Community structure unveils the path multiplicity in complex networks

Domain: physics
Published: 2026-03-06
Editorial outcome: major_revision

Abstract
Networkswithcomplextopologiesdescribenumerousnaturalandsocial
systems.Recentstudiesonpathmultiplicityhaveshownstrongheterogeneity
inshortestpathsbetweennodepairsinreal-worldnetworks.However,the
mechanismunderlyingthisphenomenonremainsunexplored.Here,wereveal
thatcommunitystructureisakeyfactorshapingpathmultiplicity.Toexplore
theintrinsicfactorsthatinfluencepathmultiplicity,wefirstintroducethe
conceptofrelativepathmultiplicityandfindthatcommunitystructureismore
stronglycorrelatedwithpathmultiplicitythanothernetworkmetrics.
Throughtargetededge-rewiringexperiments,weverifythelinkbetweenpath
multiplicityandcommunitystructure.Theunderlyingmechanismcanbe
interpretedasaninterface-driveneffectthatsharplyincreasesthenumberof
shortestpaths.Inspiredbythesefindings,weproposeatribal-structure-based
networkmodelthatreproducesphenomenaobservedinreal-worldnetworks.
Ourworkenhancestheunderstandingofnetworkorganization,withpotential
applicationsinnetworkdesignandoptimization.

Introduction
Complexnetworksformthebackboneofmanynaturalandtechno- surprisinglyshortpathlengths,hasbeenextensivelyinvestigated7,26.
logicalsystemsandarecrucialtorevealingstructuralandfunctional Beyond the length of the shortest path, the centrality of the inter-
relationshipsinawidevarietyofdomains1,2.Fromsocialinteractions connectionshasbeenwidelystudiedasameasureoftheinfluenceof
andbiologicalprocessestotechnologicalsystems,thesenetworksare nodes(edges)onthebasisoftheirfrequencyofappearanceonthe
characterized by complex connections among their components3–6.
Despite the diversity of their applications, complex networks often shortest paths between other node pairs27,28. This metric is thus
exhibitcommontopologicalproperties,includingsmall-worldeffect7,8 essential for analyzing network flow and information diffusion and
and scale-free structures9,10, which have become central themes in identifying key nodes29. Furthermore, research on navigability,
networkscience11–25.Inthestudyofcomplexnetworks,pathsbetween namely, how easily nodes can be reached through paths, has been
node pairs play a critical role in understanding the dynamics and conductedtoexplorehowdecentralizedroutingdecisionsaremadein
efficiencyofthenetwork.Traditionalresearchhaslargelyfocusedon networks26.
Despitetheadvancesinthestudyofpaths,pathmultiplicity—the
the shortest routes between nodes, which serve as a fundamental number of equidistant shortest paths between node pairs—remains
measureofnetworkconnectivityandroutingefficiency2.Thesmall- unexplored.Recently,somestudieshaverevealedtheexistenceofa
world phenomenon, in which even large networks often have universalpower-lawscalingofpathmultiplicity,indicatingthatafew
nodepairshavealargenumberofshortestpaths30.Forexample,even ThePMIreflectsthecomplexityofpathselectionwithinthenet-
intheBn-Macaque-Rhesus-Brain-1brainnetworkwithonly242nodes31, work: a large PMI suggests that the network has a hesitant-world
thelargestcountofshortestpathsbetweennodepairscanreach649, property,withmanyshortestpaths between node pairs,potentially
withanaveragevalueof11.07.Thisobservationrevealsthattheworld leading to a more complex decision-making process. To reveal the
weliveinisnotjusta“small-world”,butalsoa“hesitant-world”.The relationshipsbetweenthePMIandclassicalnetworkmetrics,wepre-
“small-world” effect means that most nodes are connected through sentcorrespondingscatterplotsalongwiththePearson’scorrelation
onlyafewsteps,eveninalargenetwork;meanwhile,the“hesitant- coefficient\rho
p
34,Spearman’scorrelationcoefficient\rho
s
35,andthequad-
world”effectimpliesthatonemayhesitateamongnumerouschoices, rantcountratio(QCR)36,37in140real-worldnetworksinFig.1a.Pear-
evenwhenthenetworkitselfissmall.
son’s correlation coefficient quantifies the strength of a linear
Althoughthemultiplicityofshortestpathsisastructuralfeature, relationship between two variables,whereas Spearman’s correlation
it actively governs how networks function. First, path multiplicity coefficient is a nonparametric rank-based measure of monotonic
shapes a network’s robustness and vulnerability. While redundant associationandisoftenpreferredwhennonlinearityoroutliersmake
shortestpathscanpreserveconnectivitywhenlinksfail,theymayalso Pearson’scoefficientundesirable.TheQCRprovidesacoarse-grained
concentratetrafficoncommonbridgesorhubs,creatingbottlenecks measure of association by dichotomizing each variable at a central	hatincreasesusceptibilitytocongestionortargetedattacks—adual location (typically the median) and quantifying how often paired
rolenotedinstudiesofnetworkflows32.Second,itaffectsspreading observationsfallinthesamecorrespondinghalf(seethe“Methods”
processesandtransport.Multiple,equallyshortroutescanaccelerate sectionfordetails).Together,Pearson’scorrelation,Spearman’scor-
the propagation of information, diseases, or resources by offering relation,andQCRoffercomplementaryperspectivesonassociationby
paralleltransmissionchannels,whilealsoredistributingloadsacross targeting linear association, monotonic rank-based association, and
thenetworkandinfluencingdiffusionoutcomes20.Finally,ithascon- median-split concordance, respectively. Consistent results across
sequencesforroutinganddecision-making.Foranyagent—whethera thesemeasuresprovidearobustnesscheckforthereportedassocia-
packet,anindividual,oranalgorithm—navigatingamongmany“opti- tion.Overall,thescatterplotsexhibitadisordereddistribution,with
mal”pathscaninduce“choiceoverload,”apsychologicaleffectknown pointsbroadlydispersed,showingnocleartrend.Moreover,itcanbe
toimpairdecisionefficiency33,potentiallydegradingtheperformance observedthatallthreecorrelationmetricsarerelativelylow,indicating
ofnavigationstrategies26.
thatthereisnosignificantcorrelationbetweenPMIandclassicalnet-
Ourrecentfinding30—astrongpower-lawdistributioninpathmul- workmetrics.
tiplicityacrossdiversereal-worldnetworks—revealsthatthepathmul- Intuitively,accordingtothedefinitionofPMI,edgedensitymay
tiplicityisafundamentalandwidespreadarchitecturalsignature.Given affectpathmultiplicity.ThesimpleexamplenetworkshowninFig.1bis
thefunctionalrelevanceoutlinedabove,acentralunansweredquestion initially a chain network with an initial PMI value of 1. As the edge
emerges: which underlying network structure produces such strong densityincreases,andthusthenetworkbecomesfullyconnected,the
heterogeneityinpathmultiplicity?Uncoveringthisstructuraloriginis correspondingPMIvaluereturnsto1.Furthermore,thechangingtrend
essentialforexplainingobservednetworkdynamicsanddesigningnet- ofPMIastheedge densitypincreasesis shownin Fig.1c forthree
workswithdesiredfunctionalproperties.Inthisstudy,weintroducethe typicalmodelnetworks:Erdős-Rényirandomnetworks(ER)38,regular
concept of relative path multiplicity to systematically compare real- ring lattices (RRL)39, and cluster Barabási-Albert scale-free networks
worldnetworkswithequivalentrandomnetworks.Usingthisapproach, (CBA)40.Unsurprisingly,onecanseethatthePMIvaluesarestrongly
weidentifythestructuralfactorsthatshapepathmultiplicityincomplex relatedtoedgedensitypwithsomeformofcomplexnonlinearrela-
networks.Furthermore,wedevelopanetworkmodelthatcanrepro- tionship. To analyze the intrinsic factors influencing the PMI, it is
ducetheempiricalcharacteristics,offeringmechanisticinsightsintothe necessarytoexcludetheeffectsofnetworksizeandedgedensityon
observedpower-lawdistributionofpathmultiplicity.

Results
Conceptofrelativepathmultiplicity
LetG(V,E)beasimpleundirectedgraphrepresentingacomplexnet-
work,whereVisthesetofnodesandE⊆V×Visthesetofedges.Let
N = ∣V∣ and M = ∣E∣ represent the numbers of nodes and edges,
respectively. The adjacency matrix of the network is denoted by
A(G)=(a ij ) N×N ,wherea ij=a ji=1ifnodesv iandv jareconnectedand
a ij=a ji=0otherwise.Theedgedensityofthenetworkisdefinedas
follows:
p=hki=(N-1), (1)
where〈k〉denotestheaveragenodedegree.
node F s o v r i a a n n y d p v a j i i r s o d f en n o o t d e e d s b ( y v i, l ij v . j T ), h t e h n e w sh e o d r e te fi s n t e p t a h t e h p le a n th gt m h u b lt e i t p w li e c e it n y m ne a t n w ’s or c k or m re e la tr t i i c o s n a c l o o e n f g fic w ie it n h t, P a e n a d rs th o e n’ Q s C c R or f r o e r la 1 t 4 io 0 n re c a o l- e w ffi o c rl i d en n t e , t S w p o e r a k r s -
amount(PMA)betweentwonodesasthenumberofshortestpathsh
ij
inFig.2(seetheSupplementaryInformationforthemetadata).Given
m co u n l n ti e p c li t c i i n t g ym th a e t m rix , a (P n M d M th ). e T m o a q t u ri a x nt H if ð y G t Þ h = e ðh o ij v Þ e N r × a N ll i m s u c l a ti l p le l d ici t t h y e of pa th th e t g h ia a n t t re c a o l n -w n o e r c l t d ed ne c t o w m or p k o s n c e a n n t b o e f d ea is c c h on n n e e t c w t o e r d k ,w to e e fo n c s u u s re on co ly n o si n ste th n e t
pathofanetwork,wedefinethepathmultiplicityindex(PMI)asthe comparisons.First,itiseasytoseethatthesenetworkmetricsarenot
significantly linearly correlated with path multiplicity, where the
averagevalueofthePMAinallpairsofnodes.Mathematically,this
relationshipisexpressedasfollows:
P P number,globalefficiency,averageshortestpathlength,andnetwork
N h
\Phi(G)= i=1 j≠i ij : (2)
N(N-1)

TheRPMIreflectsthecomplexityofpathselectionwithinthenet- diameterarecorrelatedwithpathmultiplicitywith\rho s>0.6.Further-
work: a large PMI suggests that the network has a hesitant-world more,usingQCR,weobservethattheassociationbetweenthenumber
property,withmanyshortestpaths between node pairs,potentially of communities and path multiplicity reaches a remarkably high value
leading to a more complex decision-making process. To reveal the of 0.9857, where the data points are mostly located in the first and
relationshipsbetweenthePMIandclassicalnetworkmetrics,wepre- third quadrants in the scatter plot shown in Fig. 2a. However, the
sentcorrespondingscatterplotsalongwiththePearson’scorrelation majorityoftheQCRvaluesfortheothermetricsarebelow0.9,sug-
coefficient\rho
p
34,Spearman’scorrelationcoefficient\rho
s
35,andthequad- gestingthatthenumberofcommunitiesismoresignificantlycorre-
rantcountratio(QCR)36,37in140real-worldnetworksinFig.1a.Pear- latedwiththeRPMI.Inadditiontothescatterplots,wedisplaythe
son’s correlation coefficient quantifies the strength of a linear violinplotsofeachnetworkmetric,whicharedividedintotwopartson
relationship between two variables,whereas Spearman’s correlation thebasisofthemedianvaluesoftheRPMI.Ifastrongercorrelation
coefficient is a nonparametric rank-based measure of monotonic existsbetweenanetworkmetricandtheRPMI,theoverlapbetween
associationandisoftenpreferredwhennonlinearityoroutliersmake twoviolinsubplotsshouldbesmall.AsshowninFig.2,comparedwith
Pearson’scoefficientundesirable.TheQCRprovidesacoarse-grained other network metrics, the narrow overlap corresponding to the
measure of association by dichotomizing each variable at a central number of communities further demonstrates that the community
location (typically the median) and quantifying how often paired structure41,42playsakeyroleindeterminingpathmultiplicity.
observationsfallinthesamecorrespondinghalf(seethe“Methods” Moreover,wevisualize16networksoutofthetotalof140inFig.3,
sectionfordetails).Together,Pearson’scorrelation,Spearman’scor- wheretheupperrowpresents8networkswithhighRPMIvalues,and
relation,andQCRoffercomplementaryperspectivesonassociationby thelowerrowshows8networkswithlowerRPMIvalues.Inthisway,we
targeting linear association, monotonic rank-based association, and observethatnetworkswithhigherRPMIvaluestendtohavealarger
median-split concordance, respectively. Consistent results across number of communities.For example, the Bio-DM-LC network with
thesemeasuresprovidearobustnesscheckforthereportedassocia- n=483(\Phi e (G)=21:20)displayshighlymodularstructureswithmany
tion.Overall,thescatterplotsexhibitadisordereddistribution,with well-definedcommunities(n c=34).Incontrast,networkswithlower
pointsbroadlydispersed,showingnocleartrend.Moreover,itcanbe RPMIvalues,suchasBio-DR-CXwithn=3287(\Phi e (G)=1:49),exhibit
observedthatallthreecorrelationmetricsarerelativelylow,indicating fewerdistinctcommunities(n c=4).Thiscomparisonclearlydemon-
thatthereisnosignificantcorrelationbetweenPMIandclassicalnet- stratesthatnetworkswithhigherRPMIvaluestendtohaveagreater
workmetrics.
number of communities,whereasnetworkswithlowerRPMIvaluesare
Intuitively,accordingtothedefinitionofPMI,edgedensitymay generallycharacterizedbylessmodulartopologies.
affectpathmultiplicity.ThesimpleexamplenetworkshowninFig.1bis Consideringthataspecificcommunitydetectionalgorithmmay
initially a chain network with an initial PMI value of 1. As the edge influencethecausalattributionofRPMItocommunitystructure,we
densityincreases,andthusthenetworkbecomesfullyconnected,the havetestedclassicalcommunitydetectionalgorithms,includingthe
correspondingPMIvaluereturnsto1.Furthermore,thechangingtrend LeadingEigenvectormethod43,Walktrapmethod44,Leidenmethod45,
ofPMIastheedge densitypincreasesis shownin Fig.1c forthree Label Propagation method46, Infomap method41, and Louvain
typicalmodelnetworks:Erdős-Rényirandomnetworks(ER)38,regular method47.OurexperimentalresultsarereportedintheSupplementary
ring lattices (RRL)39, and cluster Barabási-Albert scale-free networks Information section. Although these algorithms yield somewhat
(CBA)40.Unsurprisingly,onecanseethatthePMIvaluesarestrongly differentcommunitycounts,theSpearmancorrelationbetweenRPMI
relatedtoedgedensitypwithsomeformofcomplexnonlinearrela- and the number of communities remains consistently high (all
tionship. To analyze the intrinsic factors influencing the PMI, it is \rho s>0.8),andallthecorrespondingQCRvaluesareabove0.9.These
necessarytoexcludetheeffectsofnetworksizeandedgedensityon results indicate that our findings do not depend on any particular
pathmultiplicity.Thus,itinspiresustointroducetheconceptofthe communitydetectionalgorithm;accordingly,weemphasizethatour
relativepathmultiplicityindex(RPMI),denotedby\Phi e (G),whichiscal- mainconclusionsarerobusttothechoiceofacommunitydetection
culatedbynormalizingthePMIofthenetworkGwiththePMIofan algorithm.
equivalentERrandomnetworkG withthesamesizeandedgedensity Impactofcommunitystructureonpathmultiplicity
ER
asfollows:
Toverifytherelationshipbetweenpathmultiplicityandcommunity
\Phi e (G)=\Phi(G) : (3)
structure,weimplementatarget-orientededgerewiringprocedure
\Phi(G ER)
onparadigmaticmodelnetworkswith1000nodes,namelyERran-
TheRPMIallowsustoassesshowtheintrinsicstructuralproperties dom networks, Newman-Watts small-world networks (NW)48,49, and
influencepathmultiplicityrelativetoarandomizedbaseline.
Barabási-Albert scale-free networks (BA)9. As shown in Fig. 4a, we
Intrinsicfactorsofpathmultiplicity utilize the greedy principle to increase the PMI values while main-
Toinvestigatetheintrinsicfactorsinfluencingpathmultiplicityinreal- taining the same network size and edge density to observe the
worldnetworks,wepresentscatterplotsfortheRPMIandclassical changesincommunitynumberduringtheoptimizationprocess.We
node F s o v r i a a n n y d p v a j i i r s o d f en n o o t d e e d s b ( y v i, l ij v . j T ), h t e h n e w sh e o d r e te fi s n t e p t a h t e h p le a n th gt m h u b lt e i t p w li e c e it n y m ne a t n w ’s or c k or m re e la tr t i i c o s n a c l o o e n f g fic w ie it n h t, P a e n a d rs th o e n’ Q s C c R or f r o e r la 1 t 4 io 0 n re c a o l- e w ffi o c rl i d en n t e , t S w p o e r a k r s -
findthatasthePMIvalueincreases,thecommunitynumbertendsto
inFig.2(seetheSupplementaryInformationforthemetadata).Given rapidlyincrease.Forexample,intheERrandomnetworkshowninthe
m co u n l n ti e p c li t c i i n t g ym th a e t m rix , a (P n M d M th ). e T m o a q t u ri a x nt H if ð y G t Þ h = e ðh o ij v Þ e N r × a N ll i m s u c l a ti l p le l d ici t t h y e of pa th th e t g h ia a n t t re c a o l n -w n o e r c l t d ed ne c t o w m or p k o s n c e a n n t b o e f d ea is c c h on n n e e t c w t o e r d k ,w to e e fo n c s u u s re on co ly n o si n ste th n e t
firstsubplotofFig.4a,thevalueof\Phi(G)monotonicallyincreasesfrom
pathofanetwork,wedefinethepathmultiplicityindex(PMI)asthe 3.90to21.65,whereasthecommunitynumbern cexhibitsafluctuat-
ingupwardtrend,increasingfrom4to14.Wealsoimplementatarget-
averagevalueofthePMAinallpairsofnodes.Mathematically,this orientededgerewiringproceduretoincreasethecommunitynumber
relationshipisexpressedasfollows:
andobservethechangesinthePMI,asshowninFig.4b.Similartothe
P P findingsabove,weobservethatthePMIvaluealsotendstostrongly
N h
\Phi(G)= i=1 j≠i ij : (2)
increaseasthecommunitynumberisoptimized.Thesetwoexperi-
N(N-1)
mentalresultsconfirmthatthereisanotablecausalrelationship
between community structure and path multiplicity. As a point
of comparison, we perform the above target-oriented edge
rewiring experiments based on the clustering coefficient, network
diameter,andassortativity.TheresultsareprovidedintheSupple-
mentary Information and show that there is no significant causal
relationship between these topological properties and path
multiplicity.
Community-basednetworkmodel
Inspiredbytheaboveresults,wenextdevelopaTribalScale-Free(TSF)
model that can reproduce the hesitant-world features of real-world
networks.The TSF model generates networks with hierarchical and
modular structures by creating scale-free subnetworks and
interlinking them through a controlled number of intercommunity
edges.Thespecificgenerationrulesaredescribedinthe“Methods”
section. To verify the effectivenessof the proposed TSF model, we
comparetherealPMAdistributionsP(h),themaximumPMAvalues
and the PMI values \Phi(G) with the synthetic results based on four
typicalmodelnetworks,i.e.,theTSFnetwork,theERrandomnetwork,
theNWsmall-worldnetwork,andtheBAscale-freenetwork.Asshown
inFig.5a,wefindthattheTSFmodelreproducesthehesitant-world
features observed from real-world networks much better than the
other model networks. For example, in the Bio-SC-LC network with
\Phi(G) = 21.50, where the maximum PMA value is 7189, the corre-
spondingTSFnetworkhasaPMIvalueof21.46,withamaximumPMA
valueof7201.However,thePMIvaluesare4.99,6.79and8.42intheER
randomnetwork,theNWsmall-worldnetworkandtheBAscale-free
network,respectively,andthemaximumvaluesofthePMAare185,
802and398,respectively.Inaddition,inFig.5b,wepresentthePMI
valuesofrepresentativereal-worldnetworksalongwiththePMIvalues
of the reference model networks with the same network size and
similaredgedensity.WefindthattheTSFmodelcanbetterreproduce
the PMI values observed from real-world networks and thus can
capturethehesitant-worldfeature,whereastheotherthreeclassical
networkmodelssignificantlydeviatefromtheempiricalPMIvalues.

Methods
Fastalgorithmforcountingshortestpaths
The shortest path lengths between nodepairs can be calculated by
usingtheclassicalFloydalgorithm50,whichhasatimecomplexityof
O(N3).However,thisalgorithmdoesnotprovidethecountofshortest
paths between node pairs.Although Breadth-FirstSearchor Depth-
FirstSearchcanidentifyallpathsbetweenapairofnodesinO(N2)time,
calculatingthenumberofshortestpathsforallpairsofnodeswith
thesemethodsincreasesthecomplexitytoO(N4).
Path multiplicity focuses on the number of shortest paths
betweennodepairsratherthanonthenodestraversed.Inthisstudy,
we implement a fast algorithm to calculate both the number and
lengthofshortestpathsbetweenallnodepairssimultaneously.The
procedureisdescribedasfollows.
Step0.InitializethePathMultiplicityMatrixH={h ij } N×N andthe
Path Length Matrix L={l ij } N×N , setting h ij = h ji = 0 and l ij = l ji = 0.
Definek=1.
Step1.Ifh ij=h ji ≠0foralli≠j,terminatethealgorithm.Otherwise:
(a)ComputethetransitionmatrixT={t ij } N×N ,wheret ij=0ifh ij ≠0,
andt ij=1ifh ij=0.(b)UpdateHtoH=H+T⊙Ak,where ⊙denotesthe
Hadamardproduct.(c)UpdateLtoL=L+k⋅(T⊙J),whereJisanN×N
all-onesmatrix.
Step2.Incrementkby1andreturntoStep1.

Quadrantcountratio
To quantify the monotonic relationship between relative path multi-
plicityandagivennetworkmetric,weemploytheimprovedQCR,anon-
parametricmeasureofassociationderivedfromquadrantanalysis.The
QCRprovidesacoarse-grainedmeasureofassociationbydichotomizing
eachvariableatacentrallocation(typicallythemedian)andquantifying
how often paired observations fall in the same corresponding half.
Specifically,QCRiscomputedbysplittingeachvariableinto“low”and
“high”groupsusingitsmedian(oranotherrobustcentralthreshold)and
thencalculatestheproportionofpairedsamplesthatareconcordant—
i.e.,simultaneouslyhigh-highorlow-low—amongallpairs.
Givenpairedobservationsf{x i,y i }gN i=1 fortwovariablesXandY,
definexeandeyasthesamplemediansof{x i}and{y i},respectively.The
planeisdividedbytheverticallinex=exandthehorizontalliney=ey
intofourquadrants:
• QuadrantI(x≥xe,y≥ey);
• QuadrantII(x<ex,y>ey);
• QuadrantIII(x≤xe,y≤ey);
• QuadrantIV(x>xe,y<ey).

We count the number of points n, n , n , n that fallin each
I II III IV
quadrant.Ifapositivemonotonicrelationshipishypothesized,points
inQuadrantIandQuadrantIIIsharethesamesignofdeviationfrom
themediansandaredeemedconsistent,whereaspointsinQuadrants
IIandQuadrantIVareinconsistent.Thus,theQCRisdefinedas
QCR = n I +n III , (4)
N
which takes values in [0, 1]. A value close to 1 indicates thatmost
observationslieinthefirstorthirdquadrant,implyingastrongposi-
tive association, while a value near 0.5 suggests no directional pre-
ference.Whenanegativemonotonicassociationishypothesized,we
insteadtreatQuadrantsIIandIVasconsistentanddefine
QCR = n II +n IV : (5)
N
Consequently, a higher QCR indicates stronger association,
because a larger fraction of observations exhibit consistent co-
movement in the same direction relative to the central location
(whereasQCRnear0.5suggestsweakornosystematicalignment,and
valuesbelow0.5indicatepredominantlydiscordantpairing).

Communitydetectionvialeadingeigenvectormodularity
Non-overlappingcommunitiesaredetectedbymaximizingthemod-
ularityoftheundirectedgraphG(V,E)inadeterministicspectralfra-
mework. Let A(G)=(a ij ) N×N denote the adjacency matrix defined
above,k i=∑ j a ijthedegreeofnodev i,andM=∣E∣thenumberofedges,
sothat2M=∑ i,j a ij.Definethemodularitymatrix
B=A(G) - kk> , (6)
2M
wherek=(k 1 ,...,k N )>.
For a bipartition encoded by s ∈ {±1}N, the modularity can be
writtenas
Q= 1 s>Bs: (7)
4M
Letu 1 betheleadingeigenvectorofB.Aninitialsplitisobtainedby
thesignrules i=sign u 1i .Asingle-vertexrefinementisthenapplied:
iterativelymovethenodethatyieldsthelargestpositivemodularity
gainΔQ,andrepeatuntilnofurtherimprovementispossible(ΔQ≤0).
To obtain more than two groups, recursion is applied to any
acceptedvertexsubsetU⊆V(acurrentcommunity).Thenetmod-
ularitychangefromsubdividingUisevaluatedwiththegeneralized
modularitymatrix
B( ij U) =B ij - δ ij ∑ ‘2U B i‘, i,j2U, (8)
whereδ ijistheKroneckerdelta,andthesamespectral-plus-refinement
stepsareappliedwithinU.Asubdivisionisacceptedonlyifitincreases
Q (ΔQ > 0); recursion stops when all current groups are indivisible
underthiscriterion.
Matrix-vectorproductswithBareevaluatedas
Bx=A(G)x - k>x k, (9)
2M
Thisalgorithmenablesasystematicexplorationofhownetwork
metrics influence each other through structural modifications. By
maintainingthenetworksizeandedgedensity,theapproachensures
that the observed trends aresolely attributable to the optimization
process, rather than external structural changes. The experimental
results,suchasthoseshowninFig.4,demonstratetheeffectivenessof
this method in terms of revealing the causal relationship between
communitystructureandpathmultiplicity.

Target-orientededgerewiringprocedure
Agreedyalgorithmfortarget-orientededgerewiringisdesignedto
iterativelyoptimizeaspecificnetworkmetricwhilepreservingstruc-
turalpropertiessuchasthenetworksizeandedgedensity.Thealgo-
rithm is operated by modifying the edge structure of a networkto
enhanceatargetmetricandsimultaneouslyobserveitsimpactonthe
associated metric, providing insight into the integration of these
properties.Theprocedureinvolvesthefollowingsteps:
Step1:Initializethenetworkmetrics.Beginwithagivenconnected
network G0 of size N and edge density p, which serves purely as a
rewiringseedforsubsequentoptimization.Someinitialnetworksmay
alreadyhavehighbaselinemetricvalues(e.g.,communitynumberor
clusteringcoefficient),potentiallybiasingtheearlysearchtowardlocal
optima;accordingly,thefirstrecordediterationreflectsthebestout-
comeoftheinitialrewiringtrialsratherthantheoriginalnetwork,and
its metric value can be higher or lower than the initialone without
affectingsubsequentoptimization.
Step2:Identifypotentialedgerewiring.Atiterationt,startingfrom
thecurrentnetworkG(t)with∣E∣=medges,generateasetofcandidate
rewiringtrialsR(t)={r 1 (t),...,r Q (t)}.EachrewiringisappliedtoG(t),
where1≤b≤m.Eachtrialcanmodifymultipleedgeswhile
strictlypreservingN,m,andnetworkconnectivity;anytentativeswap
thatwouldviolatethesestructuralconstraintsisrejectedandreverted.
Step3:Selecttheoptimaledgerewiringwithtiehandling.Foreach
trialr q (t)2R(t),computethetargetmetricontheresultingnetworkand
identifythetrial(s)thatachievethemaximumtargetmetricvalue,i.e.,
Q max (t)=argmax 1≤q≤Q f G(t)r q (t) , (10)
whereG(t)r q (t)denotesapplyingtrialr q (t)toG(t).Ifmultipletrialsattain
the same maximum value (including the case where the maximum
equalsthetargetvalueofthepreviousacceptednetwork),randomly
selectonetrialq?2Q max (t)andusethecorrespondingrewirednetwork
asthenextstate.
Step4:Updatethenetworkandmetrics.Applytheselectedtrialto
obtain the updated network G(t+1). Recalculate both the target and
associatedmetricsonG(t+1).Recordthesevaluestotracktheoptimi-
zation progress, noting that the first recorded iteration reflects the
bestcandidatenetworkfromtheinitialtrialsratherthantheoriginal
network.
Step 5: Termination check. Repeat Steps 2-4 until a predefined
numberofvalidrewirednetworkshasbeenrecorded,where“valid”
means the rewired network satisfies the structural constraints (e.g.,
fixedNandm,andanyadditionalconstraintssuchasconnectivity).
Step 6: Analyze the results. Once the optimization is complete,
examinetherelationshipbetweentheoptimizedtargetmetricandthe
associatedmetric.Forexample,aftermaximizing\Phi(G),observehow
thecommunitynumbern cchangesacrossiterations.

TribalScale-Free(TSF)networkmodel
The TSF modelis a generative model designed tocapturethe hier-
archicalandmodularfeaturestypicalofreal-worldnetworks,adding
theseparationofdistinctcommunities(cf.the“connectedcaveman
model”7). The TSF model builds a network by integrating intra-
community scale-free subnetworks with controlled intercommunity
connections,allowingforanexaminationoftherelationshipbetween
modularity and path multiplicity. The detailed mathematical for-
mulation of the TSF model is presented below, with a step-by-step
descriptionofitsgenerationprocess.
Step1:Initialization.StartwithatotalnetworksizeNandaspecified
numberofcommunitiesn c.TheNnodesarerandomlyassignedton c
distinctcommunities,representingdifferent“tribes”.LetC idenotethe
i-thcommunity,wherei∈1,2,…,n c.Thesizeofeachcommunity∣C i∣is
determinedsuchthat
∑ n c i=1 jC i j=N: (11)
Step2:Generationofscale-freesubnets.WithineachcommunityC i,
a scale-free network is generated by using the the Goh-Kahng-Kim
(GKK)staticscale-freenetworkmodel52.TheGKKmodelconstructsa
scale-freenetworkbyassigningweightsw i=i -α i toeachnodei,where
α i ∈ [0,1),thenbyrandomlyselectingtwoverticesv ac/v baccordingto
thenormalizedweightsandaddinganedgebetweenthemunlessone
existsalready.Bytuningα i,thedegreedistributionfollowsapower-law
characterizedbyanexponentλ iviaλ i=(1+α i)/α i.Theresultingdegree
distributionadherestothefollowingform:
P(k)∼k -λ i, fork≥k min : (12)
Step 3: Intercommunity edge allocation. Once scale-free intra-
community subnets are generated, proceed with intercommunity
edgeallocation.Givenapredefinednumberofintercommunityedges
n e,twodistinctcommunities,C iandC j,areselected,andthenn eedges
are created between them. For subsequent connections, an uncon-
nectedcommunityC mischosenuniformlyatrandom,andthiscom-
munity is linked with n e edges to any of the already connected
communities.Thisprocessisrepeatediterativelyuntilallofthecom-
munitiesareconnected,ensuringtheformationofasingleconnected
componentthroughoutthenetwork.
Step4:Constructionandfinalizationofthenetwork.TheTSFnet-
workG=(V,E)consistsof∣V∣=NnodesandasetofedgesE,combining
theintracommunityedgesE intraandtheintercommunityedgesE inter
suchthatE=E intra∪E inter.Thetotalnumberofedgesinthenetworkis
expressedasfollows:
jEj= ∑ n c i=1 jE i intraj+n e (n c -1): (13)
Notably, to keep the model simple, the proposed TSF model
assumesacommonscalingexponentacrosscommunities.Inpractice,
scaling exponents can differ between communities in real-world
networks54–57,andfutureextensionsoftheTSFmodelcantherefore
allow heterogeneous, community-specific scaling exponents rather
thanenforcingasingleglobalvalue.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper offers an insightful contribution to network science by investigating the underlying mechanisms behind path multiplicity in complex networks, specifically, the observed heterogeneity in the number of shortest paths between node pairs. The authors compellingly identify community structure as a key driver of this phenomenon, a connection that has thus far remained largely unexplored. A major strength of the paper lies in its introduction of the relative path multiplicity metric, which effectively normalises for network size and edge density, allowing for a more solid analysis. The study is further strengthened by the authors’ systematic comparison of classical network metrics and their correlations with path multiplicity, where community structure stands out as the dominant factor. The use of targeted edge rewiring experiments to validate the relationship between community structure and path multiplicity is convincing. Overall, this work is a significant and original contribution, advancing our understanding of structural features in complex networks and opening up promising avenues for future research.\nAs minor, optional comments, the authors could emphasise connections between the notion of cut and community detection, as the cut size is related to the number of independent paths existing between nodes.", "reviewer_2": "General assessment:\nThe manuscript addresses an understudied yet important network property - path multiplicity - and demonstrates its empirical relevance. The analysis is strengthened using a large and diverse dataset of real-world networks, which supports the broad applicability of the findings. Moreover, the edge-rewiring experiments provide compelling evidence linking community structure to path multiplicity. The proposed TSF generative model successfully captures the observed empirical behavior, outperforming standard random-graph models and opening avenues for further theoretical work. Overall, these contributions advance our understanding of structural redundancy and resilience in complex networks.\nOn the other hand, it is increasingly recognized that purely structural metrics often have limited explanatory power without a clear link to network functions or dynamics (Gates & Rocha, 2016 for network control; Hines et al., 2010; Pahwa et al., 2012 for power grids; Zhang et al., 2015 for transportation systems). Therefore, introducing a new structural metric — especially one limited to unweighted networks — requires a strong justification of its utility beyond existing descriptors. To strengthen the paper’s impact, the authors are encouraged to further elaborate on the practical importance of path multiplicity. This could be achieved by discussing or demonstrating its role in a functional or dynamic context, such as network routing, spreading processes, or resilience to failures, showing how the metric offers truly novel insights not captured by classical measures.\nDetailed comments:\n1. A discussion/analysis on how the conclusions might change for weighted and/or directed edges would be important.\n2. I would be hesitant using the term “path hesitation” to convey the concept of path multiplicity.\n3. The causal attribution of RPHI to community structure is dependent on the specific community detection algorithm used. The authors should specify which algorithm and resolution parameter were employed.\n4. The quadrant-consistency rate (QCR) is central to the analysis but does not appear to be a standard metric in network science. The authors should provide a justification for using QCR.\n5. Figure 1 is unfortunately confusing for an introductory figure. For example, while its caption promises to show a “Relationship between the path multiplicity and classical network metrics,” the data and the demonstration do not reveal any relationship.\n6. Typos: “Path multiplicity <has> profound implications for network efficiency and route decision-making” – on page 2 and some others.\n7. Could a high RPHI also arise from a core-periphery structure? For example, in a network with few communities but a very densely connected core.\n8. The TSF model assumes a single power-law exponent, λ, across all communities. Given that real-world networks often exhibit community-specific exponents (Nature 435, 814-818, 2005, Phys. Rev. E 94, 012302, 2016), a discussion on the implications of this simplifying assumption and potential model extensions would be welcome.\n9. The rewiring method in Sec. 4.2 preserves the degree sequence and global connectivity but does not constrain other topological properties like clustering, diameter, or assortativity. Since these metrics could also influence PHI, it would be helpful to report their evolution during the optimization process.\n10. Perhaps the paper’s theoretical contribution could also be strengthened. Have the authors considered deriving an analytical relationship between path multiplicity and community structure parameters, for instance within a stochastic block model? Another valuable approach would be to use spectral methods to obtain rigorous theoretical bounds on path multiplicity.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Point-by-point Responses to Reviewers’ Comments\n² Reviewer #1:\nGeneral assessment: This paper offers an insightful contribution to network science\nby investigating the underlying mechanisms behind path multiplicity in complex\nnetworks, specifically, the observed heterogeneity in the number of shortest paths\nbetween node pairs. The authors compellingly identify community structure as a key\ndriver of this phenomenon, a connection that has thus far remained largely unexplored.\nA major strength of the paper lies in its introduction of the relative path multiplicity\nmetric, which effectively normalises for network size and edge density, allowing for a\nmore solid analysis. The study is further strengthened by the authors’ systematic\ncomparison of classical network metrics and their correlations with path multiplicity,\nwhere community structure stands out as the dominant factor. The use of targeted edge\nrewiring experiments to validate the relationship between community structure and path\nmultiplicity is convincing. Overall, this work is a significant and original contribution,\nadvancing our understanding of structural features in complex networks and opening\nup promising avenues for future research.\nResponse: We sincerely thank the reviewer for this positive and encouraging overall\nassessment of our work, and we have carefully revised the manuscript to further clarify\nthe motivation, methodology, and implications of our findings.\nComment 1：As minor, optional comments, the authors could emphasise connections\nbetween the notion of cut and community detection, as the cut size is related to the\nnumber of independent paths existing between nodes.\nResponse 1: We sincerely thank the reviewer for the insightful comment. As you\nmentioned, there is indeed a connection between the notion of cut and community\ndetection. Our study identifies community structure as the key factor associated with\npath multiplicity. The underlying mechanism might be interpreted as an interface-\ndriven effect: intercommunity edges constitute an effective cut set, and therefore, path\nmultiplicity between modules necessarily passes through boundary nodes and bridges;\nwhen multiple boundary-equivalent intramodular segments and cross-community links\nare length-equivalent, their combinations multiply, yielding a sharp increase in the\nshortest path counts.\nIn the revised version, we have added explanations in the Abstract and the first\nparagraph of the Discussion as follows.\nAbstract\nNetworks with complex topologies describe numerous natural and social systems. The\npaths that connect nodes and edges are fundamental components in network science\nand are important for exploring network connectivity, communication efficiency, and\nload balancing. Recent studies on path multiplicity have shown extreme heterogeneity\nin the number of shortest paths between pairs of nodes in real-world complex networks.\nHowever, the mechanism underlying this phenomenon remains unexplored. Here, we\nreveal that the community structure is responsible for path multiplicity. We first\nintroduce the concept of relative path multiplicity to explore the intrinsic factors that\ninfluence path multiplicity by excluding the effects of network size and edge density;\non this basis, we then further explore the relationship between classical network metrics\nand relative path multiplicity. We find that compared with other metrics, the community\nstructure is more strongly correlated with path multiplicity. Furthermore, through\ntarget-oriented edge rewiring experiments, we verify the indispensable connection\nbetween path multiplicity and the community structure. The underlying mechanism can\nbe interpreted as an interface-driven effect that yields a sharp increase in the number of\nshortest paths. Inspired by these observations, we propose a novel network model based\non a tribal structure that can be used to successfully reproduce phenomena observed in\nthe real world. Our findings improve the understanding of network structures, with\nsignificant potential applications in network design and optimization.\nDiscussion\nRecent studies have revealed that empirical networks display not only the well-known\nsmall-world property but also an intriguing hesitant-world feature, accompanied by a\nuniversal power-law governing path multiplicity. In this study, we investigated the\nmechanisms underlying path multiplicity. We first introduced the concept of the\nrelative path hesitation index (RPHI) to explore the intrinsic factors that influence path\nmultiplicity by excluding the effects of network size and edge density. We have\nuncovered a robust positive correlation between the number of communities and the\nRPHI through four-quadrant consistency analysis, revealing that networks with more\ncommunities exhibit significantly greater path multiplicity. Then, we verified this\nobservation by using the target-oriented edge rewiring procedure, where a higher path\nmultiplicity was associated with a greater number of communities, and vice versa. The\nunderlying mechanism can be interpreted as an interface-driven effect: intercommunity\nedges constitute an effective cut set, and therefore, path multiplicity between modules\nnecessarily passes through boundary nodes and bridges; when multiple boundary-\nequivalent intramodular segments and cross-community links are length-equivalent,\ntheir combinations multiply, leading to a sharp increase in the shortest path counts.\nFinally, inspired by the above results, we designed the Tribal Scale-Free (TSF) model\nto replicate the hesitant-world feature observed from empirical networks. By comparing\nthe TSF model with typical model networks such as the ER random network, the NW\nsmall-world network and the BA scale-free network, we demonstrated that the TSF\nmodel can much better reveal the power law distribution of the PHA, the maximum\nvalues of the PHA, and the PHI values observed from real-world networks.\n² Reviewer #2:\nGeneral assessment: The manuscript addresses an understudied yet important network\nproperty - path multiplicity - and demonstrates its empirical relevance. The analysis is\nstrengthened using a large and diverse dataset of real-world networks, which supports the broad\napplicability of the findings. Moreover, the edge-rewiring experiments provide compelling\nevidence linking community structure to path multiplicity. The proposed TSF generative model\nsuccessfully captures the observed empirical behavior, outperforming standard random-graph\nmodels and opening avenues for further theoretical work. Overall, these contributions advance\nour understanding of structural redundancy and resilience in complex networks.\nOn the other hand, it is increasingly recognized that purely structural metrics often have limited\nexplanatory power without a clear link to network functions or dynamics (Gates & Rocha, 2016\nfor network control; Hines et al., 2010; Pahwa et al., 2012 for power grids; Zhang et al., 2015\nfor transportation systems). Therefore, introducing a new structural metric — especially one\nlimited to unweighted networks — requires a strong justification of its utility beyond existing\ndescriptors. To strengthen the paper’s impact, the authors are encouraged to further elaborate\non the practical importance of path multiplicity. This could be achieved by discussing or\ndemonstrating its role in a functional or dynamic context, such as network routing, spreading\nprocesses, or resilience to failures, showing how the metric offers truly novel insights not\ncaptured by classical measures.\nResponse: We sincerely thank the reviewer for this positive overall assessment and\ninsightful comment on the practical importance of path multiplicity. As a structural\nmetric, path multiplicity is linked to several key aspects of network function and\ndynamics. First, it serves as a structural proxy for robustness and reliability of\nconnectivity, since multiple shortest paths can both buffer local failures and concentrate\nload on shared bottlenecks, shaping vulnerability to congestion and cascades. It also\nmodulates transport, diffusion, and spreading by providing multiple equally short\nchannels that accelerate propagation and redistribute flows or probabilities across\nroutes. Finally, by increasing ambiguity in path choice and revealing node pairs\nconnected through rich internal structure, path multiplicity is likely to affect\nnavigability, decision-making, and influence or coordination in multi-agent and control\nprocesses.\nIn the revised version, we have added a description of the practical importance of path\nmultiplicity to the third paragraph of the Introduction as follows.\nUnlike classical metrics that focus solely on single optimal paths, path multiplicity adds\na new layer of complexity when attempting to understand network structure. As a\nstructural metric, path multiplicity is linked to key aspects of network function and\ndynamics. It serves as a structural proxy for robustness and reliability of connectivity,\nsince multiple shortest paths can both buffer local failures and concentrate load on\nshared bottlenecks, shaping vulnerability to congestion and cascades. It also modulates\ntransport, diffusion, and spreading by providing multiple equally short channels that\naccelerate propagation and redistribute flows or probabilities across routes. Finally, by\nincreasing ambiguity in path choice and revealing node pairs connected through rich\ninternal structure, path multiplicity is likely to affect navigability, decision-making, and\ninfluence or coordination in multi-agent and control processes. However, the\nmechanisms behind path multiplicity remain largely unexplored. What mechanisms\nlead to the hesitant-world phenomenon? What mechanisms drive the power-law\ndistribution in path multiplicity? In this study, we introduce the concept of relative path\nmultiplicity, which allows us to systematically compare the path multiplicity of real-\nworld networks to that of equivalent random networks. With this approach, we explore\nthe underlying structural factors that contribute to the path multiplicity in complex\nnetworks. Furthermore, we develop a new network model to replicate the observed\ncharacteristics in empirical networks, offering insights into the mechanisms underlying\nthe hesitant-world phenomenon and the power-law distribution of path multiplicity.\nComment 1：A discussion/analysis on how the conclusions might change for\nweighted and/or directed edges would be important.\nResponse 1: Thank you for your helpful comments. We agree that the conclusions\nmight change for weighted and/or directed edges. Following your suggestion, we have\nadded descriptions in the third paragraph of the third paragraph of Discussion as follows.\nNotably, this paper restricts attention to undirected and unweighted networks. Considering that\nedge directionality and weights can substantially reshape path multiplicity—altering tie patterns\nand thereby inflating or suppressing the number of shortest paths—the phenomena and\nmechanisms of path multiplicity in directed and weighted networks remain to be studied in the\nfuture. Moreover, this paper draws evidence from empirical networks and simulations; a fuller\ntheoretical account remains to be developed. Promising high-impact research directions could\ninclude an analytical relationship between path multiplicity and structural parameters, spectral\napproaches to bound the path multiplicity, and cut/flow arguments via Menger’s theorem that\nlink cross-community cut size and boundary redundancy.\nComment 2：I would be hesitant using the term “path hesitation” to convey the concept of\npath multiplicity.\nResponse 2: We appreciate the concern about the term “path hesitation”. In our prior\narticle (Ye Deng et.al., PNAS Nexus, 2024), we have employed the terms “path\nhesitation amount (PHA)”, “path hesitation matrix (PHM)”, “path hesitation index\n(PHI)” to capture the intuition that, in networks with a large number of shortest paths\nbetween nodes, routing or navigation decisions become more complex and may induce\nhesitation in the decision maker. To maintain consistency in terminology, we retain this\nlegacy term. We will continue to reflect on the terminology and will adopt a more\nstandard wording in future work if a clearer community consensus emerges.\nComment 3：The causal attribution of RPHI to community structure is dependent on the\nspecific community detection algorithm used. The authors should specify which algorithm and\nresolution parameter were employed.\nResponse 3: Thank you for your helpful comments. We agree that the causal attribution of\nRPHI to community structure is dependent on the specific community detection algorithm used.\nIn this study, we employ Newman’s leading-eigenvector method for community detection. We\nhave also tested other algorithms, such as Louvain, Leiden, and Infomap, and obtained\nqualitatively similar results. Following your suggestions, we have added statements in Sec. 2.1\nand the specific algorithm in the Methods as follows.\nSec. 2.1 Concept of relative path multiplicity\nThe PHI reflects the complexity of path selection within the network: a large PHI suggests that\nthe network has a hesitant-world property, with many shortest paths between node pairs,\npotentially leading to a more complex decision-making process. To reveal the relationships\nbetween the PHI and classical network metrics, we present corresponding scatter plots along\nwith the Pearson's correlation coefficient ρ in 140 real-world networks in Fig.1a. The\ncommunity detection algorithm used in this study is Newman’s deterministic leading-\neigenvector modularity algorithm (Newman, 2006); the specific algorithmic steps and\nparameter settings are detailed in the Methods section, and we remark that our main conclusions\nare robust to the choice of community detection algorithm. For a more in-depth exploration, we\nreport the quadrant consistency rate (QCR) (Choudhary et al., 2017) as a robustness check\nalongside conventional correlations and use it to probe the dependencies that ordinary\ncoefficients may miss. The QCR measures the fraction of observations whose deviations from\ntheir respective medians share the same sign and thus captures concordance in a robust, rank-\nlike manner (see the Methods section for details). A significantly higher proportion of data\npoints within the consistent quadrants in the quadrant analysis would indicate a stronger\ncorrelation between the variables. The scattered points are clearly widely dispersed throughout\nthe plane and the values of ρ and QCR are quite low, suggesting that there are no significant\nlinear correlation between the PHI and the classical network metrics.\nMethods\nCommunity detection via leading–eigenvector modularity\nNon-overlapping communities are detected by maximizing the modularity of the undirected\ngraph G (V, E) in a deterministic spectral framework. Let 𝐴(𝐺) = '𝑎 ) denote the\n!"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Thank you for updating your manuscript following my suggestions.", "reviewer_2": "I thank the authors for considering my previous comments and providing detailed feedback. While the manuscript has certainly improved, I am afraid that the authors have not fully addressed the main concerns. Again, the concept of path multiplicity is definitely interesting. However, I do not think the overall quality and rigor of the manuscript, even after the revision, meet the standards for Nature Communications. I have the following lingering concerns:\n1. Previously, I noted that introducing a new structural metric, particularly one restricted to unweighted networks, requires strong justification. In response, the authors provided a discussion on the relevance of path multiplicity, which feels brief and generic. No references are cited to support this discussion. No justification of the necessity (or unnecessity) of considering edge weights is given. The meanings of “structural proxy” and “increasing ambiguity” remain vague in this context. The transition to the statement that “…the mechanisms behind path multiplicity remain largely unexplored…” also feels abrupt and disrupts the flow.\n2. I also noted that the correlation between RPHI and community structure is likely dependent on the specific community detection algorithm used, which the authors agreed: “…We agree that the causal attribution of RPHI to community structure is dependent on the specific community detection algorithm used…” However, this specification or limitation is not mentioned in the manuscript, which risks exaggerating the paper’s conclusion. While the authors claim that “…main conclusions are robust to the choice of community detection algorithm…” and stated in their reply that they tested algorithms such as Louvain, Leiden, and Infomap, no results are given in the Main Text, Methods, or Supplementary Information. Given the authors’ statement that “…this paper primarily focuses on the impact of community structure on path multiplicity,” it seems biased to report results for only a single algorithm. I urge the authors to include these comparative results to substantiate their claim.\n3. I asked the authors to justify why QCR was chosen. They responded by “…Its value lies in complementing standard coefficients such as Pearson’s correlation coefficient and Spearman’s correlation coefficient: when associations are monotonic but nonlinear, masked by heavy tails or outliers, or split across clusters that depress the correlation coefficient, QCR can still reveal systematic alignment between variables…” First of all, Spearman’s correlation coefficient already takes nonlinear monotonicity into account, so this argument is not convincing. I am asking for a fundamental reason why QCR is the appropriate metric for this system: why should QCR be high, and what physical or structural insight does that convey? Without a deeper justification, the use of QCR appears superficial and selective.\n4. I asked the authors to reconsider the terminology “path hesitation.” The authors insisted on keeping it, but the term remains confusing and has not been properly motivated or defined. For example, the phrase “hesitant-world” abruptly appears on page 3 without any definition regarding what it implies physically or structurally. If the authors wish to retain this terminology, it must be clearly spelled out.\nOverall, I still believe the paper will be of interest to specific readers (myself included). However, the current quality of the presentation is concerning. I hope the authors take these comments seriously and undertake a more substantial revision.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Point-by-point Responses to Reviewers’ Comments\n² Reviewer #1:\nGeneral assessment: Thank you for updating your manuscript following my suggestions.\nResponse: We sincerely appreciate your time and effort in reviewing our manuscript.\n² Reviewer #2:\nGeneral assessment: I thank the authors for considering my previous comments and providing detailed\nfeedback. While the manuscript has certainly improved, I am afraid that the authors have not fully addressed\nthe main concerns. Again, the concept of path multiplicity is definitely interesting. However, I do not think\nthe overall quality and rigor of the manuscript, even after the revision, meet the standards for Nature\nCommunications. I have the following lingering concerns. Overall, I still believe the paper will be of interest\nto specific readers (myself included). However, the current quality of the presentation is concerning. I hope\nthe authors take these comments seriously and undertake a more substantial revision.\nResponse: Thank you very much for your careful assessment and for highlighting the remaining\nconcerns. We are sorry that our previous revision did not fully address the key issues you raised.\nWe have taken your comments very seriously and undertake a substantially deeper and more\nrigorous revision to improve both the quality and clarity of the manuscript in this revision.\nComment 1: Previously, I noted that introducing a new structural metric, particularly one\nrestricted to unweighted networks, requires strong justification. In response, the authors provided\na discussion on the relevance of path multiplicity, which feels brief and generic. No references are\ncited to support this discussion. No justification of the necessity (or unnecessity) of considering\nedge weights is given. The meanings of “structural proxy” and “increasing ambiguity” remain\nvague in this context. The transition to the statement that “…the mechanisms behind path\nmultiplicity remain largely unexplored…” also feels abrupt and disrupts the flow.\nResponse 1: Thank you for your insightful and constructive feedback, which has helped us to\nsignificantly strengthen the motivation and clarity of our work. We have revised the manuscript to\nprovide a more substantial discussion on the functional importance of path multiplicity and to\nclarify the scope and logic of our study. We hope these revisions root the study of path\nmultiplicity within the broader network science discourse. Thank you for guiding us toward these\nimprovements.\n1. Justification of the practical importance of path multiplicity\nWe have expanded the discussion to explicitly link path multiplicity to well-established themes\nin network science, supported by specific citations. We now detail its role in:\n§ Robustness and Congestion: How multiple shortest paths create redundancy but also\npotential shared bottlenecks (Sole-Ribalta et al., New J. Phys., 2019).\n§ Diffusion and Transport: How they provide parallel channels that can accelerate spreading\nand redistribute flows (Gómez et al., Phys. Rev. Lett., 2013).\n§ Routing and Decision-Making: How they introduce navigational choice and potential\ndecision paralysis (Chernev et al., J. Consumer Psychol., 2015; Kleinberg, Nature, 2000).\nThe second paragraph and third paragraph of the Introduction is revised as follows:\nDespite the advances in the study of paths, path multiplicity—the number of equidistant shortest\npaths between node pairs—remains unexplored. Recently, some studies have revealed the\nexistence of a universal power-law scaling of path multiplicity, indicating that a few node pairs\nhave a large number of shortest paths (Deng et al., PNAS Nexus, 228, 2024). For example, even in\nthe Bn-Macaque-Rhesus-Brain-1 brain network with only 242 nodes (Rossi et al., AAAI, 29, 2015),\nthe largest count of shortest paths between node pairs can reach 649, with an average value of\n11.07. This observation reveals that the world we live in is not just a “small-world”, but also a\n“hesitant-world”. The “small-world” effect means that most nodes are connected through only a\nfew steps, even in a large network; meanwhile, the “hesitant-world” effect implies that one may\nhesitate among numerous choices, even when the network itself is small.\nAlthough the multiplicity of shortest paths is a structural feature, it actively governs how networks\nfunction. First, path multiplicity shapes a network's robustness and vulnerability. While redundant\nshortest paths can preserve connectivity when links fail, they may also concentrate traffic on\ncommon bridges or hubs, creating bottlenecks that increase susceptibility to congestion or targeted\nattacks—a dual role noted in studies of network flows (Sole-Ribalta et al., New J. Phys. 21, 2019).\nSecond, it affects spreading processes and transport. Multiple, equally short routes can accelerate\nthe propagation of information, diseases, or resources by offering parallel transmission channels,\nwhile also redistributing loads across the network and influencing diffusion outcomes (Gómez et\nal., Phys. Rev. Lett. 110, 2013). Finally, it has consequences for routing and decision-making. For\nany agent—whether a packet, an individual, or an algorithm—navigating among many “optimal”\npaths can induce “choice overload,” a psychological effect known to impair decision efficiency\n(Chernev et al., J. Consumer Psychol. 25, 2015), potentially degrading the performance of\nnavigation strategies (Kleinberg, Nature. 406, 2000).\n2. Justification of the necessity of considering edge weights\nWe agree that this is a critical direction for extending the understanding of path multiplicity. We\nhave carefully considered your point and have refined our discussion to focus specifically on how\nweights influence path multiplicity and to outline meaningful next steps for research in weighted\nnetworks. We have studied in particular the following:\n§ Impact of Edge Weights: In weighted networks, the definition of the \"shortest path\" shifts\nfrom a topological distance to a sum of edge weights. This dramatically reduces the number\nof qualifying paths. Consequently, the extreme multiplicity observed in unweighted\ntopologies is likely suppressed in many weighted scenarios, often resulting in a single,\nunique shortest path.\n§ Current Knowledge Gap: This highlights a significant gap in our understanding. Since\nmost real-world networks are inherently weighted, the multiplicity of near-optimal or\nfunctionally equivalent routes becomes the more relevant operational concept. The current\ntheory of path multiplicity cannot directly capture this.\n§ Constructive Future Direction: A promising avenue is to generalize this concept. One\ncould define a \"tolerance parameter\" (ε) that allows paths with total weight within (1+ε) of\nthe absolute minimum to be considered as \"tolerant shortest paths.\" Analyzing\nthe multiplicity of these ε-tolerance paths could bridge our structural insights with\nweighted reality, revealing how network architecture provides robustness and alternative\nrouting options under realistic, non-exact conditions.\nThe second paragraph of the Discussion is revised as follows:\nThis study identifies community structure as the key driver of path multiplicity in unweighted\nnetworks. A crucial next step is to extend this insight to weighted networks—the mode of most\nreal-world systems, from transportation to social interactions. Edge weights fundamentally alter\nthe shortest path landscape: the condition for multiplicity (multiple paths with identical total\nweight) becomes mathematically strict, often reducing many shortest paths to a single optimal\nroute. This shift reveals an important gap and opportunity, as practical applications often concern\nnear-optimal alternatives rather than perfectly equivalent paths. To bridge this gap, future work\ncould generalize the concept by introducing such a tolerance parameter ε ≥ 0, defining the paths\nwithin (1+ε) of the minimum weight as “tolerant shortest paths.” Analyzing the multiplicity of\nsuch paths would clarify how network architecture—particularly the community structure—\nsupplies robust routing options under realistic conditions, directly linking our topological findings\nto challenges in resilient infrastructure, adaptive traffic routing, and the analysis of neural or social\nnetworks with continuous link strengths.\n3. Refinement of the Introduction: We have reorganized the Introduction to make it clearer and\naccessible. The logical flow now progresses from (a) the empirical discovery of a power-law\nin path multiplicity, to (b) its recognized functional implications, to (c) the outstanding\nquestion of its structural origin, which our study directly addresses.\nThe third paragraph of the Introduction is revised as follows:\nOur recent finding (Deng et al., PNAS Nexus, 228, 2024)—a strong power-law distribution in path\nmultiplicity across diverse real-world networks—reveals that the path multiplicity is a fundamental\nand widespread architectural signature. Given the functional relevance outlined above, a central\nunanswered question emerges: which underlying network structure produces such extreme\nheterogeneity in path multiplicity? Uncovering this structural origin is essential for explaining\nobserved network dynamics and for designing networks with desired functional properties. In this\nstudy, we introduce the concept of relative path multiplicity to systematically compare real-world\nnetworks with equivalent random networks. Using this approach, we identify the structural factors\nthat shape path multiplicity in complex networks. Furthermore, we develop a new network model\nthat can reproduce the empirical characteristics, offering mechanistic insights into the observed\npower-law distribution of path multiplicity.\nComment 2: I also noted that the correlation between RPMI and community structure is likely\ndependent on the specific community detection algorithm used, which the authors agreed: “…We\nagree that the causal attribution of RPMI to community structure is dependent on the specific\ncommunity detection algorithm used…” However, this specification or limitation is not mentioned\nin the manuscript, which risks exaggerating the paper’s conclusion. While the authors claim that\n“…main conclusions are robust to the choice of community detection algorithm…” and stated in\ntheir reply that they tested algorithms such as Louvain, Leiden, and Infomap, no results are given\nin the Main Text, Methods, or Supplementary Information. Given the authors’ statement that\n“…this paper primarily focuses on the impact of community structure on path multiplicity,” it\nseems biased to report results for only a single algorithm. I urge the authors to include these\ncomparative results to substantiate their claim.\nResponse 2: We sincerely thank you for this helpful comment. We agree that any empirical\ncorrelation between RPMI and community structure can depend on how communities are\noperationalized, i.e., the specific community detection algorithm used. In our initial revision, we\ndid not make this limitation sufficiently explicit in the manuscript text, which could indeed leave\nroom for overstatement. To address this concern, we have now performed a comprehensive\nanalysis based on various community detection methods and reported the corresponding\ncomparative results in the Supplementary Information.\nSpecifically, we performed the correlation analysis between RPMI and community structure using\nsix different algorithms, including Leading Eigenvector (Newman, M.E et al, 2006), Walktrap\n(Pons, P et al, 2006), Leiden (Traag, V.A et al, 2019), Label Propagation (Raghavan, U.N et al,\n2007), Infomap (Rosvall, M et al, 2008), and Louvain (Blondel, V.D et al, 2008). The resulting\ncorrelation plots are now provided in the Supplementary Information. All community detection\npackages/functions and parameter settings used in this study, together with a brief documentation,\nare available at https://github.com/gituserbnu/path-multiplicity.git. In addition, we have revised\nthe Main Text to explicitly state that our main conclusions are robust to the choice of community\ndetection algorithm.\nIn the revised version, we have added descriptions in Sec. 2.2 and Supplementary Information as\nfollows.\nSec. 2.2 Intrinsic Factors of Path Multiplicity\nConsidering that a specific community detection algorithm may influence the causal attribution of\nRPMI to community structure, we have tested classical community detection algorithms, including\nthe Leading Eigenvector method (Newman et al., Phys. Rev. E. 74(3), 036104, 2006), Walktrap\nmethod (Pons et al., J. Graph Algorithms Appl. 10(2), 191–218, 2006), Leiden method (Traag et\nal., Sci. Rep. 9(1), 1–12, 2019), Label Propagation method (Raghavan et al., Phys. Rev. E. 76(3),\n036106, 2007), Infomap method (Rosvall et al., Proc. Natl. Acad. Sci. 105(4), 1118–1123, 2008),\nand Louvain method (Blondel et al., J. Stat. Mech.: Theory Exp. 2008(10), 10008, 2008). Our\nexperimental results are reported in the Supplementary Information. Although these algorithms\nyield somewhat different community counts, the Spearman correlation between RPMI and the\nnumber of communities remains consistently high (all ρ > 0.8), and all the corresponding QCR\n!\nvalues are above 0.9. These results indicate that our findings do not depend on any particular\ncommunity detection algorithm; accordingly, we remark that our main conclusions are robust to\nthe choice of community detection algorithm.\nSec. Supporting Information\nTable R1. Community counts of 140 real-world networks under multiple community detection\nalgorithms. Columns correspond to the number of communities detected by each algorithm.\nLeading Label\nNo Name N Walktrap Leiden Infomap Louvain\nEigenvector Propagation\nbn-cat-mixed-\n1 65 3 4 4 1 1 4\nspecies_brain_1\nbn-macaque-\n2 91 2 1 3 1 1 4\nrhesus_brain_2\nbn-macaque-\n3 rhesus_cerebral- 91 3 1 3 1 1 3\ncortex_1\nbn-macaque-\nrhesus_interarea\n4 93 1 1 1 1 1 1\nl-cortical-\nnetwork_2\nbn-mouse-\n5 kasthuri_graph_ 987 21 117 20 82 109 22\nv4\nbn-\n6 213 2 2 3 1 1 3\nmouse_brain_1\nbn-\n7 1076 4 10 5 1 1 5\nmouse_retina_1\nbn-\n8 mouse_visual- 29 6 3 3 4 5\n3\ncortex_1-2\n9 aves-barn-\nswallow- 17 2 4 3 1 1 2\ncontact-network\naves-barn-\n10 swallow-non- 17 2 2 2 1 1 2\nphysical\naves-sparrow-\n11 52 4 5 4 1 1 4\nsocial\naves-sparrow-\n12 31 3 4 3 3 2 3\nsocial-2009\naves-sparrow-\n13 40 4 3 3 1 1 3\nsocial-2010\naves-\n14 sparrowlyon- 43 3 7 3 1 1 3\nflock-season2\naves-\n15 sparrowlyon- 27 2 2 2 1 1 2\nflock-season3\naves-thornbill-\n16 63 3 1 3 1 1 3\nfarine\n17 bio-CE-GT 878 22 61 17 29 51 17\n18 bio-CE-LC 993 91 59 27 95 104 27\n19 bio-CE-PG 734 3 1 4 2 1 4\n20 bio-DM-HT 2831 243 336 45 292 310 43\n21 bio-DM-LC 483 34 58 18 47 52 18\n22 bio-DR-CX 3287 4 170 7 2 143 5\n23 bio-HS-HT 2499 50 293 20 49 204 19\n24 bio-SC-LC 1999 15 276 10 6 106 10\n25 bio-SC-TS 101 2 1 2 1 1 3\n26 bio-celegans 453 8 35 9 6 36 10\n27 bio-celegans-dir 453 8 35 10 6 37 11\n28 bio-grid-mouse 791 66 92 26 76 84 26\n29 bio-grid-plant 1272 137 104 28 97 102 27\n30 bio-grid-worm 3343 36 269 27 103 277 31\n31 bio-yeast 1458 152 174 32 172 175 31\nbio-yeast-\n32 1458 152 174 32 174 180 32\nprotein-inter\n33 ca-AstroPh 17903 611 2264 38 347 898 37\n34 ca-CSphd 1025 138 66 33 136 137 32\n35 ca-CondMat 21363 979 2306 56 1552 1337 54\n36 ca-Erdos992 4991 430 197 36 330 296 33\n37 ca-GrQc 4158 494 448 42 375 367 40\n38 ca-HepPh 11204 206 2631 43 368 758 39\n39 eco-everglades 69 3 2 3 1 1 4\neco-foodweb-\n40 128 4 2 4 1 1 4\nbaydry\neco-foodweb-\n41 128 4 2 4 1 1 5\nbaywet\n42 eco-mangwet 97 3 2 4 1 1 4\n43 eco-stmarks 54 4 8 4 1 1 4\nENZYMES_g1\n44 95 11 13 9 16 14 9\n45 50 6 5 5 9 7 5\n46 55 6 9 8 7 10 6\n98\n56 5 6 5 7 9 5\n48 57 6 6 6 6 10 6\n49 64 7 6 6 8 9 6\n50 52 7 8 7 9 9 7\n51 60 5 9 7 4 10 7\n52 66 8 8 9 16 11 9\n53 74 7 12 6 13 12 5\n54 28 6 5 5 5 5 5\n41\n55 econ-mahindas 1258 10 195 8 3 120 9\n56 econ-poli 2343 216 88 45 216 208 45\n57 econ-wm3 257 5 8 6 4 19 4\n58 ia-email-univ 1133 14 49 11 15 77 11\nia-hospital-\n59 75 3 3 4 1 1 4\nward-proximity\nia-hospital-\n60 ward-proximity- 75 3 3 4 1 1 5\nattr\n61 ia-infect-hyper 113 4 5 5 1 1 6\n62 ia-reality 6809 337 68 45 81 91 43\nia-\n63 18 2 2 2 2 1 2\nsouthernwomen\nia-workplace-\n64 92 4 4 4 1 4 4\ncontacts\n65 power-1138-bus 1138 56 87 27 133 134 25\n66 power-494-bus 494 29 37 21 67 69 19\n67 power-662-bus 662 27 42 19 60 76 19\n68 power-685-bus 685 17 21 17 50 61 16\n69 power-US-Grid 4941 141 364 43 497 486 39\npower-\n70 1723 76 50 25 174 178 25\nbcspwr09\npower-\n71 5300 100 69 37 375 385 34\nbcspwr10\ncopresence-\n72 95 2 2 2 1 1 2\nInVS13\nhospital-ward-\n73 75 3 3 4 1 1 4\nproximity\n74 infect-hyper 113 4 5 5 1 1 6\nprimary-school-\n75 242 4 5 6 2 1 6\nproximity\nroad City of\n76 6105 120 177 55 526 630 55\nOldenburg\n77 road beijing 5036 101 92 41 444 448 45\n78 road-chesapeake 39 3 2 3 1 1 3\n79 road-euroroad 1039 25 67 25 90 134 22\n80 road-minnesota 2640 40 46 32 199 267 31\n81 tech-routers-rf 2113 135 278 22 97 176 20\n82 email-dnc 1833 18 67 17 32 78 13\n83 email-univ 1133 14 49 9 5 72 13\n84 inf-USAir97 332 4 27 6 7 20 7\n85 inf-euroroad 1039 25 67 22 108 125 22\n86 inf-openflights 2905 29 194 25 63 159 27\n87 inf-power 4941 141 364 42 504 490 40\nlondon_overgro\n88 369 21 45 17 55 56 19\nund\nlondon_undergr\n89 271 19 31 16 42 43 17\nound\n90 web-EPA 4253 113 206 27 112 207 30\nweb-indochina-\n91 11358 1105 996 95 847 929 71\n2004\n92 web-spam 4767 28 595 24 56 308 32\nweb-webbase-\n93 16062 1327 387 81 682 537 71\n2001\n94 soc-dolphins 62 4 4 5 5 6 5\n95 soc-firm-hi-tech 33 3 5 4 2 4 3\n96 soc-hamsterster 2000 28 202 28 57 155 22\n97 soc-karate 34 4 5 4 2 3 4\n98 soc-tribes 16 3 3 3 1 1 3\n99 soc-wiki-Vote 889 12 42 9 7 74 10\n100 fb-pages-food 620 62 54 18 29 54 18\nfb-pages-\n101 5908 205 260 29 141 241 30\npolitician\nfb-pages-public-\n102 11565 295 889 42 179 724 34\nfigure\n103 fb-pages-tvshow 3892 508 443 49 238 290 46\n104 socfb-Reed98 962 4 78 6 1 19 6\n105 socfb-nips-ego 2888 17 6 8 10 11 8\n106 rt-retweet 96 10 10 8 14 16 8\n107 rt-twitter-copen 761 42 108 21 82 100 20\n108 rt_assad 2139 50 217 30 139 190 28\n109 rt_damascus 3052 399 90 27 149 199 29\n110 rt_obama 3212 254 102 40 192 236 39\n111 rt_occupy 3225 268 277 45 270 309 40\nrt_voteonedirect\n112 2280 66 105 32 69 123 32\nion\n115 brock200-1 200 3 3 5 1 1 5\n116 gen200-p0-9-44 200 2 1 4 1 1 4\n117 johnson8-4-4 70 5 1 5 1 1 5\n118 san200-0-9-1 200 2 2 2 1 1 2\nbio-\n119 297 4 22 6 1 7 6\ncelegansneural\n120 chesapeake 39 3 2 3 1 1 4\n121 delaunay_n10 1024 14 14 16 44 58 13\n122 delaunay_n11 2048 18 23 18 109 110 19\ninsecta-ant-\n123 trophallaxis- 41 3 6 4 1 1 4\ncolony1\ninsecta-ant-\n124 trophallaxis- 39 2 6 4 1 1 5\ncolony2\nmammalia-\n125 raccoon- 24 2 2 2 1 1 2\nproximity\nreptilia-tortoise- 7\n126 121 9 16 9 5 14\nnetwork-bsv\nreptilia-tortoise-\n127 31 5 5 4 1 2 4\nnetwork-cs\nreptilia-tortoise-\n128 496 54 39 23 56 62 21\nnetwork-fi\nreptilia-tortoise-\n129 6 2 2 2 1 1 2\nnetwork-hw\nreptilia-tortoise-\n130 41 4 7 5 2 7 5\nnetwork-lm\nreptilia-tortoise-\n131 22 3 4 3 3 3 4\nnetwork-pv\n135 citeseer 2110 165 270 40 197 243 36\n136 cora 2485 149 163 26 169 242 30\n137 gene 814 67 91 22 85 105 23\ninternet-\n138 industry- 219 6 32 8 2 23 7\npartnerships\n139 webkb-wisc 251 13 31 12 10 35 12\n140 tech-WHOIS 7476 102 453 24 43 297 19\nFig. S1. Correlations between the relative path multiplicity and community number estimated by the Leading Eigenvector method\nin 140 real-world networks. The metadata and results for all 140 networks are provided in the Supplementary Information. Each\npanel presents a scatter plot on the left, illustrating the correlation between the relative path multiplicity and community\nnumber(detected by Leading Eigenvector method) based on the given metric and their RPMI values. The networks are categorized\ninto four quadrants, divided by the median values of community number 𝑛 and the RPMI 𝛷#(𝐺).\n!\nThe right-hand violin plots show\nthe distribution, mean and median values of community number, which are divided into two parts based on the median value of the\nRPMI. If there are more overlap between two violin subplots, it suggests a weaker correlation between community number and the\nRPMI. All community detection packages/functions and parameter settings used in this study, together with brief documentation,\nare available at https://github.com/gituserbnu/path-multiplicity.git.\nFig. S2. Correlations between the relative path multiplicity and community number estimated by the Walktrap method in 140 real-\nworld networks. The metadata and results for all 140 networks are provided in the Supplementary Information. Each panel presents\na scatter plot on the left, illustrating the correlation between the relative path multiplicity and community number(detected by\nWalktrap method) based on the given metric and their RPMI values. The networks are categorized into four quadrants, divided by\nthe median values of community number 𝑛 and the RPMI 𝛷#(𝐺).\n!\nThe right-hand violin plots show the distribution, mean and\nmedian values of community number, which are divided into two parts based on the median value of the RPMI. If there are more\noverlap between two violin subplots, it suggests a weaker correlation between community number and the RPMI. All community\ndetection packages/functions and parameter settings used in this study, together with brief documentation, are available at\nhttps://github.com/gituserbnu/path-multiplicity.git.\nFig. S3. Correlations between the relative path multiplicity and community number estimated by the Leiden method in 140 real-\nworld networks. The metadata and results for all 140 networks are provided in the Supplementary Information. Each panel presents\na scatter plot on the left, illustrating the correlation between the relative path multiplicity and community number(detected by\nLeiden method) based on the given metric and their RPMI values. The networks are categorized into four quadrants, divided by\nthe median values of community number 𝑛 and the RPMI 𝛷#(𝐺).\n!\nThe right-hand violin plots show the distribution, mean and\nmedian values of community number, which are divided into two parts based on the median value of the RPMI. If there are more\noverlap between two violin subplots, it suggests a weaker correlation between community number and the RPMI. All community\ndetection packages/functions and parameter settings used in this study, together with brief documentation, are available at\nhttps://github.com/gituserbnu/path-multiplicity.git.\nFig. S4. Correlations between the relative path multiplicity and community number estimated by the Label Propagation method in\n140 real-world networks. The metadata and results for all 140 networks are provided in the Supplementary Information. Each panel\npresents a scatter plot on the left, illustrating the correlation between the relative path multiplicity and community number(detected\nby Label Propagation method) based on the given metric and their RPMI values. The networks are categorized into four quadrants,\ndivided by the median values of community number 𝑛 and the RPMI 𝛷#(𝐺).\n!\nThe right-hand violin plots show the distribution,\nmean and median values of community number, which are divided into two parts based on the median value of the RPMI. If there\nare more overlap between two violin subplots, it suggests a weaker correlation between community number and the RPMI. All\ncommunity detection packages/functions and parameter settings used in this study, together with brief documentation, are available\nat https://github.com/gituserbnu/path-multiplicity.git.\nFig. S5. Correlations between the relative path multiplicity and community number estimated by the Infomap method in 140 real-\nworld networks. The metadata and results for all 140 networks are provided in the Supplementary Information. Each panel presents\na scatter plot on the left, illustrating the correlation between the relative path multiplicity and community number (detected by\nInfomap method) based on the given metric and their RPMI values. The networks are categorized into four quadrants, divided by\nthe median values of community number 𝑛 and the RPMI 𝛷#(𝐺).\n!\nThe right-hand violin plots show the distribution, mean and\nmedian values of community number, which are divided into two parts based on the median value of the RPMI. If there are more\noverlap between two violin subplots, it suggests a weaker correlation between community number and the RPMI. All community\ndetection packages/functions and parameter settings used in this study, together with brief documentation, are available at\nhttps://github.com/gituserbnu/path-multiplicity.git.\nFig. S6. Correlations between the relative path multiplicity and community number estimated by the Louvain method in 140 real-\nworld networks. The metadata and results for all 140 networks are provided in the Supplementary Information. Each panel presents\na scatter plot on the left, illustrating the correlation between the relative path multiplicity and community number (detected by\nLouvain method) based on the given metric and their RPMI values. The networks are categorized into four quadrants, divided by\nthe median values of community number 𝑛 and the RPMI 𝛷#(𝐺).\n!\nThe right-hand violin plots show the distribution, mean and\nmedian values of community number, which are divided into two parts based on the median value of the RPMI. If there are more\noverlap between two violin subplots, it suggests a weaker correlation between community number and the RPMI. All community\ndetection packages/functions and parameter settings used in this study, together with brief documentation, are available at\nhttps://github.com/gituserbnu/path-multiplicity.git.\nComment 3: I asked the authors to justify why QCR was chosen. They responded by “…Its value\nlies in complementing standard coefficients such as Pearson’s correlation coefficient and\nSpearman’s correlation coefficient: when associations are monotonic but nonlinear, masked by\nheavy tails or outliers, or split across clusters that depress the correlation coefficient, QCR can still\nreveal systematic alignment between variables…” First of all, Spearman’s correlation coefficient\nalready takes nonlinear monotonicity into account, so this argument is not convincing. I am asking\nfor a fundamental reason why QCR is the appropriate metric for this system: why should QCR be\nhigh, and what physical or structural insight does that convey? Without a deeper justification, the\nuse of QCR appears superficial and selective.\nResponse 3: Thank you for your thoughtful comment, which has helped us better explain our\nchoice of the QCR in our study. We acknowledge that Spearman’s correlation captures monotonic\nnonlinear relationships, and in the revised manuscript we have now included Spearman’s analysis\nas you suggested. At the same time, we retained QCR because it provides a different and\nmeaningful perspective for our data.\nWhile Spearman’s correlation assesses rank-order agreement across the full range of values, QCR\nmeasures how often two variables lie on the same side of their respective medians—that is, whether\nthey tend to be “both high” or “both low” simultaneously. This makes QCR particularly\ninformative when the relationship is driven by coordinated shifts between states, rather than by a\nsmooth trend across all values. In our study, a high QCR indicates that the network metric and\nPMI frequently move together into similar high or low regimes, reflecting state-like coordination\nin the system’s behavior—an insight that complements what Spearman’s correlation reveals.\nAdditionally, QCR is inherently robust to outliers and heavy-tailed distributions, as it depends only\non whether an observation is above or below the median, not on its exact magnitude or rank.\nTherefore, QCR was not chosen as a replacement for Spearman’s correlation, but as a\ncomplementary tool specifically suited to reveal coordinated state changes—a pattern that\nconventional correlation measures often miss.\nIn the revised manuscript, we simultaneously present Pearson’s correlation coefficient (ρₚ),\nSpearman’s correlation coefficient (ρₛ), and the QCR, and provide a clearer explanation of the\ndistinct information each metric offers. The changes can be found in Sec. 2.1 and 2.2. We\nappreciate your suggestion, which has strengthened the justification of our methodological\napproach.\nSec. 2.1 Concept of relative path multiplicity\nThe PMI reflects the complexity of path selection within the network: a large PMI suggests that\nthe network has a hesitant-world property, with many shortest paths between node pairs,\npotentially leading to a more complex decision-making process. To reveal the relationships\nbetween the PMI and classical network metrics, we first present corresponding scatter plots along\nwith the Pearson's correlation coefficient ρ (Pearson et al., Proc. R. Soc. Lond. 58, 240–242,\n\"\n1895), Spearman's correlation coefficient ρ (Spearman et al., Am. J. Psychol. 15(1), 72–\n!\n101,1904), and the quadrant count ratio (QCR) (Holmes et al., Teach. Stat. 23, 2001; Choudhary,\net al., Wiley Online Library, 2017) in 140 real-world networks in Fig.1a. Pearson’s correlation\ncoefficient quantifies the strength of a linear relationship between two variables, whereas\nSpearman’s correlation coefficient is a nonparametric rank-based measure of monotonic\nassociation and is often preferred when nonlinearity or outliers make Pearson’s coefficient\nundesirable. The QCR provides a coarse-grained measure of association by dichotomizing each\nvariable at a central location (typically the median) and quantifying how often paired observations\nfall in the same corresponding half (see the Methods section for details). Together, Pearson’s\ncorrelation, Spearman’s correlation, and QCR offer complementary perspectives on association by\ntargeting linear association, monotonic rank-based association, and median-split concordance,\nrespectively. Consistent results across these measures provide a robustness check for the reported\nassociation. Overall, the scatter plots exhibit a disordered distribution, with points broadly\ndispersed and showing no clear trend. Moreover, we can observe that all three correlation metrics\nare relatively low, indicating that there is no significant correlation between PMI and classical\nnetwork metrics.\nFig. 1. Potential factors influencing path multiplicity. a, The scatter plots show the correlation between path multiplicity and\nclassical network metrics in 140 real-world networks. As a reference, the median values of the PMI Φ(𝐺) and classical network\nmetrics are shown as dashed lines. These scatter plots reveal that the points are widely distributed throughout the plane in all four\nquadrants, suggesting that there are no significant correlations between path multiplicity and classical network metrics. The\nmetadata and results for all 140 networks are provided in the Supplementary Information. The community detection algorithm used\nin this study is Newman’s deterministic leading eigenvector modularity algorithm (Newman et al., Phys. Rev. E. 74, 2006); the\nspecific algorithmic steps and parameter settings are detailed in the Methods section. b, The example simple network is initially a\nchain network with an initial PMI value of 1. As the edge density increases to a fully connected network, the corresponding PMI\nvalue eventually returns to 1. c, The PMI is plotted as a function of the edge density 𝑝 for three classical network models, ER\nrandom networks, regular ring lattices, and CBA scale-free networks, with different network sizes 𝑁=100,500,1000. For all of\nthe model networks, the PMI changes with the edge density 𝑝 in some kind of complex non-linear relationship forms.\nSec. 2.2 Intrinsic Factors of Path Multiplicity\nTo investigate the intrinsic factors influencing path multiplicity in real-world networks, we present\nscatter plots for the RPMI and classical network metrics along with Pearson's correlation\ncoefficients, Spearman's correlation coefficient，and the QCR for 140 real-world networks in\nFig.2 (see the Supplementary Information for the metadata). Given that real-world networks can\nbe disconnected, we focus only on the giant connected component of each network to ensure\nconsistent comparisons. First, it is easy to see that these network metrics are not significantly\nlinearly correlated with path multiplicity, where the highest Pearson's correlation coefficient is\nρ = 0.3164. By inspecting Spearman’s correlation coefficients, we find that the community\n\"\nnumber, global efficiency, average shortest path length, and network diameter are correlated with\npath multiplicity with ρ > 0.6. Furthermore, using QCR, we observe that the association between\n!\nthe number of communities and path multiplicity reaches a striking value of 0.9857, where the\ndata points are mostly located in the first and third quadrants in the scatter plot shown in Fig. 2a.\nHowever, the majority of the QCR values for the other metrics are below 0.9, suggesting that the\nnumber of communities is more significantly correlated with the RPMI. In addition to the scatter\nplots, we display the violin plots of each network metric, which are divided into two parts on the\nbasis of the median values of the RPMI. If a stronger correlation exists between a network metric\nand the RPMI, the overlap between two violin subplots should be small. As shown in Fig. 2,\ncompared with other network metrics, the narrow overlap corresponding to the number of\ncommunities further demonstrates that the community structure (Rosvall et al., Proc. Natl. Acad.\nSci. 105, 2008; Girvan et al., Proc. Natl. Acad. Sci. 99, 2002) plays a key role in determining path\nmultiplicity.\nFig. 2. Correlations between the relative path multiplicity and classical network metrics in 140 real-world networks. The metadata\nand results for all 140 networks are provided in the Supplementary Information. The metrics include: a, the community number 𝑛 ,\n!\nb, the average degree 〈𝑘〉, c, the average shortest path length 𝐿, d, the global efficiency 𝐸 ,\n\"#$% \ne, the diameter 𝐷, f, the assortativity\ncoefficient 𝑟, g, the clustering coefficient 𝐶, and h, the k-shell index 𝑘 .\n&\nEach panel (a-h) presents a scatter plot on the left,\nillustrating the correlation between the relative path multiplicity and the classical network metrics based on the given metric and\ntheir RPMI values. The networks are categorized into four quadrants, divided by the median values of the given metric and the\nRPMI 𝛷#(𝐺). The right-hand violin plots show the distribution, mean and median values of the corresponding network metric,\nwhich are divided into two parts based on the median value of the RPMI. If there are more overlap between two violin subplots, it\nsuggests a weaker correlation between a network metric and the RPMI.\nComment 4: I asked the authors to reconsider the terminology “path hesitation.” The authors\ninsisted on keeping it, but the term remains confusing and has not been properly motivated or\ndefined. For example, the phrase “hesitant-world” abruptly appears on page 3 without any\ndefinition regarding what it implies physically or structurally. If the authors wish to retain this\nterminology, it must be clearly spelled out.\nResponse 4: We sincerely thank the reviewer for the helpful comment. We agree that the previous\nterminology “path hesitation” was insufficiently motivated and could lead to unnecessary\nconceptual ambiguity. In the revised manuscript, we have thoroughly removed this terminology\nand adopted the standard term “path multiplicity” across all definitions, descriptions, and analyses.\nCorrespondingly, all symbols and abbreviations previously derived from the hesitation\nterminology have been systematically updated. For example, the quantity formerly denoted as\nPHA (“path hesitation amount”) is now renamed PMA (“path multiplicity amount”), reflecting its\nprecise mathematical meaning as the multiplicity of shortest paths between node pairs.\nLikewise, we have renamed PHI (“path hesitation index”) as PMI (“path multiplicity index”) to ensure\nconsistent terminology throughout the manuscript. Overall, we have conducted a manuscript-wide\nrevision to ensure that all notation consistently aligns with the standard terminology. Also, we\nhave modified the second paragraph of the Introduction. We are grateful to the reviewer for this\ncomment, which has significantly improved the clarity and precision of the manuscript.\n² Reviewer #3:\nComment 1: I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This\nis part of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse 1: Thank you for your insightful comments. We have provided a point-by-point response to every\ncomment in the reports and indicated the corresponding revisions with page/line references; all changes are\nhighlighted in the manuscript.\nIn my opinion, the authors have not yet fully addressed the concerns and comments\nraised in the first round of review. Therefore, I believe another major revision is\nnecessary. My specific reasons are as follows:\n1. In the previous review, we emphasized that introducing a new structural metric,\nparticularly one restricted to unweighted networks, requires strong justification of its\nutility beyond existing descriptors. ---- However, the authors provided only a brief\ndiscussion of potential applications without any concrete analysis, limiting the\npersuasiveness of their argument. It would greatly enhance the manuscript if the authors\ncould demonstrate the metric's practical relevance through at least one specific\napplication scenario.\n2. If demonstrating practical application proves challenging, the manuscript should\ninstead clarify and strengthen its theoretical contributions. Currently, these theoretical\ncontributions remain limited and somewhat ambiguous:\n2.1 The proposed Relative Path Hesitation Index (RPHI), which normalizes the\nobserved PHI using an equivalent Erdős–Rényi (ER) random graph, is listed as one of\nthe major contributions to this paper. However, employing random-graph null\nmodels as baselines for normalizing network metrics is already a common and\nintuitive practice widely documented in network science literature.\n2.2 Another main finding in this paper as claimed by the author \"We find that\ncompared with other metrics, the community structure is more strongly correlated with\npath multiplicity\" --- which remains largely descriptive and correlation-based. In\nparticular, the authors computed their proposed metric alongside several established\nmetrics across multiple networks and identified correlations. Such an approach,\nalthough clear and straightforward, offers limited theoretical depth. Moreover, in the\nrebuttal, the authors responded by stating, \"deriving an analytical relationship between\npath multiplicity and community-structure parameters remains a challenging open\nproblem and appears to require additional theoretical tools,\" without directly\naddressing the concern. To significantly strengthen the theoretical grounding, the\nauthors should provide clearer theoretical analysis distinguishing their proposed metric\nfrom existing community-based metrics. Without this clarification, it remains unclear\nwhy the proposed metric is preferable to long-established measures directly based on\ncommunity structure (why not choose classical metric directly?). This naturally leads\nto the third issue as follows:\n2.3 The Tribal Scale-Free (TSF) model described in the manuscript appears closely\nrelated conceptually to existing network models designed to integrate community\nstructure with realistic, heterogeneous degree distributions. Specifically, it resembles\nwell-known models such as the Degree-Corrected Stochastic Block Model (DC-SBM)\nand models of Hierarchical Modular Networks (HMN), as well as earlier constructive\nmodels like the \"Connected Caveman\" model. The authors need to explicitly\nclarify how the TSF model is theoretically distinct from these classical models, or\nclearly state whether it is primarily an incremental extension of these existing\nframeworks.\n2.4 As for the performance of the TSF model that is designed with explicit modularity,\noutperforms ER, NW, and BA models, which lack strong inherent community\nstructure. This result is expected with no surprises. A more rigorous evaluation would\ninvolve comparing the TSF model against established models for generating modular\nnetworks, such as the Stochastic Block Model (SBM). The authors should discuss how\nthe TSF model compares to SBM variants and whether the specific internal structure of\nthe TSF modules (scale-free via Configuration Model) is necessary for high RPHI, or\nif modularity alone (as in a standard SBM) is sufficient.\n2.5 The authors propose an \"interface-driven effect\" as the mechanism, where\ncombinations of equivalent path segments across community boundaries multiply the\nshortest path counts. While plausible, the analysis focuses mainly on the correlation\nwith the number of communities. However, community structure is more than just\nhow many modules there are. I'm thinking of modularity, distribution of community\nsizes, or cut sizes (number of edges crossing communities), which may be the quantities\nmore directly related to the “interface-driven effect”."}, {"round": 3, "reviewer_comments": {"reviewer_2": "The paper has been substantially revised. I have no further comments and would like to thank the authors for considering the suggestions.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": ""}]

Decision letter