorsalattentionnetwork(DAN)incontrols,andincreasedFCinthedefault
modenetwork(DMN)inpatients.Thesimulationprimarilyexaminedgroupdifferencesin
small-worldness(globalmetric)andwithin-andbetween-moduledegree(nodalmetrics).
AdditionaldetailsareprovidedintheMethods.
Resultsofglobalinsensitivitysimulation.Consistentwithcommonmethodsinthe
field,weappliedproportionalthresholding(PT)between7.5%and25%toeachgraph.We
computedsmall-worldness,𝜎,ateachthreshold(seeMethods),thentestedforgroup
differencesinthiscoefficient.AsdepictedinFigure4,wedidnotobserveanysignificant
groupdifferencesinsmall-worldnessatanyedgedensity,Mt=.36,ps>.3.
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Graphtheoryasatooltounderstandbraindisorders
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Figure4.Groupdifferencesinsmallworldness(𝜎)asafunctionofedgedensity.Thecentralbarofeachrectangledenotesthemedian𝜎statistic(patient–control),whereastheupperandlowerboundariesdenotethe90thand10thpercentiles,respectively.
However,consistentwiththestructureofoursimulations,wefoundlargegroup
differencesinwithin-andbetween-moduledegree(Figure5).Inamultilevelregressionof
within-networkdegreeongroup,density,andmodule,wefoundasignificantDMNincrease
inpatientsirrespectiveofdensity,B=1.37,t=33.25,p<.0001.Likewise,controlshad
significantlyhigherwithin-networkdegreeintheFPNandDAN,ts=21.67and13.52,
respectively,ps<.0001.Thesefindingsweremirroredingroupanalysesofbetween-
networkdegreeintheDMN,FPN,andDAN,ps<.0001(seeFigure5).
1.08
1.12
1.16
1.20
1.24
10 15 20 25Density (%)
Smal
l wor
ldne
ss (σ
)
Groupcontrolspatients
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Graphtheoryasatooltounderstandbraindisorders
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Figure5.Groupdifferencesinz-scoreddegreestatisticsat10and20%density.Degreedifferencesforconnectionsbetweenmodulesaredepictedinthetoprow,whereaswithin-moduledifferencesareinthebottomrow.Dotsdenotethemeanzstatisticacrossnodes,whereasthelinesrepresentthe95%confidenceintervalaroundthemean.NonsignificantgroupdifferencesintheVisualnetworkaredepictedforcomparison,whereasconnectivityintheDAN,FPN,andDMNwasfocallymanipulatedinthesimulation.DAN=DorsalAttentionNetwork;FPN=Fronto-ParietalNetwork;DMN=DefaultModeNetwork.
Discussionofglobalinsensitivitysimulation.Intheglobalinsensitivitysimulation,
weinducedlargegroupchangesinFCattheleveloffunctionalmodulesthatrepresent
canonicalresting-statenetworks(e.g.,theDMN).Thesimulationdifferentiallymodulated
FCwithinandbetweenregionsoftheDAN,FPN,andDMN.Ingroupanalysesofwithin-and
between-moduledegreecentrality,wedetectedtheselargeshiftsinFC.However,despite
robustdifferencesinnetworkstructure,thetwogroupswereverysimilarinthesmall-
worldpropertiesoftheirgraphs.
●●
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10% Density 20% Density
Between−Module
Within−M
odule
−0.5 0.0 0.5 −0.5 0.0 0.5
DMN
FPN
DAN
Visual
DMN
FPN
DAN
Visual
Degree (z)
Mod
ule Group
●
●
controlspatients
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Graphtheoryasatooltounderstandbraindisorders
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Aswiththewhack-a-nodesimulation,wedidnotseektotesttherangeofconditions
underwhichthesefindingswouldhold.Rather,theglobalinsensitivitysimulationprovides
aproofofconceptthatresearchersshouldbeawarethattheabsenceofgroupdifferences
atahigherlevelofthegraph(here,globaltopology)doesnotsuggestthatthenetworksare
otherwisesimilaratlowerlevels(here,modularstructure).Aswehavenotedabove,in
graphanalysesofcase-controlresting-statenetworks,weencourageresearcherstostate
theirstudygoalsintermsthatclearlymatchthehypothesestothescaleofthegraph.
Intheglobalinsensitivitysimulation,failingtodetectdifferencesinsmall-worldness
shouldnotbeseenasanomnibustestofmodularornodalstructure.Likewise,ifgroup
differencesaredetectedatthegloballevel,theremaybesubstantialvalueininterrogating
finerdifferencesinthenetworks,evenifthesewerenothypothesizedapriori.
GeneralSummaryandConclusions
Theoverarchinggoalofthisreviewwastopromotesharedstandardsforreporting
findingsinclinicalnetworkneuroscience.Oursurveyoftheresting-statefunctional
connectivityliteraturerevealedthepopularityandpromiseofgraphtheoryapproachesto
networkorganizationinbraindisorders.Thispotentialisevidentinlarge-scaleinitiatives
foracquiringandsharingresting-statedataindifferentpopulations(e.g.,theHuman
ConnectomeProject;Barchetal.,2013).Publiclyavailableresting-statedatacanalso
supportreproducibilityeffortsbyservingasreplicationdatasetstocorroboratespecific
findingsinanindependentsample(e.g.,Jalbrzikowskietal.,2017).Wesharethefield’s
enthusiasmforsuchworkandanticipatethatwithfurthermethodologicalrefinementand
standardization,thecouplingofnetworkscienceandbrainimagingcanprovidenovel
insightsintotheneurobiologicalbasisofbraindisorders.
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Graphtheoryasatooltounderstandbraindisorders
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Ourreview,however,suggestedthattheheterogeneityofmethodsispreventingthe
fieldfromrealizingitspotential.Graphanalysesacrossclinicalstudiesvariedsubstantially
intermsofbrainparcellation,FCquantification,andtheuseofthresholdingmethodsto
defineedgesingraphs.Thesedecisionsarefundamentaltographtheoryandprecede
analysesatspecificlevelssuchasglobaltopology.Inaddition,althoughitwasnotafocusof
ourreview,therewassubstantialvariationinwhatnetworkmetricswerereportedacross
studies.Thelackofstandardizationinmethodsatmultipledecisionpointshas
multiplicativeconsequences:thelikelihoodthatanytwostudiesusedthesameparcellation
scheme,FCdefinition,thresholdingstrategy,andnetworkmetricwasremarkablylow.
Thismakesformalmeta-analysesoftheclinicalnetworkneuroscienceliteraturevirtually
impossibleatthepresenttime,detractingfromeffortstodistinguishdistinct
pathophysiologicalmechanismsortoidentifytransdiagnosticcommonalities.Inaddition,
methodologicalheterogeneityingraphanalysesundercutsthevalueofdatasharingefforts
thathavemademassivedatasetsavailabletothenetworkneurosciencecommunity.For
theimmensepotentialofdatasharingtoberealized,standardizationmustoccurnotonly
indataacquisition,butalsoindataanalysis,withasharedframeworktoguidehypothesis-
to-scalematchingingraphs.
Tomoveforward,webelievethatthefieldshouldworktowardaprincipled,
commonapproachtographanalysesofRSFCdata.Thisisachallengingproposition
becauseoftherapidandexcitingdevelopmentsinfunctionalbrainparcellation(e.g.,
Schaeferetal.,inpress),FCdefinition(e.g.,Cassidyetal.,2015),edgethresholding(van
denHeuveletal.,2017),andnetworkmetrics(e.g.,Vargas&Wahl,2014).Such
developmentshighlightboththeenthusiasmfor,andrelativeinfancyof,network
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Graphtheoryasatooltounderstandbraindisorders
29
neuroscienceasafield.Althoughwearesensitivetotheimportanceofcontinuingtorefine
functionalparcellationsofthebrain,wealsoseegreatvalueindevelopingfield-standard
parcellationstopromotecomparability.Indeed,manyaspectsofnetworkstructure(e.g.,
homogeneityoffunctionalconnectivitypatternswithinaregion)arelargelyconvergent
aboveacertainlevelofdetail(likely200-400nodes)inthefunctionalparcellation
(Craddocketal.,2012;Schaeferetal.,inpress).Likewise,theoptimalapproachfor
quantifyingfunctionalconnectivityisanopenquestion(Smithetal.,2011),yetinthe
absenceofmethodologicalconvergence,graphswereoftennotcomparableacrossstudies.
Arelateddilemmawasthatin57%ofstudies,littleornodetailwasprovidedabout
hownegativeFCvalueswereincorporatedintographanalyses.Thiswasespecially
troublinginsofarasglobalsignalregressiontendstoyieldanFCdistributioninwhich
approximatelyhalfofedgesarenegative(Murphyetal.,2009).Furthermore,FCestimates
atthelowendofthedistributionmayhavedifferenttopologicalpropertiessuchasreduced
modularity(Schwarz&McGonigle,2011).Evenamongthe21%ofstudiesinwhich
negativeedgeswereexplicitlydropped,itremainsunclearwhatconsequencesthis
decisionhasonsubstantiveconclusionsaboutgraphstructure.Inrecentyears,therehave
beenadvancesinquantifyingcommongraphmetricssuchasmodularityinweighted
networksthatincludenegativeedges(Rubinov&Sporns,2011),aswellasincreasingcalls
forweighted,notbinary,graphanalyses(Bassett&Bullmore,2016).Regardless,we
believethatgreaterclarityinreportingofnegativeFCwillpromotecomparisonsamong
clinicalstudies.
Methodologicalheterogeneityalsoresultedinveryfewgraphstatisticsthatwere
reportedincommonacrossstudies,anessentialingredientforexaminingthe
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Graphtheoryasatooltounderstandbraindisorders
30
reproducibilityoffindings.Networksthatareresilienttothedeletionofspecificedges
oftenhaveahighlyskeweddegreedistribution(Callaway,Newman,Strogatz,&Watts,
2000)thatmayrelatetosmall-worldnetworkproperties(Achardetal.,2006).Wepropose
thatstudiesshouldroutinelydepictthisdistribution.Likewise,broadmetricssuchasedge
density,meanFC,transitivity,andcharacteristicpathlengthprovideimportant
informationaboutthebasicpropertiesofgraphsthatcontextualizemoredetailed
inferentialanalyses.Thechallengeofdevelopingreportingstandardsinclinicalnetwork
neuroscienceechoesthebroaderconversationinneuroimagingaboutreproducibility,
especiallytheimportanceofdetailandtransparencyintheanalyticapproach(Nicholset
al.,2017).
Inadditiontothegeneralissuesofstandardizinggraphanalysesandreporting
procedures,ourreviewexaminedtwocriticalissuesingreaterdetail.First,weconsidered
thepotentialbenefitsandrisksofusingproportionalthresholding(PT),acommon
procedureforequatingthenumberofedgesbetweengraphs.Second,wearticulatedthe
valueofconsideringthetelescopinglevelsofgraphstructuresinordertomatcha
hypothesistothecorrespondingscaleofthegraph.
RoughlyonethirdofthestudiesincludedinourreviewappliedPT,thresholding
graphsatsingleormultipleedgedensities.Althoughthisalignswithpreviousguidance
(Achard&Bullmore,2007;vandenHeuvel,Stam,Boersma,&HulshoffPol,2008),its
applicationinclinicalstudiesisoftenconceptuallyproblematic.Manybrainpathologies
appeartoaffecttargetedregionsornetworks,whileleavingtheconnectivityofother
regionslargelyundisturbed.Forexample,althoughfrontolimbiccircuitryisheavily
implicatedinmooddisorders(Price&Drevets,2010),visualnetworkslargelyappear
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Graphtheoryasatooltounderstandbraindisorders
31
unaffected.Thereisgrowingevidencethatbraindisordersalterthestrengthoffunctional
couplingandpotentialthenumberoffunctionalconnections(Hillary&Grafman,2017).
Consequently,ifsomeregionsareaffectedbythepathology,butothersaresimilartoa
matchedcontrolpopulation,PTmayerroneouslyremoveoraddconnectionstographsin
onegroupinordertomaintainequalaveragedegreebetweengroups.Furthermore,ifa
brainpathologyaltersthedensityoffunctionalconnections—forexample,neurological
disruptionisassociatedwithhyperconnectivity(Hillaryetal.,2015)—PTwillpreclude
theinvestigatordetectingdensitydifferencesbetweengroups.If,intruth,thegroupsdiffer
inedgedensity,artificiallyequatingdensityalsodetractsfromtheinterpretabilityofgraph
analyses(cf.vandenHeuveletal.,2017).
Inadditiontotheseconceptualproblems,ourwhack-a-nodesimulation
demonstratedthatPTmayresultinthedetectionofspuriousgroupdifferences(Figure2a).
Altogether,applyingPTinclinicalstudiesmaybeamethodologicaldoublejeopardy,
characterizedbyreducedsensitivitytopathology-relateddifferencesinconnectiondensity
andtheriskofidentifyingnodaldifferencesbetweengroupsthatareastatisticalartifact.
TheserisksmakePTespeciallyunappealingwhenoneconsidersthatFC-based
thresholdingandweightedanalysesaccuratelydetectedgroupdifferences(Figure2b,d)
whilealsoallowingedgedensitytovary.
Wethereforehavetworecommendationsforedgethresholdingincase-control
comparisons.First,weightedanalysesorFCthresholdingshouldtypicallybepreferredto
PTifoneisinterestedinnodalstatistics.Second,toruleoutthepossibilitythatnodal
findingsreflectglobaldifferencesinmeanFC,onecouldincludemeanFCasacovariatein
weightedanalysesorper-subjectdensityinanalysesofFC-thresholdedbinarygraphs.
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Graphtheoryasatooltounderstandbraindisorders
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Crucially,weproposethatthesebetreatedassensitivityanalysesconductedonlyafter
establishinganodalgroupdifference.Thatis,ifoneidentifiesgroupdifferencesinFC-
thresholdedgraphs(e.g.,greaterdegreeinanteriorcingulatecortexamongpatients),does
includingedgedensityasacovariateabolishthisfinding?Ifso,itsuggeststhatdifferences
inglobaltopologymayaccountforthenodalfinding.However,oneshouldnotinclude
densityasacovariateinFC-thresholdedgraphsasafirststeptoidentifywhichnodesdiffer
betweengroups,asthiscouldfallpreytothe‘whackanode’problem(i.e.,spuriousnodal
effects).
Thesecondcriticalissuewasthatmanystudiesprovidedlimitedtheoretical
justificationforthealignmentbetweenagivenhypothesisandthecorrespondinggraph
analysis.Amajorityofstudies(69%)testedwhethergroupsdifferedinglobalmetricssuch
assmall-worldness,butmostpathologies(e.g.,braininjury,Alzheimer’sdisease)primarily
affectregionalhubswithinnetworks(Crossleyetal.,2014).Ourglobalinsensitivity
simulationfocusedontheimportanceofmatchinghypothesestographanalyses,or
telescoping.Wedemonstratedthatglobalgraphmetrics,specificallysmall-worldness,may
notbesensitivetogroupdifferencesinmoduleornodecentrality.Metricssuchas
modularityandnodalcentralityoffervitalinformationaboutregionalbrainorganization
thatcanbeinterpretedinthecontextofalterationsinaveragedegree.Thegeneralpointis
thatonecannotgeneralizefindingsfromonelevelofagraphtoanother,norshouldnull
effectsatonelevelbeviewedassuggestingthatthegroupsaresimilaratotherlevels.By
conceptualizingadataanalysisplanintermsofthetelescopingstructureofgraphs,
researcherscanclearlydelineateconfirmatoryfromexploratoryanalyses,whichis
consistentwiththespiritofreproduciblescienceinneuroimaging(Poldracketal.,2017).
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Graphtheoryasatooltounderstandbraindisorders
33
Insummary,thereisaneedforacommonframeworktoinformgraphtheory
analysesofRSFCdataintheclinicalneurosciences.Anyrecommendationsshouldemerge
organicallyfromascientificcommunitywhoseinvestigatorsvoluntarilyadoptprocedures
thatmaximizesensitivitytohypothesizedeffectsandsimultaneouslypermitgraph
comparisonsamongstudies(cf.theCOBIDASeffortinneuroimagingmorebroadly;Nichols
etal.,2017).Weanticipatethatacommonmethodologicalframeworkwillpromote
hypothesis-drivenresearch,alignmentbetweentheoryandgraphanalysis,reproducibility,
datasharing,meta-analyses,andultimatelymorerapidprogressofclinicalnetwork
neuroscience.
Methods
PubmedSearchSyntax
Neurologicaldisordersquery:
(graphORgraphicalORgraph-theor*ORtopology)AND(brainORfMRIORconnectivity
ORintrinsic)AND(resting-stateORrestingORrest)AND(neurologicalORbraininjuryOR
multiplesclerosisORepilepsyORstrokeORCVAORaneurysmORParkinson'sORMCIor
Alzheimer'sORdementiaORHIVORSCIORspinalcordORautismORADHDOR
intellectualdisabilityORDownsyndromeORTourette)AND"humans"[MeSHTerms]
Mentaldisordersquery:
(graphORgraphicalORgraph-theor*ORtopology)AND(brainORfmriORconnectivityOR
intrinsic)AND(resting-stateORrestingORrest)AND(clinicalORpsychopathologyOR
mentaldisorderORpsychiatricORneuropsychiatricORdepressionORmoodORanxiety
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Graphtheoryasatooltounderstandbraindisorders
34
ORaddictionORpsychosisORbipolarORborderlineORautism)AND"humans"[MeSH
Terms]
GeneralApproachtoNetworkSimulations
Toapproximatethestructureoffunctionalbrainnetworks,weidentifiedayoung
adultfemalesubjectwhocompleteda5-minuteresting-statescaninaSiemens3TTrio
Scanner(TR=1.5s,TE=29ms,3.1x3.1x4.0mmvoxels)withessentiallynohead
movement(meanframewisedisplacement[FD]=.08mm;maxFD=.17mm).We
preprocessedthedatausingaconventionalpipeline,including:1)motioncorrection(FSL
mcflirt);2)slicetimingcorrection(FSLslicetimer);3)nonlineardeformationtotheMNI
templateusingtheconcatenationoffunctional→structural(FSLflirt)andstructural→
MNI152(FSLflirt+fnirt)transformations;4)spatialsmoothingwitha6mmFWHMfilter
(FSLsusan);5)andvoxelwiseintensitynormalizationtoameanof100.Afterthesesteps,
wealsosimultaneouslyappliednuisanceregressionandbandpassfiltering,wherethe
regressorsweresixmotionparameters,averageCSF,averageWM,andthederivativesof
these(16totalregressors).Thespectralfilterretainedfluctuationsbetween.009and.08
Hz(AFNI3dBandpass).Wethenestimatedfunctionalconnectivity(FC)of264functional
ROIsfromthePowerandcolleaguesparcellation,whereeachregionwasdefinedbya5mm
radiusspherecenteredonaspecificcoordinate.Theactivityofaregionovertimewas
estimatedbythefirstprincipalcomponentofvoxelsineachROI,andthefunctional
connectivitymatrix(264x264)wasestimatedbythepairwisePearsoncorrelationsof
timeseries.
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Graphtheoryasatooltounderstandbraindisorders
35
This264x264adjacencymatrix,W,servedasthegroundtruthforallsimulations,
withspecificwithin-person,between-person,andbetween-groupalterationsapplied
accordingtoamultilevelsimulationofvariationacrossindividuals(detailsofsimulation
parametersprovidedinTableS1).Morespecifically,toapproximatepopulation-level
variabilityinacase-controldesign,wesimulatedresting-stateadjacencymatricesfor50
‘patients’and50‘controls’byintroducingsystematicandunsystematicsourcesof
variabilityforeachsimulatedparticipant.Systematicsourceswereintendedtotest
substantivehypothesesaboutproportionalthresholding(whack-a-nodesimulation)and
insensitivitytoglobalversusmodulardifferences(globalinsensitivitysimulation),whereas
unsystematicsourcesreflectedwithin-andbetween-personvariationinedgestrength.
Inthismodel,thesimulatededgestrengthbetweentwonodes,iandj,foragiven
subject,s,is:
𝑟%&' = 𝑤%& + 𝑔%&' +𝛼%&' + 𝑒%&' (1)
where𝑤%& istheedgestrengthfromthegroundtruthadjacencymatrixW.Globalvariation
inmeanFCisrepresentedby𝑔%&',whichreflectscontributionsofbothbetween-and
within-personvariation:
𝑔%&' = 𝑏' + 𝑢%&' (2)
where𝑏'representsnormallydistributedbetween-personvariationinmeanFC:
𝒃~𝑁(0, 𝜎7) (3)
with𝜎7controllingthelevelofbetween-personvariationinthesample.Theterm𝑢%&'
representswithin-personvariationofthisedgearoundthepersonmeanFC,𝑏'.Within-
personvariationinFCacrossalledgesisassumedtobenormallydistributed:
𝒖..'~𝑁(0, 𝜎:) (4)
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Graphtheoryasatooltounderstandbraindisorders
36
with𝜎: scalingthedegreeofwithin-personFCvariationacrossalledges.
Node-specificshiftsinFCarerepresentedby𝛼%&',whichincludesbothbetween-
personandwithin-nodecomponents.Morespecifically,themodulationofFCbetween
nodesiandjisgivenby:
𝛼%&' = 𝑎%' + 𝑣%&' (5)
wherebetween-personvariationinFCfornodeiis:
𝒂%.~𝑁(𝜇?@, 𝜎?@) (6)
with𝜇?% and𝜎?% capturingthemeanandstandarddeviationinFCshiftsfornodeiacross
subjects,respectively.EdgewiseFCvariationofanodeiacrossitsneighbors,j,isgivenby:
𝒗𝒊.𝒔~𝑁(0, 𝜎D%) (7)
where𝜎D@ representsthestandarddeviationofFCshiftsacrossneighborsofi.Whennodesi
andjwerebothmanipulated,theshiftswereappliedsequentiallysuchthatFCfortheedge
betweeniandjwasnotallowedtohavecompoundingchanges.Thatis,weset𝛼%&' = 0fori
>j.Finally,𝑒%&'representstherandomvariationinFCfortheedgebetweeniandjfor
subjects.Thisvariationwasassumedtobenormallydistributedacrossalledgesfora
subject:
𝒆..'~𝑁(0, 𝜎F) (8)
where𝜎F controlsthestandarddeviationofedgenoiseacrosssubjects.
Whack-a-NoteSimulationMethods
Asmentionedabove,proportionalthresholding(PT)isoftenappliedincase-control
studiestoruleoutthepossibilitythatnetworkdifferencesbetweengroupsreflect
differencesinthetotalnumberofedges.Importantly,ingraphsofequalorderN(i.e.,the
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Graphtheoryasatooltounderstandbraindisorders
37
samenumberofnodes),proportionalthresholdingequatesboththedensity,D,and
averagedegree, 𝑘 betweensubjects:
𝐷 = IJK(KLM)
, 𝑘 = IJK 𝑘 = 𝐷(𝑁 − 1) (9)
AsnotedbyvandenHeuvelandcolleagues(2017),whenoneappliesPT,differences
inaverageconnectivitystrengthcanleadtotheinclusionofweakeredgesinmoresparsely
connectedgroups.Furthermore,weakedgesestimatedbycorrelationaremorelikelyto
reflectanunreliablerelationshipbetweennodes.Thus,ifonegrouphaslowermean
functionalconnectivity,PTcouldintroducespuriousconnections,potentiallyundermining
groupcomparisonsofnetworktopology.
Thatis,whengroupsareotherwiseequivalent,thesumofincreasesindegreein
hyperconnectednodesforagroupmustbeoffsetbyequal,butopposite,decreasesin
degreeforothernodesinthatgroup.Thisphenomenonholdsbecauseofthemathematical
relationshipbetweenaveragedegreeandgraphdensity(Eq.9).Forsimplicity,ourfirst
simulationrepresentsthescenariowheretherearemeaningfulFCincreasesinpatientsfor
selectednodesandunreliableFCdecreasesinothernodes.Thisunreliabilityisintendedto
representsamplingvariabilitythatcouldleadtoerroneousfalsepositives.
Structureofwhack-a-nodesimulation.Asdescribedabove,wesimulated50
patientsand50controlsbasedonagroundtruthFCmatrix.Weestimated100replication
sampleswithequallevelsofnoiseinbothgroups(seeTableS1).Ineachreplication
sample,weincreasedthemeanFCinthreenodes,selectedatrandom,byr=0.14.Wealso
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Graphtheoryasatooltounderstandbraindisorders
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appliedsmallerdecreasesofr=-.04tothreeotherrandomlytargetednodes1.Thislevelof
decreasewaschosensuchthatgroupdifferencesinanalysesofnodalstrength(i.e.,
computedonweightedgraphs)werenonsignificantonaverage(Mp=.19,SDp=.02).We
appliedPTtobinarizegraphsineachgroup,varyingdensitybetween5%and25%in1%
increments.Likewise,forFCthresholding,webinarizedgraphsatrthresholdbetweenr=
.2and.5in.02increments.Finally,weretainedweightedgraphsforallsimulatedsamples
toestimategroupdifferencesinnodalstrength.Toensurethateffectswerenotattributable
toparticularnodes,weaveragedgroupstatisticsacrossthe100replicationsamples,where
thetargetednodesvariedrandomlyacrosssamples.
Ineachreplicationsample,weestimateddegreecentralityforthePositive
(hyperconnected),Negative(weaklyhypoconnected),andComparatornodes.Comparators
werethreerandomlyselectednodesineachsamplethatwerenotspecificallymodulated
bythesimulation.Theseservedasabenchmarktoensurethatsimulationsdidnotinduce
groupdifferencesincentralityfornodesnotspecificallytargeted.Toquantifytheeffectof
PTversusFCthresholdinginbinarygraphs,weestimatedgroupdifferences(patient–
control)ondegreecentralityusingtwo-samplet-testsforeachPositive,Negative,and
Comparatornode.Forweightedanalyses,weestimatedgroupdifferencesinstrength
centrality.
GlobalInsensitivitySimulationMethods:Hypothesis-to-scalematching
1ResultsarequalitativelysimilarusingothervaluesforgroupshiftsinFC.
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Graphtheoryasatooltounderstandbraindisorders
39
Wesimulatedadatasetof50‘controls’and50‘patients’inwhichthefronto-parietal
network(FPN)anddorsalattentionnetwork(DAN)weremodulatedincontrolscompared
tothegroundtruthmatrix,W.Inthissimulation,thepatientgrouphadincreasedFCinthe
defaultmodenetwork(DMN),acommonfindinginneurologicaldisorders(Hillaryetal.,
2015).Incontrols,weincreasedFCstrengthforedgesbetweenFPN/DANregionsandother
networks,Mr=0.2,SD=0.1.Per-modulevariationinbetween-networkFCchangeswas
assumedtobenormallydistributedwithineachsubject,SD=0.1.Wealsoincreased
controls’FConedgeswithintheFPNandDAN,Mr=0.1,between-subjectsSD=0.05,
within-subjectsSD=.05.Inpatients,between-networkFCforDMNnodeswasincreased,M
r=0.2,between-subjectsSD=0.1,within-subjectsSD=0.1.Likewise,within-networkFCin
theDMNwasincreased,Mr=0.1,between-subjectsSD=.05,within-subjectsSD=.05.That
is,weappliedsimilarlevelsofFCmodulationtotheFPN/DANincontrolsandtheDMNin
patients,althoughthesechangeslargelyaffecteddifferentedgesinthenetworksbetween
groups.
Wecomputedthesmall-worldnesscoefficient,𝜎,accordingtotheapproachof
HumphriesandGurney(2008):
𝜎 =𝐶Q/𝐶S?TU𝐿Q/𝐿S?TU
Here,𝐶Qrepresentsthetransitivityofthegraph,whereas𝐶S?TU denotesthe
transitivityofarandomgraphwithanequivalentdegreedistribution.Likewise,𝐿Qand
𝐿S?TU representthecharacteristicpathlengthofthetargetgraphandrandomlyrewired
graph,respectively.Valuesof𝜎above1.0correspondtoanetworkwithsmall-world
properties.Togeneratestatisticsforequivalentrandomgraphs,weappliedarewiring
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Graphtheoryasatooltounderstandbraindisorders
40
algorithmthatretainedthedegreedistributionofthegraphwhilepermuting347,160
edges(10permutationsperedge,onaverage).Thisalgorithmwasappliedtothetarget
graph100timestogenerateasetofequivalentrandomnetworks.Transitivityand
characteristicpathlengthwerecalculatedforeachofthese,andtheiraverageswereused
incomputingthesmall-worldnesscoefficient,𝜎.Wealsoanalyzedwithin-andbetween-
networkdegreecentralityforeachnode,z-scoringvalueswithineachmoduleanddensity
toallowforcomparisons.
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