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Graphtheoryasatooltounderstandbraindisorders

Graphtheoryapproachestofunctionalnetworkorganizationinbraindisorders:

Acritiqueforabravenewsmall-world

MichaelN.Hallquist1,2

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: fhillary@psu.edu.

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

.CC-BY-NC-ND 4.0 International licenseis made available under aThe copyright holder for this preprint (which was not peer-reviewed) is the author/funder. It. https://doi.org/10.1101/243741doi: bioRxiv preprint

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.

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

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.

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.

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.

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%)

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

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.

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,

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,

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

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

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,

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

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).

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

0.05 0.15 0.25Density

Density thresholda)

−505

0.4 0.3 0.2Pearson r

FC Thresholdb)

−505

0.4 0.3 0.2Pearson r

FC threshold, covary densityc)

−505

Node TypePositiveNegativeComparator

Weightedd)

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=

●●

● ●● ● ●

Comparator

Negative

Positive

0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25

−0.4

−0.2

Density threshold

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

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

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

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

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.

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.

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).

10 15 20 25Density (%)

ss (σ

Groupcontrolspatients

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.

●●

10% Density 20% Density

Between−Module

Within−M

−0.5 0.0 0.5 −0.5 0.0 0.5

Visual

Degree (z)

ule Group

controlspatients

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.

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

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

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

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.

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).

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

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.

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)

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

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

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.

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

algorithmthatretainedthedegreedistributionofthegraphwhilepermuting347,160

edges(10permutationsperedge,onaverage).Thisalgorithmwasappliedtothetarget

graph100timestogenerateasetofequivalentrandomnetworks.Transitivityand

characteristicpathlengthwerecalculatedforeachofthese,andtheiraverageswereused

incomputingthesmall-worldnesscoefficient,𝜎.Wealsoanalyzedwithin-andbetween-

networkdegreecentralityforeachnode,z-scoringvalueswithineachmoduleanddensity

toallowforcomparisons.

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