Table Of ContentBenchmark model to assesscommunity structure inevolvingnetworks
Clara Granell,1 Richard K. Darst,2 Alex Arenas,1,3 Santo Fortunato,2 and SergioGo´mez1
1Departamentd’EnginyeriaInforma`ticaiMatema`tiques, UniversitatRoviraiVirgili,43007Tarragona, Spain
2Complex Systems Unit, Department of Computer Science,
Aalto University School of Science, P.O. Box 12200, 00076 Aalto, Finland
3Institut Catala` de Paleoecologia Humana i Evolucio´ Social, 43007 Tarragona, Spain
Detectingthetimeevolutionofthecommunitystructureofnetworksiscrucialtoidentifymajorchangesinthe
internalorganizationofmanycomplexsystems,whichmayundergoimportantendogenousorexogenousevents.
This analysis can be done in two ways: considering each snapshot as an independent community detection
5 problemortakingintoaccountthewholeevolutionofthenetwork.Inthefirstcase,onecanapplystaticmethods
1
onthetemporalsnapshots,whichcorrespondtoconfigurationsofthesysteminshorttimewindows,andmatch
0
afterwardsthecommunitiesacrosslayers.Alternatively,onecandevelopdedicateddynamicprocedures,sothat
2
multiplesnapshotsaresimultaneouslytakenintoaccountwhiledetectingcommunities,whichallowsustokeep
l memory of the flow. To check how well a method of any kind could capture the evolution of communities,
u
suitablebenchmarksareneeded. Hereweproposeamodelforgeneratingsimpledynamicbenchmarkgraphs,
J
basedonstochasticblockmodels. Inthem,thetimeevolutionconsistsofaperiodicoscillationofthesystem’s
9 structurebetweenconfigurationswithbuilt-incommunitystructure. Wealsoproposetheextensionofquality
1 comparisonindicestothedynamicscenario.
]
h PACSnumbers: 89.75.Fb,89.75.Hc
p
-
c 1. INTRODUCTION gap. Weproposeamodel,derivedfromtheclassicstochastic
o blockmodels[18–21],thatgeneratesthreeclassesofdynamic
s
benchmarkgraphs. Theobjectiveistoprovidetime-evolving
. The analysis and modeling of temporal networks has re-
s networks,suchthatateachsnapshotthepartitionintocommu-
c ceived a great deal of attention lately, mainly due to the in-
i creasingavailabilityoftime-stampednetworkdatasets[1–5]. nitiesiswelldefinedaccordingtothemodel. Tokeepthings
s
simple we consider a periodic evolution such that the same
y Arelevantissueiswhetherandhowthecommunitystructure
historyrepeatsitselfin cyclesandisinvariantundertimere-
h of networks [6] changes in time. Communities reveal how
p networksareorganizedandfunction,hencemajorchangesin versal. Theanalysisofthecommunitystructureevolutionfor
[ the designed benchmarks reveals that approaches exploiting
theirconfigurationmightsignalimportantturnsintheevolu-
the flow of system configurationsmightbe more accurate in
2 tionofthesystemasawhole, possiblyanticipatingdramatic
detectingtheevolvingcommunitystructurethanmethodsthat
v developmentssuchasrapidgrowthordisruption.
8 consider the snapshots independently. Note that in real data
Indeed, there has been a great deal of activity aroundthis
0 sets this evolutioncan be sharp andbursty, howeverin these
topic in recent years [7–17]. However, most investigations
8 casesthechallengeoffindingthecommunitystructureisnot
lack strength on the validation part, which typically consists
5 welldefined,becausetherangeoftimescalesmakesthemeso-
0 incheckingwhethertheresultsofthealgorithm“makesense”
scopicstructureclearlydisconnected.
. in one or more real networks whose community structure is
1 The paper is structured as follows. In Section 2 we de-
usually unknown. Actually, it is not obviouswhat exactly it
0 scribe the model to generate the benchmark networks, Sec-
5 means to test an algorithm for detecting evolving communi-
tion 3 introduces measures of comparison between dynamic
1 ties. Oneideacouldbethatofcorrectlyidentifyingthecom-
clusterings,Section4showsanexampleoftheapplicationof
: munitystructureofthesystemateachtimestamp. However,
v adynamicmultislicealgorithmontheproposedbenchmarks.
during the evolution of the system several events that affect
i Section5givesasummaryandreportsourconclusions.
X thenetworkstructuremayoccur,suchasthecreationordele-
r tion of nodesor links or link rewiring, and it is not possible
a
todetecttheseeventsbyobservingasingletime-stampednet-
work,theyrequiretakingintoaccountthewholepicturetobe 2. MODELDESCRIPTION
properlyunderstood.
Toexplicitlykeeptrackofthehistoryofthesystem,anop- Themodelweproposeforgeneratingnetworkswithevolv-
tion is to consider multiple snapshots at once. For instance, ing community structure is based on the classic stochastic
intheevolutionaryclusteringapproach[8]thegoalistofind block model (SBM) [18]. It works as follows. A network
apartitionthatisdescriptiveofthestructureofagivensnap- is divided into a number q of subgraphs and the nodes of
shotaswellascorrelatedtothestructureoftheprevioussnap- thesamesubgrapharelinkedwithaprobabilitypin,whereas
shots. Furthermore,the addedvalue of any approachshould nodes of different subgraphs are linked with a probability
be the ability to promptly detect changes in the community pout. Such probabilities match the link densities within and
structure of the network. It would be possible to verify this between subgraphs. Supposing subgraphs of equal size, if
if there were suitable benchmark graphswith evolving clus- pin > (q −1)pout the resulting subgraphsare communities,
ters,butthosearestillmissing. Thispaperaimsatfillingthis asthe(expected)linkdensitywithinsubgraphsexceedstheir
2
connectivitytotherestofthegraph. Thegenerationofsam- (a) Grow / Shrink
ples from this model has a built-in efficiency: If there are
mmax pairs of nodes, the actual number of edges is drawn
from a binomial distribution with parameters mmax and p.
Then,wesimplyplacethisnumberofedgesrandomlytogen-
erateasamplefromourensemble. t=0 t=τ/4 t=τ/2 t=3τ/4 t=τ
Themodelimplementsthetwofundamentalclassesofdy-
namicprocesses: growingorshrinkingandmergingorsplit- (b) Merge / Split
tingofcommunities.Bycombiningthesetworeversibletypes
of processes one can capture the most common behaviors
of dynamic communities in real systems. We are then able
to generate three standardized benchmarks: One consists in
communitiesthat grow and shrink in size (keeping fixed the t=0 t=τ/2 t=τ
totalnumberofnodesofthenetwork),whilethesecondcon- (c) Mixed
siders communities that merge and split. The third one is a
mixedversionoftheprevioustwo,andconsistsofacombina-
tionofthelastfouroperations.
A. Grow-shrinkbenchmark
Thisprocessmodelsthemovementofnodesfromonecom-
munitytoanother. Atalltimes,twocommunitiesarekeptin t=0 t=τ/4 t=τ/2 t=3τ/4 t=τ
a SBM ensemble with intracommunity link density pin and
intercommunity link density pout. However, the number of FIG.1: (Coloronline)Schematicrepresentationofthebenchmarks.
nodesinthetwocommunitieschangesovertime. Inthebasic (a)Grow-shrinkbenchmarkwithq = 2. Webeginwithtwoequal-
process,wehaveatotalof2nnodesintwocommunities. In sizedcommunities,andoveraperiodofτ nodesmovefromthebot-
thebalancedstate,thesearesplitintotwoequalcommunities tomcommunity to thetop, thenfromthe toptobottom, then back
tothesymmetricstate. (b)Merge-Splitbenchmarkwithq = 2. We
ofnnodes,whichwecallAandB. Attheextremes,afrac-
beginwithtwocommunities,andoveraperiodofτ welinearlyadd
tionf ofnodesincommunityAwillswitchtocommunityB.
edgesuntilthereisonecommunitywithuniformlinkdensity, then
Ifwetakenl asthesizeofcommunityA,thenthenumberof reversetheprocess. (c)Mixed benchmark withq = 4, combining
nodesinthecommunityB isnr = 2n−nl. Then,attimet themergingandgrowingprocesses.
thenumberofnodesincommunityAis
nA =n−nf[2x(t+τ/4)−1] (1) andindependentprobabilitypinandbetweenjandeachnode
withtheτ/4phasefactorspecifyingequalsizedcommunities incommunityAwithequalandindependentprobabilitypout,
thustheensembleismaintained. Conveniently,alledgescan
att=0. Thefunctionx(t)isthetriangularwaveform
bepre-computedandstoredtoallowastrictlyrepeatingpro-
∗ ∗ cess,withthestateattimetbeingidenticaltothestateattime
2t , 0 ≤t <1/2
x(t)= ∗ ∗ (2) t+τ,inanalogytothemergingprocess.
(cid:26)2−2t , 1/2 ≤t <1
A special case that we need to cope with is the situation
(with t∗ ≡ (t/τ + φ)mod1), which controls the time pe- wheref isveryhighandpin isverylow. Whenthishappens,
riodicity. The constant φ is a phase factor with φ = 0 a community shrinks too much and it may become discon-
for the q = 2 case and specified otherwise in the case of nected. In order to preserve the ensemble, we do not take
q > 2. With this formulation, we get communities of sizes actionstototallyeliminatethispossibility,butweensurethat
(n,n), (n −nf,n+nf), (n,n), and (n +nf,n− nf) at n(1−f)pin ≫ 2toreducetheprobabilityofdisconnection.
t/τ mod1 = 0, 1, 2,and 3,respectively. Inpractice,all2n However,ifadisconnectionoccurs,theprocessisabortedand
4 4 4
nodesaresortedinsomearbitraryorder,andthefirstnAnodes re-run. Figure1(a)isasketchofthegrow-shrinkbenchmark
areputintocommunityA,andtheothersintocommunityB. forthecaseq =2.
Saythesenodesarei=0toi=2n−1.
After the communitysizes are decided, the edgesmustbe
placed, taking into account that it is necessary that we keep B. Merge-splitbenchmark
thetwocommunitiesintheproperSBMensemblewithequal
and independentlink probability pin at all times. The inde- This process models the mergingof two communities. In
pendence of pairs provides a hint on how to do this. When thissetup, wehavea setof2nnodes, dividedintotwo com-
a node j is movedfrom communityA to B, all the existing munitiesofnnodeseach.Eachofthetwoinitialcommunities
edgesofnodej areremoved.Thenanedgeisaddedbetween hasalinkdensityofpin,wherethoselinksareplacedatinitial-
j and each node in the destination communityB with equal izationandkeptunmodifiedovertime. Therearetwoextreme
3
states: the unmergedand the mergedstate. In the unmerged differentprocesses,anedgeexistswithaprobabilityofpout.
state,allpossiblepairsofnodesbetweenthetwocommunities Figure1(c)exemplifiesthemixedbenchmarkwhenq =4.
haveanedgewith probabilitypout. Thismeansthatthenet-
workstillhasaconnectedcomponent,butthenodesformtwo
communities. Inthemergedstate,allpossiblepairsofnodes 3. TIME-DEPENDENTCOMPARISONMEASURES
betweenthesetwocommunitieshaveanedgewithprobability
pin,whichimpliesthatallpairsofnodesinthenetworkhave Theassessmentoftheperformanceofanyclusteringalgo-
the same link density pin, the previoustwo communitiesare rithm requires the use of measures to define the distance or
nowindistinguishable,andthuswehaveonelargecommunity similarity between any pair of partitions. The list of avail-
with2nnodes. able measures is long, includinge.g. the Jaccard index [23],
The merge-split process is a periodic interpolation of the the Rand index [24], the adjusted Rand index [25], the nor-
mergedandunmergedstates. Thenumbersofintercommunity malizedmutualinformation[26],thevanDongenmetric[27]
edgesintheunmergedstatemumandinthemergedstatemm andthe normalizedvariationofinformationmetric [28]. All
are first picked from a binomial distribution consistent with of them have in common the possibility of being expressed
the binomialdistributionparametersn2 and pout or pin. All intermsoftheelementsoftheso-calledconfusionmatrixor
possible intercommunity edges are placed in some arbitrary contingencytable, thus we focusfirst on its calculation. Let
butrandomorder,andthefirst C = {Cα|α = 1,...,r} and C′ = {Cα′′|α′ = 1,...,r′}
betwopartitionsofthedatainrandr′ disjointclusters. The
∗
m (t)=[1−x(t)]mum+x(t)mm (3) αα′thcomponentofthecontingencytableM accountsforthe
edgesare selected to be activeat time t. The effectiveintra- numberofelementsintheintersectionofclustersCαandCα′′,
ceotemrmx(utn)itiysltihnektdreiannsgituylaisrpw∗inatveer(fot)rm=fmro∗m(t)E/qn.2(.2T).heInpaprraamc-- mαα′ =|Cα∩Cα′′|. (5)
tice, this means that at time t/τ mod1 = 0 the communi-
Thesizesoftheclusterssimplyreadnα =|Cα|= α′mαα′
tiesareunmergedandatt/τ mod1 = 1/2thecommunities and n′α′ = |Cα′′| = αmαα′ and the total numPber of ele-
aremerged,withlinearinterpolation(ofthenumberofedges) mentsis N = αnαP= α′n′α′ = α α′mαα′. With
between these points. Since the possible edges are ordered
thesedefinitionsPathand,onPecancalculPatetPheJaccardindex,
only at initialization, the process is strictly periodic, that is,
the edges present at time t are identical to those present at mαα′
timet+τ.
(cid:18) 2 (cid:19)
One may think that the communities are fully merged at J = Xα Xα′ , (6)
′
the extreme of this process, where the intercommunity link nα nα′ mαα′
+ −
density is p∗inter = pin (at t = τ/2). However, due to the Xα (cid:18) 2 (cid:19) Xα′ (cid:18) 2 (cid:19) Xα Xα′ (cid:18) 2 (cid:19)
detectabilitylimitofcommunitiesinstochasticblockmodels,
thisisnotthecase[22]. Evenwhenpout <pin,itcanbethat thenormalizedmutualinformationindex,
theconfigurationisindistinguishablefromonelargecommu-
nity.Following[22],atthepoint Nmαα′
−2Xα Xα′ mαα′log nαn′α′
pin−p∗inter =rn1 (pin+p∗inter) (4) NMI= nαlognα + n′α′logn′α′ , (7)
N N
Xα Xα′
we consider the communities to be merged into one for all
practical purposes. While this limit is strictly speaking only andthenormalizedvariationofinformationmetric,
accurateinthesparseandinfinite-sizelimit,itisanadequate
approximation.Aschematicrepresentationofthemerge-split −1 mαα′ (mαα′)2
NVI= log , (8)
benchmark,forq =2isshowninFig.1(b). logN Xα Xα′ N nαn′α′
where,byconvention,0log0=0.
C. Mixedbenchmark In the case of evolvingnetworkswe have to comparetwo
sequences of partitions {C(t)|t = 1,...,T} and {C′(t)|t =
Thisprocessisacombinationofthemergingandgrowing 1,...,T},ataskthatcanbeperformedindifferentways.The
processes.Inthisprocess,thereisatotalof4nnodeswithtwo simplestsolutionistheindependentcomparisonofpartitions
merging-splittingcommunities(2nnodes)andtwo growing- ateachtimestep,bymeasuringthesimilarityordistancebe-
shrinking communities (2n nodes). The intra-community tweenC(t)andC′(t)foreachvalueoft,thusobtaining,e.g.,
links are managed with the same processes as above with a Jaccard index J(t) for each snapshot, see Fig. 2(a). How-
phase factors of φ = 0 for both. If there are q = 4a > 4 ever, this procedure discards the evolutionary nature of the
total communities, then the pairs of communities involved communities: We would like to quantify not only the static
in merging and growing process have phase factors φ = resemblance of the communities but also if they evolve in a
0, 1, 2,...a−1. Between the pairs of nodes that belong to similarway.
a a a
4
Planted Algorithm clusterevolutionsDα(t;σ)andDα′′(t;σ). Likewise,wehave
A A A A A A
1 1
2 A B A 2 B B B nα(t;σ)=|Dα(t;σ)|= mαα′(t;σ), (10)
3 B B C 3 B B C Xα′
4 C C C 4 C C D n′α′(t;σ)=|Dα′′(t;σ)|= mαα′(t;σ), (11)
t=1 t=2 t=3 t=1 t=2 t=3 Xα
(a) A A A
and
A B A A B C
B B C
C C C A 1 1 N = nα(t;σ)= n′α′(t;σ)= mαα′(t;σ).
Xα Xα′ Xα Xα′
A A A B 1
(12)
B B B
B B C C 1 Finally,wemayuseEqs.(6)–(8)tocalculatethecorrespond-
C C D ing windowed Jaccard index J(t;σ), windowed normalized
mutualinformationindexNMI(t;σ), and windowednormal-
(b) A A A AA BB CC ized variation of information metric NVI(t;σ), respectively.
A B A Of course, the windowed measures reduce to the standard
B B C AA 1 static ones when σ = 0, and are able to capture differences
C C C
AB 1 in theevolutionofcommunitiesthatcannotbedistinguished
A A A usingtheirclassicalversions(seetheAppendix).
B B B BB 1
We will see in the nextsection how the plots of NVI(t;σ)
B B C
CC 1 arevaluableto comparedifferentalgorithmsandtodetectin
C C D
which moments of the time evolution they differ. Neverthe-
less,itisalsoconvenienttohaveasinglenumbertoquantify
FIG.2:(Coloronline)Constructionofthecontingencytablesmαα′.
theoveralldeviation.Asimplesolutionistheuseoftheaver-
Ontopwerepresentthreesteps(columns)ofthetimeevolutionofa
networkoffournodes(rows),andthepartitionsincommunitieswe agesquarederrors,whichisexpressedasfollows:
wanttocompare,e.g.theplantedpartitionsfromthebenchmarkand
T
thoseobtainedbyacertainalgorithm. Tocomparethesetwoparti- 1
2
tionings, wecandoitasitisdepictedin(a),whichtakesonlyone EJ(σ) = [J(t;σ)−1] , (13)
T
snapshotatatime(σ=0),orasin(b),buildingacontingencytable Xt=1
wheretheentriesconsidertwosnapshotsatthesametime(σ = 1). T
1
Afterward,themeasures(NVI,NMIorJaccardindex)arecalculated E (σ) = [NMI(t;σ)−1]2, (14)
NMI
fromthesetables. T Xt=1
T
1
2
E (σ) = NVI(t;σ) . (15)
NVI
T
Xt=1
Ourproposalconsistsinthedefinitionofwindowedforms
of the differentindices and metrics, obtainedby considering
Forsimplicityandforitssuperiormathematicalproperties
sequences of consecutive partitions, i.e. time windows of a
(see[28])wehavechosentouseonlytheNVImetricsinthe
predefined duration σ. In Fig. 2(b) we show the compari- restofthisarticle. SeeSupplementalMaterialfortheresults
sonbetweenindividualsnapshotsandsequencesoflength2.
usingthe normalizedmutualinformationandthe Jaccardin-
Forexample,letusconsiderthetimewindowformedbytime
dex.
stepsfromttot+σ. Everynodebelongstoadifferentcluster
at each snapshot, and this evolution can be identified as one
oftheitemsinD(t;σ) = C(t)×C(t+1)×···×C(t+σ) 4. RESULTS
for the first sequence of partitions, and D′(t;σ) = C′(t) ×
···×C′(t+σ)forthesecondone, wherethemultiplication
Here we show an example of the application of a com-
sign denotes the Cartesian product of sets. Since the num-
munity detection algorithm, designed to take into account
berofnodesisN,thereareatmostN differentnonvoidsets
the evolutionof complexnetworks, to revealthe community
Dα(t;σ) ∈ D(t;σ) and the same for Dα′′(t;σ) ∈ D′(t;σ). structureinourbenchmarks.Thechosenmethodisthemulti-
Forexample,inFig.2(b),thecombinationsofpartitions(ex-
slice algorithmin [12], whichextendsthedefinitionofmod-
cludingemptysets)areD(t=1;σ =1)={AA,AB,BB,CC}
ularity to multilayer networks. In their representation, each
andD′(t = 1;σ = 1)= {AA,BB,CC}. Next,wemaydefine
layer(slice) consistsofa singlenetworkata particulartime.
theelementsofthecontingencytableforthistimewindowas
Theslicesareconnectedbetweenthembyjoiningeachnode
with its counterpart in the next and previous layer, and this
mαα′(t;σ)=|Dα(t;σ)∩Dα′′(t;σ)|, (9) linkhasaspecifiedweightω, equalforalllinksofthiskind,
whichactsasa tuningparameter. For ω = 0, noconnection
between slices is considered and the algorithm is performed
which accountsfor the numberof nodes followingthe same statically.Asthisvalueincreases,moreconsiderationisgiven
5
to the communities across layers. The formulation includes NVIincreasewiththesizeofthetimewindowforω =0and
an additional parameter γ, which accounts for the tuning of ω = 0.5, but in a larger amount when the parameter corre-
theresolutionatwhichcommunitiesarefound,inthemanner spondsto thestatic versionofthemultislice algorithm. This
of[29]. Inthiswork,wehaveusedthecodeavailablein[30], meansthatthe intersliceweightishelpingto findthe persis-
settingthe resolutionparameterγ to1 andvaryingtheinter- tenceofnodesintheircommunities,asexpected.Themerge-
slicecouplingω. split benchmark shows an almost identical bad performance
Thebenchmarksusedtoputtotestthisalgorithmaregen- forthethreevaluesofωatwindowsofsize1,butω =2does
erated using the model proposed in this paper. For the sake notmakeitworsewhenthesizeofthewindowincreases,un-
ofsimplicity,wegeneratethreesimplestandardbenchmarks, liketheothertwo.Themixedbenchmarkisquiteneutral,with
one for each basic procedure: grow-shrink, merge-split and justasmalldifferencefromω = 2. Finally,theNVIsquared
mixed. The grow-shrink benchmark consists in a network errors reported in Table I and calculated using Eq. (15) are
with q = 2 communities, where each community has ini- inperfectagreementwiththisanalysis. Theresultsusingthe
tiallyn = 32nodes(thereforethetotalsizeofthenetworkis NMIandJaccardindices(seetheSupplementalmaterial)also
N =64),withpin =0.5,pout =0.05,f =0.5,andτ =100 support these observations. Thus, we may conclude that, in
timesteps.Themerge-splittesthasavariablenumberofcom- thiscase,theuseofmemorytotracktheevolutionofcommu-
munities;inthispaperweusetheparametersq = 2commu- nities is convenient, but the trade-offbetween the continuity
nitiesofsizen = 32each,withpin = 0.5,pout = 0.05,and of the community structure and its static relevance must be
τ = 100. Themixedbenchmark,acombinationoftheprevi- carefullyadjusted.
oustwo, hasq = 4communitiesofn = 32nodeseach,and
theotherparametersaresetasinthepreviouscases.
5. CONCLUSIONS
Figure 3 shows the planted partitions for the three bench-
marks and the results from the multislice algorithm at three
different interslice couplings: In the extreme case ω = 0 We have presented a simple model based on the stochas-
slices are considered independently, ω = 0.5 is an interme- tic block model that allows for the construction of time-
diate value that provides good results, and ω = 2 provides dependent networks with evolving community structure. It
an example of the partitioning obtained when using strong is useful for benchmarkingpurposes in testing the ability of
coupling between layers. It can be seen that for ω = 0 we community detection algorithms to track properly the struc-
obtainadifferentpartitionforeachtimestep, andtheresults tural evolution. We have also introduced extended time-
aremostlycorrect,exceptforthoseconfigurationsofthesizes dependentmeasuresforthecomparisonofdifferentpartitions
of the communities where the preference of modularity for inthedynamiccase,whichallowfortheobservationofdiffer-
equal-sized communities hampers the process (see the first encesbetweentheoutcomeofthealgorithmsandtheplanted
columnofFig.3). Highervaluesofω requesthigherconsis- partitionsthroughtime.
tencythroughtime,whichimpliesthatthenumberofmisclas- Ourcodeforbenchmarkgenerationandthetime-dependent
sifiedindividualsnapshotsisreduced.Wehavealsocompared comparisonindicesisavailableat[32]andreleasedunderthe
themultislicemethodwithatemporalstabilityapproach[31] GNUGeneralPublicLicense.
and the results obtainedare verysimilar to the results of the
multislicealgorithmobtainedatω =0.5.
Acknowledgments
Toquantitativelyevaluatetheresults,weusethewindowed
measuresintroducedintheprevioussection. Wecalculatethe
measuresbetweenthepartitionsobtainedbythealgorithmand This work was partially supported by MINECO through
theplantedones,forthreevaluesofthetimewindow. When Grant No. FIS2012-38266; and by the EC FET-Proactive
the time window is of size 1 (σ = 0), each snapshotis con- ProjectPLEXMATH(GrantNo.317614).A.A.alsoacknowl-
sideredindependently,thatis,wehavecomputedthemeasure edgespartialfinancialsupportfromtheICREAAcademiaand
between the planted partition at t and the algorithm’s result the JamesS. McDonnellFoundation. R.K.D. andS.F. grate-
att,repeatingthisprocessuntilt = τ. Instead,withthetime fully acknowledge MULTIPLEX, Grant No. 317532 of the
windowofsize2(σ =1),weevaluatetheevolutionofthepar- EuropeanCommission,andthecomputationalresourcespro-
titionsduringtwoconsecutivetimesteps,followingthesame videdbyAaltoUniversityScience-ITproject.
processbutcomparingtheplantedpartitionsat[t,t+1]with
thealgorithm’sresultsat[t,t+1]. Thisformulationismore
restrictive,asweimpose,inadditiontotheconditionthatthe Appendix
nodes must belong to the same community, that their evolu-
tionduringtwoconsecutivetimestepsisalsothesame. Sim- Distinguishingcommunityevolutionswithwindowedmeasures
ilarly,wehavealsoanalyzedtimewindowsofsize5(σ = 4)
to checkthequalityofthe detectedcommunityevolutionsat Figure 5 shows an example in which, according to the
longerranges. plantedpartitions,theeightnodesofanetworkaredividedin
Figure4showstheresultsfortheNVI.Weobservethat,for two communitiesof four nodeseach and these partitionsre-
thegrow-shrinkbenchmark,theerrorislargeforω = 0, but mainconstantthroughoutthethreetimesstepsofthenetwork
becomesalmost zero at ω = 2. Moreover,the valuesof the evolution.Twodifferentcommunitydetectionalgorithmsfind
6
Grow-Shrink Merge-Split Mixed
60 60 120
50 50 100
d
ed40 40 80
ntde i30 30 60
ao
Pln20 20 40
10 10 20
0 0 0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
60 60 120
50 50 100
0 d40 40 80
= e i
ω od30 30 60
n
20 20 40
10 10 20
0 0 0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
60 60 120
50 50 100
5
= 0. de id3400 3400 6800
ω no
20 20 40
10 10 20
0 0 0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
60 60 120
50 50 100
2 d40 40 80
= e i
ω od30 30 60
n
20 20 40
10 10 20
0 0 0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
time time time
FIG.3:(Coloronline)Resultsoftheapplicationofthemultislicecommunitydetectionmethodtothethreebenchmarksproposed(incolumns).
Thefirstrowcorrespondstotheplantedpartitionofeachbenchmark,whilethethreeremainingrowsarethepartitionsobtainedbythemultislice
algorithmfordifferentvaluesoftheintersliceparameterω,whichistheweightofthecouplingbetweendifferentinstancesofthesamenodes
acrosslayers. Whenω = 0theslicesaredisconnectedandthenthecommunitydetectionanalysisisdoneforeachsliceseparately. Asthis
valueincreases,moreimportanceisgiventotheevolvingnatureoftheproblem, andcommunitiesacrossslicesarefound. Ineachplot,the
verticalaxiscorresponds totheindexofnodesinthenetwork, whilethehorizontalaxisrepresentsthetime. Thecolorofeachpair{node,
time}isthelabelofthecommunityatwhichthenodeisassignedatthatspecifictime.
thecommunitiesevolutionsrepresentedinFigs.5(a)and5(b), in time, with a constant value of the NVI equal to 0.2856.
which are characterized by the assignment of just one node However,ifwetakeintoaccountatimewindowofsize3,the
to the wrong communityat each time step. In Fig. 5(a) this twoevolvingcommunitystructuresdetectedbythealgorithms
node is the fourth one during the three time steps, while in aredifferent,yieldingstructurallydifferentcontingencytables
Fig.5(b)theyarethesecond,thethird,andthesixth,respec- and values of the NVI equal to 0.2856 and 0.3852, respec-
tively. Sincethenatureofthemistakeisthesameatalltime tively. Therefore,the conclusionis thatwindowedmeasures
steps, thecomparisonoftheplantedandalgorithmpartitions give complementaryinformationfor the comparison of time
withatimewindowofsize1generatesequivalentcontingency evolving community structures due to their capacity to take
tables,thusthestandardcomparisonmeasuresdonotchange intoaccountseveralsnapshotsatthesametime.
[1] L.Kovanen,M.Karsai,K.Kaski,J.Ke´rtesz,andJ.Sarama¨ki, [2] P.HolmeandJ.Sarama¨ki,Phys.Rep.519,97 (2012).
J.Stat.Mech.(2011)P11005. [3] N.Perra,A.Baronchelli,D.Mocanu,B.Gonc¸alves,R.Pastor-
7
Grow-Shrink Merge-Split Mixed
1.0
ω = 0.0
1 ω = 0.5
= 0.8 ω = 2.0
w
ndo VI0.6
wi N
e 0.4
m
Ti 0.2
0.0
1.0
ω = 0.0
2 ω = 0.5
= 0.8 ω = 2.0
w
o 0.6
d VI
n N
wi 0.4
e
m
Ti 0.2
0.0
1.0
ω = 0.0
5 ω = 0.5
= 0.8 ω = 2.0
w
o 0.6
d VI
n N
wi 0.4
e
m
Ti 0.2
0.0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
time time time
FIG.4: (Coloronline)Plotsofthenormalizedvariationofinformation(NVI)betweentheplantedpartitionandtheresultsofthemultislice
algorithminFig.3,forthreedifferentinterslicecouplingsandforthethreebenchmarksproposed. TheNVIiscomputedusingtheproposed
evolvingformulationandforthreedifferentwindowsizes:1,2and5.Thereisacolumnforeachbenchmark,andarowforeachtimewindow
size.
.
Satorras, and A. Vespignani, Phys. Rev. Lett. 109, 238701 Onnela,Science328,876(2010).
(2012). [13] C.Granell,S.Go´mez,andA.Arenas,Chaos21,016102(2011).
[4] M.Starnini,A.Baronchelli,A.Barrat,andR.Pastor-Satorras, [14] P. Ronhovde, S. Chakrabarty, D. Hu, M. Sahu, K. K.
Phys.Rev.E85,056115(2012). Sahu, K. F. Kelton, N. A. Mauro, and Z. Nussi-
[5] A.Barrat,B.Fernandez,K.K.Lin,andL.-S.Young,Phys.Rev. nov, Sci. Rep. 2, 329 (2012), ISSN 2045-2322, URL
Lett.110,158702(2013). http://dx.doi.org/10.1038/srep00329.
[6] S.Fortunato,Phys.Rep.486,75(2010). [15] D.S.Bassett,M.A.Porter,N.F.Wymbs,S.T.Grafton,J.M.
[7] J. Hopcroft, O. Khan, B. Kulis, and B. Selman, Proc. Natl. Carlson,andP.J.Mucha,Chaos23,013142(2013).
Acad.Sci.USA101,5249(2004). [16] P.Bro´dka,S.Saganowski,andP.Kazienko,Soc.NetworkAnal.
[8] D. Chakrabarti, R. Kumar, and A. Tomkins, KDD ’06: Pro- Min.3,1(2013),ISSN1869-5450.
ceedings of the 12th ACMSIGKDD International Conference [17] M.DeDomenico,A.Lancichinetti,A.Arenas,andM.Rosvall,
onKnowledgeDiscoveryandDataMining(ACM,NewYork, Phys.Rev.X5,011027(2015).
2006),pp.554–560. [18] P.Holland, K.B.Laskey, andS.Leinhardt, Soc. Networks5,
[9] G.Palla,A.-L.Baraba´si,andT.Vicsek,Nature(London)446, 109(1983).
664(2007). [19] M.GirvanandM.E.Newman,Proc.Natl.Acad.Sci.USA99,
[10] J.Ferlez,C.Faloutsos,J.Leskovec,D.Mladenic,andM.Gro- 7821(2002).
belnik,Proceedingsofthe2008IEEE24thInternationalCon- [20] A. Lancichinetti, S. Fortunato, and F. Radicchi, Phys. Rev. E
ferenceonDataEngineering(IEEEComputerSociety,Wash- 78,046110(2008).
ington,DC,2008),pp.1328–1330. [21] R. Guimera` and M. Sales-Pardo, Proc. Natl. Acad. Sci. USA
[11] P.RonhovdeandZ.Nussinov,Phys.Rev.E80,016109(2009). 106,22073(2009).
[12] P.J.Mucha, T.Richardson, K.Macon, M.A.Porter,andJ.P. [22] A. Decelle, F. Krzakala, C. Moore, and L. Zdeborova´, Phys.
8
Time NVIsquarederror
Multislice window Grow-shrink Merge-split Mixed
1 0.0065 0.0851 0.0015
ω=0.0 2 0.0201 0.2146 0.0015
5 0.0658 0.4427 0.0016
1 0.0023 0.0808 0.0014
ω=0.5 2 0.0067 0.2019 0.0014
5 0.0242 0.4278 0.0015
1 0.0006 0.0878 0.0023
ω=2.0 2 0.0005 0.1113 0.0024
5 0.0006 0.1922 0.0029
TABLEI:TheNVIsquarederror,foreachmethodtestedandeachbenchmarkinFig.3,consideringthreedifferenttimewindows.
(a) (b)
Planted Algorithm Planted Algorithm
1 A A A C C C 1 A A A C C C
2 A A A C C C 2 A A A DD C C
3 A A A C C C 3 A A A C DD C
4 A A A D D D 4 A A A CC CC CC
5 B B B D D D 5 B B B D D D
6 B B B D D D 6 B B B D D CC
7 B B B D D D 7 B B B D D D
8 B B B D D D 8 B B B D D D
t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3
Time window = 1 Time window = 1
t=1 C D t=2 C D t=3 C D t=1 C D t=2 C D t=3 C D
A 3 1 A 3 1 A 3 1 A 3 1 A 3 1 A 4 0
B 0 4 B 0 4 B 0 4 B 0 4 B 0 4 B 1 3
0.2856 0.2856 0.2856 0.2856 0.2856 0.2856
Time window = 3 Time window = 3
CCC DDD CCC CDC DCC DDC DDD
AAA 3 1 AAA 2 1 1 0 0
BBB 0 4 BBB 0 0 0 1 3
0.2856 0.3852
FIG.5: (Coloronline)Exampleofthecomparisonofaplantedevolvingcommunitystructurewiththeresultsfromtwodifferentalgorithms.
TheNVIvaluesareshownbelowthecorrespondingcontingencytablesfortimewindowsofsizes1and3.
.
Rev.Lett.107,065701(2011). [29] J. Reichardt and S. Bornholdt, Phys. Rev. Lett. 93, 218701
[23] P.Jaccard,NewPhytol.11,37(1912). (2004).
[24] W.M.Rand,J.Am.Stat.Assoc.66,846(1971). [30] URLhttp://netwiki.amath.unc.edu/GenLouvain/GenLouvain
[25] L.HubertandP.Arabie,J.Classif.2,193(1985),ISSN0176- [31] G.PetriandP.Expert,Phys.Rev.E90,022813(2014).
4268. [32] URLhttp://rkd.zgib.net/proj/multiplex/.
[26] A. Strehl and J. Ghosh, J. Mach. Learn. Res. 3, 583 (2002),
ISSN1532-4435.
[27] S.V.Dongen, Ph.D.thesis,DutchNationalResearchInstitute
Supplementalmaterial
forMathematicsandComputerScience,UniversityofUtrecht,
2000,(2000.
[28] M.Meila˘,J.Multivar.Anal.98,873(2007). AdditionalTablesIIandIIIandFigs.6and7.
9
Time Jaccardsquarederror
Multislice window Grow-shrink Merge-split Mixed
1 0.0720 0.3345 0.0307
ω=0.0 2 0.1365 0.4840 0.0303
5 0.2336 0.6272 0.0325
1 0.0293 0.3193 0.0276
ω=0.5 2 0.0546 0.4608 0.0282
5 0.1105 0.6013 0.0303
1 0.0019 0.3326 0.0360
ω=2.0 2 0.0014 0.3605 0.0374
5 0.0147 0.4488 0.0421
TABLEII:Jaccardsquarederror,foreachmethodtestedandeachbenchmark,consideringthreedifferenttimewindows.
Time NMIsquarederror
Multislice window Grow-shrink Merge-split Mixed
1 0.0337 0.4932 0.0067
ω=0.0 2 0.0621 0.4806 0.0063
5 0.1022 0.4855 0.0059
1 0.0143 0.4896 0.0060
ω=0.5 2 0.0262 0.4753 0.0059
5 0.0479 0.4790 0.0055
1 0.0065 0.4951 0.0094
ω=2.0 2 0.0041 0.4891 0.0094
5 0.0041 0.4825 0.0100
TABLEIII:NMIsquarederror,foreachmethodtestedandeachbenchmark,consideringthreedifferenttimewindows.
10
Grow-Shrink Merge-Split Mixed
1.0
1
= 0.8
w ex
ndo d ind0.6
e wi ccar0.4
m Ja ω = 0.0
Ti 0.2 ω = 0.5
ω = 2.0
0.0
1.0
2
= x0.8
w de
o n0.6
wind card i0.4
me Jac ω = 0.0
Ti 0.2 ω = 0.5
ω = 2.0
0.0
1.0
5
= x0.8
w de
o n0.6
wind card i0.4
me Jac ω = 0.0
Ti 0.2 ω = 0.5
ω = 2.0
0.0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
time time time
FIG.6:PlotsoftheJaccardIndexbetweentheplantedpartitionandtheresultsofthemultislicealgorithmforthreedifferentinterslicecouplings
andforthethreebenchmarksproposed.TheJaccardindexiscomputedusingtheproposedevolvingformulationandforthreedifferentwindow
sizes:1,2and5.Thereisacolumnforeachbenchmark,andarowforeachtimewindowsize.
