Detecting Cohesive and 2-mode Communities in Directed and Undirected Networks

1.6 MB·

by  Jaewon Yang

Detecting Cohesive and 2-mode Communities in
Directed and Undirected Networks
Jaewon Yang, Julian McAuley, Jure Leskovec
Stanford University
{jayang, jmcauley, jure}@cs.stanford.edu
ABSTRACT
4
1
Networksareagenerallanguageforrepresentingrelationalinfor-
0
mation among objects. An effective way to model, reason about,
2
and summarize networks, is to discover sets of nodes with com-

n monconnectivitypatterns.Suchsetsarecommonlyreferredtoas
a network communities. Research onnetwork community detection
J haspredominantlyfocusedonidentifyingcommunitiesofdensely

9 connectednodesinundirectednetworks.
2 In this paper we develop a novel overlapping community de-
tection method that scales to networks of millions of nodes and

] edges and advances research along two dimensions: the connec-
I
S tivity structure of communities, and the use of edge directedness
. forcommunitydetection.First,weextendtraditionaldefinitionsof Figure1:Twotypesofnetworks(directedandundirected)and
s networkcommunitiesbybuildingontheobservationthatnodescan two types of communities (cohesive and 2-mode). While re-
c
bedenselyinterlinkedintwodifferentways:Incohesivecommu- searchhaspredominantlyfocusedonundirected-cohesivecom-
[
nitiesnodeslinktoeachother,whilein2-modecommunitiesnodes munities(topleft),wedevelopamethodthatcandetectcohe-

1 linkinabipartitefashion,wherelinkspredominatebetweenthetwo siveaswellas2-modecommunitiesinbothdirectedandundi-
v partitionsratherthaninsidethem.Ourmethodsuccessfullydetects rectednetworks.
5 both2-modeaswellascohesivecommunities,thatmayalsoover-
7 laporbehierarchicallynested.Second,whilemostexistingcom-
3 munitydetectionmethodstreatdirectededgesasthoughtheywere worksatthelevelofgroupshelpsustodiscoverfunctionalrolesof
7 undirected, our method accounts for edge directions and is able proteinsinprotein-proteininteractionnetworks[36],politicalfac-
. tions in a network of bloggers [1], social circles in online social
1 toidentifynovelandmeaningfulcommunitystructuresinbothdi-
networks[31],oreventopicsinwordassociationnetworks[2].
0 rectedandundirectednetworks,usingdatafromsocial,biological,
Onewaytounderstandnetworksatthelevelofgroupsistoiden-
4 andecologicaldomains.
tify sets of nodes with similar connectivity patterns. Traditional
1 CategoriesandSubjectDescriptors:H.2.8[DatabaseManage-
methods aim to find network communities, which are defined as
: ment]:DatabaseApplications–Datamining
v groupsofnodeswithmanyconnectionsamongthegroup’smem-
GeneralTerms:Algorithms,theory,experimentation.
i bers, butfew totherestof thenetwork [2,11,14,34].However,
X Keywords:Networkcommunities,Overlappingcommunitydetec-
densecommunitiesarebutonekindofgroupstructureinnetworks,
tion,2-modecommunities.
r and there may be other structures that help us to understand net-
a
1. INTRODUCTION worksbetter.Forexample,consideraTwitterfollowernetworkand
the“community”ofcandidatesinthe2012U.S.presidentialelec-
Networks are a powerful way to model relational information
tion. Thiscommunity isnot densely interlinked, inthesense that
amongobjectsfromsocial,natural,andtechnologicaldomains.Net-
thecandidatesdonotfolloweachother;thuswewouldnotbeable
workscanbestudiedat variouslevelsof resolutionranging from
tofindthiscommunityifweweretousetraditionalmethodsthat
wholenetworkstoindividualnodes.Arguablythemostusefullevel
search for densely connected sets of nodes. However, such com-
ofresolutionisatthelevelofgroupsofnodes.Studyinggroupsof
munitiescanbeidentifiedbecausetheyformaroundnodeswhose
nodes allows us to identify and analyze modules or components
edgeshavesimilarendpoints.Continuingourexample,presidential
of networks. For example, understanding the organizationof net-
candidates form a community in Twitter not because they follow
eachotherbutbecauseacommonsetof“fans”followsthem.
Permissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalor
Thuscommunitiescanbecharacterizedbytheconnectivitystruc-
classroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributed
turebetweenthemembersandalsobytheconnectivitystructureof
forprofitorcommercialadvantageandthatcopiesbearthisnoticeandthefullcitation
onthefirstpage.CopyrightsforcomponentsofthisworkownedbyothersthanACM thememberstotherestofthenetwork.Werefertothesecommu-
mustbehonored.Abstractingwithcreditispermitted.Tocopyotherwise,orrepublish, nitiesas2-modecommunities.Forexample,incaseof“fans”link-
topostonserversortoredistributetolists,requirespriorspecificpermissionand/ora ingto“celebrities”membersofacommunitymaybelinkedtothe
[email protected]. samesetofendpoints,eveniftheydonotlinktoeachother.Sim-
WSDM’14,February24–28,2014,NewYork,NewYork,USA.
ilar examples also exist beyond social networks; for example, in
Copyright2014ACM978-1-4503-2351-2/14/02...$15.00.
protein-protein interaction networks, some protein complexes act
http://dx.doi.org/10.1145/2556195.2556243.
asbridgesorregulators,i.e.,theydonotinteractamongthemselves bothsendsandreceiveslinksfromothermembers);2-modecom-
butregulate/interactwiththesamesetofproteins[36]. munitiesaremodeledwithunidirectionalmembershipswheresome
Anothercommonassumptionmadebymanypresentcommunity members mostlysend/create links (i.e.,fans) whileothersmostly
detection methods is that networks are undirected [34, 43]. This receivethem(i.e.,celebrities).
implies that relationships between connected nodes are symmet- Having defined the node-community affiliation model we then
ric or reciprocal. However, indirected networks relationships are developamethodtofitthemodel toagivennetwork.Ourmodel
asymmetric,aswithourpreviousexampleabout“fans”whofollow fittingprocedurebuildsonthatoftheBigCLAMcommunitydetec-
“celebrities”.Eventhoughmethodscanoftenbeadaptedtohandle tionmethod[45].Althoughwesolveamorecomplexproblemthan
directednetworks,thisisoftendoneinanad-hocfashion(e.g.,by BigCLAM(i.e.,wefindboth2-modeaswellascohesivecommuni-
treating directed edges as though they were undirected) and can ties),weemploysimilarapproximationtechniques.Untilrecently,
lead to unexpected or undesirable results [14, 26, 39]. Moreover, methodsforoverlappingcommunitydetectioncouldonlyprocess
byignoringedgedirectednessimportantinformationmaybelost, networkswithuptoaround10,000nodes[16].Incontrast,CoDA
especially if relationships are predominantly non-reciprocal as in caneasilyhandlenetworksthataretwoordersofmagnitudelarger:
predator-preynetworks[26]orinsocialnetworkslikeTwitter. millionsofnodes,tensofmillionsofedges.Moreover,CoDAcan
Presentwork:Detectingcohesiveand2-modecommunitiesin beeasilyparallelizedwhichfurtherincreasesthescalability.
directedandundirectednetworks.Hereweconsidernewnotions Present work: Experimental results. We evaluate CoDA on a
ofcommunitylinkingstructurethatgobeyondthinkingofcommu- numberofnetworksfromvariousdomains.Weconsidersocial,bi-
nitiesasinternallywell-connected setsof nodes. Our workstems ological,communication,andecologicalnetworks.WetestCoDA
fromsocialnetworkliteratureonstructuralequivalence[9],where onnetworkswithexplicitlylabeledground-truthcommunities[31,
ithasbeennotedthatsocialhomogeneity(i.e.,socialcommunities) 44] as well as on networks where communities can be manually
arisesnotonlybetweennodesthatlinktoeachother(i.e.,internal examined.
groupconnectivity),butalsobetweennodesthatlinktotherestof ExperimentsdemonstratethatCoDA’sabilitytodetect2-mode
the network inacoordinated way (i.e.,external group connectiv- as well as cohesive communities leads to improved performance
ity).Inparticular,weconsiderdifferentnotionsof“communities” overtheexistingstate-of-the-art.Forexample,whendetectingso-
that are depicted in Figure 1. We differentiate between cohesive cial circles in the Google+ online social network, CoDA gives a
communities(Fig.1,Cohesive) and2-modecommunities(Fig.1, relativeimprovementinaccuracyof36%overLinkclustering[2]
2-mode)wherenodeslinkinabipartitefashionwithlinkspredom- (28%overMMSB[3],25%overcliquepercolation[34]and21%
inantlyappearingbetweenpartitionsratherthaninsidethem. overDEMON[11]).
Whileexistingcommunitydetectionmethodstypicallyfocuson Moreimportantly,CoDAfacilitatesnoveldiscoveriesaboutthe
Undirected-Cohesive or Directed-Cohesive communities [11, 14, community structure of networks. For example, we find that 2-
26,34,43],thefocusof ourpaper isondeveloping methodsthat modecommunitiesinfoodwebsofpredatoryrelationsbetweenor-
candetectcommunitiesofallfourdifferenttypesdepictedinFig- ganisms correspond to groups of predators who rely on similar
ure1.Bymodelingeachofthesedefinitionsinconcert,weareable groups of prey. Interestingly we find that in scientific paper cita-
tocapturethecomplexstructurepresentinnetworks. tionnetworks,protein-proteininteractionnetworks,aswellasweb
Presentwork:CommunitiesthroughDirectedAffiliations.We graphs, the majority of detected communities are 2-mode. How-
presentCoDA(CommunitiesthroughDirectedAffiliations),ameth- ever,insocialnetworkswhereedgessignifyreciprocalfriendships,
od for overlapping community detection that scales to networks cohesive communities are more frequent. In Twitter or Google+,
with millions of nodes and tens of millions of edges. CoDA ex- where relationships are asymmetric, 2-mode communities repre-
hibits the following three properties: (1) It naturally detects both sentasignificantportionofthenetwork(20%inTwitterand30%
cohesively connected as well as 2-mode communities. (2) CoDA inGoogle+).
allowscohesiveand2-modecommunitiestooverlaporbehierar- Furtherrelatedwork.Whilethereexistanumberofdifferentdef-
chically nested. (3) CoDA naturallyallowsfor community detec- initions of network communities [14], traditionally, communities
tionindirectedaswellasundirectednetworks. havebeenthoughtofasdenselyconnectedsetsofnodes[2,12,34,
We develop our community detection method by first present- 37].Incontrast,thenotionofstructuralequivalence suggeststhat
ingagenerativemodelofnetworkswhereedgesarisefromaffili- nodeswithsimilarconnectivitypatternsmaybeconsideredacom-
ationsofnodestocohesiveand2-modecommunities.Thenwefit munityeveniftheydonotlinktoeachother[9,17,23].Ourwork
themodeltoagivennetworkandthusdiscovercommunities. herebuildsonbothnotionsofnetworkcommunitiesandattempts
Our model starts with a bipartite affiliation graph [25, 43, 45, toresolvethembyusingasingle,unifiedmodel.
47],wherenodesoftheunderlyingnetworkrepresentone‘layer’of Detectingcommunitiesofdenselyconnectedsetsofnodesisan
thebipartitegraphandcommunitiesrepresenttheother.Edgesbe- extensivelyresearchedarea[14,30,35,42]withaplethoraofdif-
tweennetwork-nodesandcommunity-nodesintheaffiliationgraph ferent algorithms and heuristics. For example, separate methods
representmembershipsofnodestocommunities.However,ourap- have been proposed for detecting communities in undirected net-
proachhasasimplebutcriticalinnovation:whilemembershipsof works that are disjoint [4, 13, 21, 38, 40], overlapping [3, 6, 11,
nodestocommunitieshavepreviouslybeenmodeledasundirected, 34,43],orhierarchicallynested[2,44].Ontheotherhand,detec-
wemodelthemembershipsasdirected. tion of 2-mode communities has been much less researched. An
Thoughsimpleonthesurface,thismodificationleadstosubstan- exceptionhereisTrawling[23],whichisamethodforextracting
tialchangesinthemodelingcapabilityofaffiliationnetworkmod- 2-modecommunitiesinlargedirectednetworks.Thecriticaldiffer-
els.Inparticular,adirectedaffiliationbetweenanodeandacom- encewithourworkhereisthatTrawlingonlyidentifiescomplete
munitymodelswhetherthenodesendsorreceives(orboth)links bipartite subgraphs of a given directed network. In contrast, our
toothermembersofthecommunity.Directedaffiliationsallowus methodisabletoidentifycohesiveaswellasbipartitecommuni-
tosimultaneouslymodelcohesiveaswellas2-modecommunities. tiesindirectedaswellasundirectednetworks.
Incohesivecommunitiesnodeaffiliationsarebidirectional(anode
Conceptually CoDA is related to existing work on block mod-
els,whichareinprinciplecapableofdetectingcohesiveaswellas
2-mode communities [3,16,18].Our workdiffersfromsuchap-
proaches intermsofhow communitiesoverlapandarehierarchi-
callynested.WealsoemphasizethescalabilityofCoDAcompared
totheseapproaches.
CoDA is an example of an affiliation network model [25, 43,
(a) Nodecommunityaffiliations (b) NetworkG
45,47].Whileexistingaffiliationnetworkmodelscanonlymodel
undirectedcohesivecommunities,thecrucialdifferencehereisour
Figure 2: (a) Directed node community affiliation graph.
abilitytomodeldirectednetworksand2-modecommunities.
Squares: communities, Circles: nodes of network G. Affilia-
Therestofthepaper isorganizedasfollows.Section2defines
tions from nodes to communities indicate that nodes create
theaffiliationnetworkmodelandSection3discussesthemodelfit-
edgestoothermembersinthosecommunities,whileaffiliations
tingprocedure.WepresentexperimentalresultsinSections4and5,
from communities to nodes indicate that nodes receive edges
andconcludeinSection6.
from others. Community A is cohesive, while B is a 2-mode
community.(b)NetworkGcorrespondingtomodelin(a).
2. DIRECTED COMMUNITY
AFFILIATIONS
members of the community, or both. Therefore, we assume that
Westartbypresentingastochasticgenerativemodelofnetworks nodesindirectednetworkscanhavetwo“types”ofcommunityaf-
in which the probability of an edge appearing between a pair of filiation:“Outgoing”affiliationsfromnodestocommunitiesmean
nodes depends on the community affiliations of thise nodes. We thatinthenetworkthenodesendsedgestoothermembersofthe
thendevelopanefficientmodelfittingprocedurewhichallowsfor community.And,“incoming”affiliationsfromcommunitiestonodes
detectingcommunityaffiliationsofnodesinagivennetwork. meanthatnodesreceiveedgesfromothercommunitymembers.We
Wedescribeourmodelinthecontextofdirectednetworksand modelthisusingdirectedmembershipsbetweennodesandcommu-
then show how it can straightforwardly be adapted to undirected nities:outgoingmembershipsandincomingmemberships.
networks.OurmodelbuildsuponBigCLAM,anaffiliationmodel Formally,wedenoteabipartiteaffiliationgraphasB(V,C,M),
foroverlappingnetworkcommunities[45].However,whereasBig- whereV isthesetofnodesoftheunderlyingnetworkG,Ctheset
CLAMfocusesonfindingonlycohesivecommunitiesinundirected ofcommunities,andMasetofdirectededgesconnectingnodesV
networks,ourworkhereaimstofind2-modecommunitiesaswell andcommunitiesC.Anoutgoing membershipedge ofnodeu
ascohesivecommunitiesinbothdirectedandundirectednetworks. V to community c C is denoted as (u,c) M and, and a∈n
∈ ∈
DirectedAffiliationNetworkModel.Webeginwiththeintuition incomingmembershipisdenotedas(c,u) M.
∈
thatadesirablemodelofcommunitiesindirectednetworksshould Now,giventheaffiliationgraphB(V,C,M),weneedtospecify
exhibit twoproperties. First,communitiesshouldbemodelednot aprocessthatgeneratestheedgesEoftheunderlyingdirectednet-
onlyintermsoftheirinternalconnectivity, butalsointermshow workG(V,E).Tothisendweconsiderasimpleparameterization
membersconnect tonon-members. Second, themodelshould ac- whereweassignasingleparameterpctoeverycommunityc C.
∈
countforasymmetries,i.e.,directedness,ofedgesbetweennodes. Theparameterpcmodelstheprobabilityofadirectededgeforming
We later demonstrate that accounting for these two properties is fromamembernodeuwithanoutgoingmembershiptocommunity
important.Perhapssurprisingly,ourmethodgivesimprovedperfor- ctoanothermembervofcwithanincomingmembership.Inother
manceevenwhenmodelingcommunitiesinundirected networks. words, wegenerateadirectededge betweenapairof nodes with
Thisis due tothe fact that when edge directions arenot explicit, probabilitypciftheyareconnectedinBwitha2-stepdirectedpath
relationships inthenetwork maystillbe(implicitly) asymmetric, viacommunityc.Eachcommunityccreatesedgesindependently.
andidentifyingsuchasymmetriesleadstoimprovedperformance. However,iftwonodesareconnectedbymorethanonecommunity,
We proceed by formulating a simple conceptual model of net- duplicateedgesarenotincludedinthegraphG(V,E).
worksthatwerefertoasaDirectedAffiliationNetworkModel.Our
workbuildsonafamilyofaffiliationnetworkmodels[8],however, DEFINITION1 (DIRECTEDAFFILIATIONNETWORKMODEL).
existing affiliation models are typically designed to handle cohe- LetB(V,C,M)beadirectedbipartitegraphwhereV isasetof
sivecommunitiesinundirectednetworks[25,43,45,47];herewe nodes,C isasetofcommunities,andM isasetofdirectededges
extendsuchmodelsinordertocapturecohesiveaswellas2-mode between V and C. Also,let pc bea set of probabilitiesfor all
{ }
communitiesindirectedaswellasundirectednetworks. c C. Given B(V,C,M) and pc , the model generates a di-
∈ { }
To represent node community memberships, we consider a bi- rectedgraphG(V,E)bycreatingadirectededge(u,v)fromnode
partite affiliation graph where the nodes of the network (bottom u V tonodev V withprobabilityp(u,v):
∈ ∈
layer) connect to communities (top layer) to which they belong
(Figure2(a)).Edgesoftheunderlyingnetwork(Figure2(b))then p(u,v)=1− Y (1−pk), (1)
ariseduetosharedcommunityaffiliationsofnodes. k∈Cuv
Considerforamomentanundirectednetwork;whenanodebe- whereCuv C isasetofcommunitiesthroughwhichuhasa2-
longstoacommunityinsuchanetworkittypicallymeansthatthe stepdirected⊂pathtov(Cuv = c(u,c),(c,v) M ).IfCuv =
node has(undirected) edges toother membersof thecommunity. ∅thenwesetp(u,v)=1/V .{ | ∈ }
| |
Thistypeofcommunityaffiliationcanbemodeledusingabipar-
titegraphofnodesandcommunitieswhereundirectedaffiliations OurDirectedAffiliationNetworkModelandtheunderlyinggen-
areformedbetweennodesandcommunities[25,43,45,47]. eratednetworkareillustratedinFigure3.Directedaffiliationsare
Indirectednetworks,however,weneedarichernotionofcom- abletoexplaintheoverlappingnatureofcohesiveaswellas2-mode
munityaffiliation(Figure2(a)):anodemaycreateedgestoother communities. For example, imagine a Twitter network among a
members of a community, and it also receive edges from other community of music fans (A), a community of movie fans (B),
3. COMMUNITY DETECTION
Given an unlabeled, directed network G(V,E), our goal is to
identifycohesiveaswellas2-modecommunities.Weachievethis
byfittingourDirectedAffiliationNetworkModeltoG(V,E),i.e.,
byfinding anaffiliationgraph B andparameters pc thatmaxi-
{ }
mizethedatalikelihood. Fornow, weassumethatthenumber of
communitiesKisgiven;wewilllaterdiscusshowtoautomatically
determineK.WeaimtosolvethefollowingMaximumLikelihood
Estimationproblem:
argmax X logp(u,v)+ X log(1 p(u,v)), (2)
−
P,{pc} (u,v)∈E (u,v)6∈E
wheretheedgeprobabilityp(u,v)isdefinedinEq.1.
Eq.2leadstoachallengingoptimizationproblem.Specifically,it
involvesacombinatorialsearchoverallpossibleaffiliationgraphs
B[43].Therefore,wedevelopanapproximatealgorithmforopti-
mizingEq.2.Weachivethisbyrelaxingtheoriginalproblemby
changingbinarymembershipsintoreal-valuedmemberships.
WebuildontheintuitionfromtheBigCLAM[45]optimization
procedureandbeginbyintroducingvariablestorepresentthemem-
berships ofthenodes. Asnotedearlier,wedistinguishnodes’ in-
Figure3:Affiliationgraph(top)oftheDirectedAffilationNet-
comingmembershipsandoutgoingmemberships.Inparticular,let
workModelthatcorrespondstothenetworkadjacencymatrix
(bottom).Itcontainstwooverlappingcohesive(A,B)andtwo Mucindicatewhetherthenodeubelongstocommunitycwithan
overlapping 2-mode (C, D) communities. Black edges in the outgoingmembership,andLvcindicatewhethernodevhasanin-
comingmembershipforc.NowEq.1canberepresentedas:
affiliationgraphdenotebidirectionalcommunitymemberships
andrededgesdenoteunidirectionalmemberships. p(u,v)=1 Y (1 pc)=1 Y(1 pc)MucLvc,
− − − −
c∈Cuv c
agroupoffamoussingers(C),andagroupoffamousactors(D).
By applying the change of variables 1 pc = exp( αc) with
Members incommunities A and B build bi-directional social re- − −
αc 0,theequationbecomeslinearinM,L,andαc:
lationships inside their respective communities. Somenodes may ≥
belongtobothcommunitiesAandBastheyareinterestedinboth p(u,v)=1 exp( XMucαcLvc).
− −
moviesandmusic.Asforone-directionalrelationships,wecaneas- c
ily see that music fans would follow singers (C) and movie fans
wouldfollowactors(D).Together,theserelationswouldformthe WethenfurthersimplifytheequationbylettingM˜uc =√αcMuc
adjacencymatrixatthebottomofFigure3.Ourmodelcapturesthis andL˜vc =√αcLvc.
cmoumnpitlyexafcfiolmiamtiounnigtyraspthruacbtuorveevtehreyandajtaucraelnlcyy,amsashtroixw,nwihnetrheegcroemen- p(u,v)=1−exp(−XM˜ucL˜vc).
c
nodesrepresentmusicfans(A),bluenodesaremoviefans(B),red
nodesarefansofbothmoviesandmusic,ivorynodesaresingers So far, we have not used any approximations and the problem
(C), and purple nodes areactors (D). Affiliationsbetween nodes isstillcombinatorialsincethevariablesremainrestricted:M˜uc
andcohesivecommunitiesAandBflowinbothdirectionsbecause √αc,0 andL˜vc √αc,0 . ∈
{ } ∈{ }
membersofthosecommunitieshavereciprocalrelationshipswith However,notethatwecaninterpretM˜uc asthestrengthofthe
eachother,whereasfansandcelebritiesbelongingto2-modecom- membershipofnodeutocommunityc.ThustheconditionM˜uc
∈
munitiesC andDhaveedgesflowinginonlyonedirection(fans √αc,0 simply means that if node u belongs to c, it would be
{ }
followcelebrities,celebritiesarefollowedbyfans). connectedtoothermembernodesincwiththefactor√αc,which
More generally, our model has two important advantages over determinespc.ThesameargumentalsoappliestoL˜vc.
existingapproaches[25,43,45,47]:First,CoDAcanmodelnatural NowwereplaceM˜ucandL˜vcwithnonnegativecontinuousval-
overlapsbetweencommunities.Ithasbeenshownthatcommunity uedmemberships Fuc andHvc, respectively. Theadvantage here
affiliationmodelsforundirectednetworks[43]canmodelcommu- isthatnoweachnodecanpickthe“strength”ofitsmembershipto
nity overlaps accurately, which traditional models of overlapping agivencommunity:AhighvalueofFucmeansthatthenodeuhas
communities fail to capture [2, 3, 34]. The model also captures manyoutgoingedgestowardsothermembersofc,whilehighHvc
realistic community overlaps because its modeling power gener- meansthatnodevhasmanyincomingedgesfromothermembers
alizes that of other community affiliation models for undirected ofc.Nowwecawrite:
networks, i.e.,CoDA can model overlaps betweencohesive com-
munities in addition to 2-mode communities. The second advan- p(u,v)=1−exp(−FuHvT).
tageofourmodelisitsabilitytomodel2-modecommunities.By AndwetransformedEq.2intoacontinuousoptimizationproblem:
modelingsuchcommunities,wecanbettercapturetheinteraction
Fˆ,Hˆ =argmaxl(F,H) (3)
between groups of nodes. Thisis asignificant improvement over { }
F,H≥0
currentmethodsthatmodelonlyinteractionswithincommunities.
where
l(F,H)= X log(1−exp(−FuHvT))− X FuHvT.
(u,v)∈E (u,v)6∈E
Inotherwords,inordertodetectnetworkcommunitieswefitour alocallyminimalneighborhoodk,weinitializeFu′k =1ifu′has
modelbyestimatingnon-negativeaffiliationmatricesFˆ,Hˆ ∈RN×K anoutgoingedge(orFu′k =0otherwise),andsetHu′k =1ifu′
thatmaximizethelikelihoodl(F,H)=logP(G|F,H). hasanincomingedge(orHu′k =0otherwise).
Solvingtheoptimizationproblem.TosolvetheprobleminEq.3, Choosing the number of communities. To automatically deter-
we adopt a block coordinate ascent approach: We update Fu for minethenumberofcommunitiesK,wefollowtheapproachpro-
eachuwithH fixedandupdateHv foreachv withF fixed,i.e., posedin[3].Wedivideallnodepairsinto80%trainingand20%
weupdateeitherincomingoroutgoingmembershipsofonenode testset.VaryingK,wefitCoDAwithKcommunitiesonthetrain-
whilefixingtheothertypeofmemberships.Thisapproachhasthe ing pairs and measure the likelihood for the test pairs. We then
advantagethateachsubproblemofupdatingFuandHvisconvex. selectK withthehighesttestsetlikelihood.Forasmallnetworks
For brevity we describe only how to update Fu. Updating Hv is withfewerthan100edges,wefindthatadifferentcriterionworks
analogous.Foreachuwesolve: betterinpractice.HerewechooseK soastoachievethesmallest
valueoftheBayesianInformationCriterion:
argmaxl(Fu), (4)
Fuc≥0 BIC(K)= 2l(Fˆ,Hˆ)+NKlog E .
− | |
where
Parallelization and implementation details. Our approach also
l(Fu)= X log(1−exp(−FuHvT))− X FuHvT, naturallyallowsforparallelization,whichfurtherincreasesscala-
v∈N(u) v6∈N(u) bility of CoDA. When updating Fu for each node u (Eq. 4), we
observethateachsubproblemisseparablesinceallothervariables
where (u)isasetofneighborsofu.Tosolvethisconvexprob-
N inEq.4(H)remainfixed.Thatis,updatingthevalueofFu fora
lem,weuseprojectedgradientascentwiththefollowinggradient:
specificnodeudoesnotaffectupdatesofFv forallothernodesv.
∇l(Fu)=v∈XN(u)Hv1−exepx(p−(F−uFHuvHT)vT) −v6∈XN(u)Hv onInfodtthheesemipnaeprtahalrolaedll.liezUle.pdTdhvaitesirnspgiaorHnaluoleffloiCzraoetDiaocAnh,dnwooeedsesnouoltvceaafnfEebqce.tp4thaefroafirllnemalliuzrleetdispuillnet
thesameway.AsweshowinSection4,parallelizationonasingle
Wecomputethestepsizeusingbacktrackinglinesearch.Aftereach
sharedmemorymachinebooststhespeedofCoDAbyafactorof
update,weprojectFuintoaspaceofnonnegativevectorsbysetting
20 (the number of threads) used when analyzing a 300,000 node
Fuc =max(Fuc,0).
network.Last,wealsoexperimentedwithotheroptimizationtech-
Naivecomputationof l(Fu)takestimeO(V ).However,we
∇ | | niques such as the cyclic coordinate descent method (CCD) [19]
reducethecomputationalcomplexitytothedegreeofu,O( (u)),
which significantly increases the scalability of our appro|aNch. W|e whichoptimizesFuc foreachuandeachcbyNewton’smethod,
butwefoundthatblockcoordinateascentconvergesthefastest.
achievethisbycomputingthesecondtermPv6∈N(u)HvinO(|N(u)|) AparallelC++implementationofCoDAispubliclyavailableat
bystoring/cachingPvHv: http://snap.stanford.edu.
X Hv =(XHv Hu X Hv). CoDAforundirectednetworks.Sofar,wehavediscussedCoDA
− −
v6∈N(u) v v∈N(u) underthecontextofdirectednetworks.However,CoDAcaneasily
beappliedtoundirectednetworksaswell.Wemakeasimpleob-
Given that real-world networks are extremely sparse ( (u)
servation:undirectednetworksmodelsymmetricrelationshipsand
|N | ≪
N),wecanupdateFu forasinglenodeuinnear-constant time. thus anundirected relationshipisequivalent totwodirected rela-
The update rule for Hv can be similarly derived and takes near- tionships, one each way. Thus, given an undirected network, we
constanttimeO( (v)).Inpractice,weiterativelyupdateFu,Hu simplyconvertthenetworkintoadirectedonebyregardingevery
|N |
foreachuandstopiteratingoncethelikelihooddoesnotincrease
edgeasreciprocal,andthenapplyCoDAtodetectcommunities.
(by0.01%)afterweupdateFu,Huforallu. Now,CoDAwilleasilydetectcohesivecommunitiesinthiscon-
Determiningcommunityaffiliationsofnodes.Fromthereal-val- verted network as edges in cohesive communities are reciprocal.
uedFˆ,Hˆ thatweestimate,wewanttodetermine“hard”commu- Detecting 2-mode communities is also simple. Consider the case
nityaffiliationsofnodes.WeachievethisbythresholdingFucand where we are given an undirected 2-mode community X where
Huc withaconstantδ,i.e.,weregarduhasanoutgoingmember- nodes in group A are connected to nodes in group B. Once we
ship to community c if Fuc δ, and an incoming membership convert X intoadirectednetworkwithreciprocal edges between
≥
fromcifHuc δ. AandB,CoDAwillestimatetwo2-modecommunitiesfromthis
Wechooset≥hevalueofδsothateverypairofmembersincom- communityX:Xˆ foredgesfromAtoB,andXˆ foredgesfrom
1 2
munity c has edge probability higher than the background edge BtoA.Thus,CoDAisabletocorrectlydiscoverX,withthecaveat
probability1/V (seeEq.1): thatitdiscoversittwice(bothXˆ andXˆ correspondtoX).
| | 1 2
1
1 exp( δ2) 4. EXPERIMENTS
V ≤ − −
| |
WeevaluatetheperformanceofCoDAandcompareittostate-
Thisinequalityleadstoδ=p log(1 1/V ).Wenotethatwe of-the-artcommunitydetectionmethodsonarangeofdirectedas
− − | |
alsoexperimentedwithothervaluesofδandfoundthatthischoice well as undirected networks. We measure the quality of commu-
forδworkswellinpractice. nitydetectionbycomputingthedetectionaccuracybasedongold-
Algorithm initialization. To initialize F,H, we employ locally standardground-truthcommunities.Wealsoevaluatethescalabil-
minimalneighborhoods,whichprovidegoodseed-setsforcommu- ityofthemethodsbymeasuringruntimeasnetworksizeincreases.
nity discovery [15]. A neighborhood N(u) of a node u is a set
4.1 DatasetDescription
consistingofthenodeuanditsneighbors,andN(u)issaidtobe
“locallyminimal”ifN(u)haslowerconductancescorethanN(v) Webeginbybrieflydescribingthenetworksthatweconsiderin
foranyotherneighborvofu[15].Foranodeu′belongingtosuch this study. Overall, we consider 5 undirected and 9 directed net-
F1score Jaccardsimilarity
Method Google+ Twitter Facebook Google+ Twitter Facebook Average
MMSB[3] 0.324(0.033) 0.262(0.005) 0.374(0.042) 0.214(0.026) 0.169(0.004) 0.266(0.036) 0.268
Cliquepercolation[34] 0.331(0.036) 0.246(0.006) 0.429(0.051) 0.240(0.032) 0.163(0.005) 0.342(0.050) 0.292
Linkclustering[2] 0.304(0.016) 0.334(0.003) 0.372(0.027) 0.226(0.016) 0.238(0.003)2 0.275(0.024) 0.291
BigCLAM[43] 0.324(0.017) 0.344(0.005) 0.442(0.042) 0.217(0.014) 0.234(0.004) 0.325(0.038) 0.315
DEMON[11] 0.343(0.029) 0.308(0.005) 0.418(0.046) 0.255(0.027) 0.210(0.005) 0.311(0.041) 0.307
NMF[29] 0.333(0.019) 0.318(0.004) 0.406(0.038) 0.242(0.026) 0.221(0.004) 0.301(0.050) 0.303
CoDA,undirected 0.414(0.027)1 0.348(0.005)2 0.470(0.042)1 0.314(0.026)1 0.237(0.004) 0.357(0.039)1 0.3572
CoDA,directed 0.406(0.025)2 0.363(0.005)1 0.470(0.042)1 0.314(0.024)1 0.250(0.004)1 0.357(0.039)1 0.3601
Table2:PerformanceonFacebook,Google+,andTwitter.Higherisbetter.Standarderrorsareshowninparentheses.Thebestand
secondbestmethodsareannotatedas‘1’and‘2’.
Dataset Directed N E C S A Last,wenotethattheabovebaselinesrepresentthecurrentstate-
Google+ ✓ 250,469 30,230,905 437 143.51 0.25 of-the-art in community detection. However, we also considered
Twitter ✓ 125,120 2,248,406 3,140 15.54 0.39 other baselines, including those that make use of node features
Facebook ✗ 4,089 170,174 193 28,76 1.36 [20],networktopology[39],orboth[5,31];howeverexperiments
Enron ✓ 45,266 185,172 4,572 63.93 6.46 demonstratethatnoneofthesealternativesoutperformsCoDA.
LiveJournal ✗ 3,997,962 34,681,189 287,512 22.31 1.59
Youtube ✗ 1,134,890 2,987,624 8,385 13.50 0.10 Evaluation.Toevaluatetheperformanceoftheabovemethodswe
quantify the degree of correspondence between the ground-truth
Table 1: Dataset statistics. Directed: Yes/no, N: number of and the detected communities. To compare a set of ground-truth
nodes,E:numberofedges,C:numberofground-truthcom- communities C∗ to a set of predicted communities C, we adopt
munities,S:averageground-truthcommunitysize,A:ground- an evaluation procedure previously used in [43, 45],where every
truthcommunitymembershipspernode.Furtherdatasetsused detected(ground-truth)communityismatchedwithitsmostsimilar
inthisstudyaredescribedinTable5. ground-truth(detected)counterpartcommunity:
1 1
wmournkiscaftrioomn,ainwfoidrmeasptieocnt,rubmioloofgdicoamlaaninds.ecWoelocgoicnaslidneertwsoocrikasl,.1com- 2|C∗|Ci∗X∈C∗Cmja∈xCδ(Ci∗,Cj)+ 2|C|CXj∈CCmi∗∈aCx∗δ(Ci∗,Cj),
Networkswithground-truthcommunities.Fortheexperiments where δ(Ci∗,Cj) is some measure of the similarity between the
in this section, we consider a subset of 6 publicly available net- communities Ci∗ and Cj. We consider two standard measures of
workswherewehaveexplicitground-truthmembershipsofnodes the similarity between sets, namely the F score and the Jaccard
1
to communities [44]. The availability of ground-truth allows us similarity.Thus,weobtainavaluebetween0and1,where1indi-
toquantifythequalityofcommunitydetectionmethodsquantita- catesperfectrecovery.
tively.Table1showsthestatisticsofthenetworksandtheground-
4.3 Detecting Social Circles
truth communities. The networks come from three different do-
mains:Thefirstthreenetworksarethecollectionofego-networks First we consider the problem of discovering users’ social cir-
fromonlinesocialnetworksofFacebook,TwitterandGoogle+[31], cles[31].Circles(or‘lists’inFacebookandTwitter)giveusersa
theEnronemailcommunicationnetwork[22],andLiveJournaland meansofcategorizingtheirimmediateneighbors,orinthecaseof
Youtube social networks [33]. Wedescribe thenature of ground- directednetworks,theuserswhomtheyfollow.Thustheproblem
truthcommunitiesineachofthesedatasetsinmoredetaillater. ofautomaticallyidentifyingusers’socialcirclescanbeposedasa
communitydetectionproblemoneachuser’sego-network[31].
4.2 Experimental Setup
InTable2weevaluatetheperformanceofCoDAandbaselines
onsocialcircledetection.Acrossallthreedatasetsandbotheval-
Baselines.Forcomparisonweconsiderthefollowingbaselinemet-
uationmetrics,CoDA(thelastrow)isthebestorsecond-bestper-
hods: MMSB (Mixed Membership Stochastic Blockmodels) [3],
former. On average, CoDA outperforms MMSB by 34%, Clique
whichcandetectbothcohesiveand2-modecommunitiesinundi-
percolationby23%,Linkclusteringby24%,BigCLAMby14%,
rected networks and is extremely slow; Clique Percolation, [34]
DEMONby17%,andNMFby19%.
Link Clustering [2], BigCLAM [43, 45] are state of the art over-
The 3 data sets possess very different reasons for community
lapping cohesive community detection techniques for undirected
(i.e., social circle) formation: Facebook is an undirected network
networks; DEMON [11] is a scalable local community detection
and in Facebook circles are driven by dense mutual friendships
methodfordirectednetworks;NMF[29]isastate-of-the-artnon-
among users with homogeneous backgrounds [31]; therefore, we
negative matrix factorization approach which can be used for di-
wouldexpectcohesivecommunitiesinFacebook.Google+andTwit-
rectednetworks.Weusepubliclyavailableimplementationsofeach
ter are directed networks and as such circles are not necessarily
ofthemethods.
based on friendship, because edges in these networks denote fol-
Some methods require input parameters. MMSB and NMF re-
lowerrelationships:Thefractionofreciprocatededgesisonly29%
quiresthenumberofcommunitiesK.WeusetheBayesinforma-
inGoogle+and54%inTwitter.Forexample,asocialcircleinTwit-
tioncriterionsuggestedbytheauthors[3]tochooseK.DEMON
termightconsistofauthorswhopublishinthesamegenre,orcan-
requires ε, the threshold value for merging two communities. As
didatesinthesameelection.AswewillseelaterinSection5,many
thereexistsnostandardcriterionfor ε,wesetε sothat DEMON
socialcirclesinGoogle+andTwitterfollowsuch2-modestructure.
detectsthesamenumberofcommunitiesasCoDAdoes.
Regardlessofverydifferentnatureofthedatasets,CoDAisthe
1WeusethepubliclyavailabledatafromtheStanfordLargeNet- best performing method in each of them. This result means that
workCollection:http://snap.stanford.edu. CoDA recovers 2-mode circles in Google+ or Twitter as well as
Method F score Jaccardsimilarity Average RelativeF score AbsoluteF score
1 1 1
Method LiveJournal Youtube LiveJournal Youtube
MMSB N/A N/A N/A
Cliquepercolation N/A N/A N/A Metis 100% 200% 0.12 0.028
Linkclustering 0.195 0.294 0.245 Graclus 100% 185.7% 0.12 0.026
BigCLAM 0.478 0.358 0.418 BigCLAM 121.0% 278.1% 0.14 0.039
DEMON 0.464 0.350 0.407 DEMON N/A 100% N/A 0.014
NMF N/A N/A N/A CoDA 129.4% 307.1% 0.15 0.043
CoDA,undirected 0.538 0.431 0.485
CoDA,directed 0.617 0.516 0.567
Table 4: Relative accuracy (compared to the worst perform-
ingmethod)ofdetectedcommunitiesonlargescalesocialnet-
Table3:PerformanceofrecipientdiscoveryontheEnronnet-
works.
work.Algorithmsthatdonotscaletothesizeofthedatasetare
labeledas“N/A”.
10000
cohesivecirclesinFacebook,i.e.,CoDAcandetectbothkindsof 8000
communitiesmoreaccuratelythanthebaselines.
Directedvs.undirectednetworks.Tofurtherexaminetheperfor- ec.) 6000
S
manaenxcpeeoruimtmenetthwoidthotnhedigroeaclteodfaunnddeurnsdtairnedcitnegdwnehtewthoerrksCwoDeApeirsfosrtimll me ( 4000 CPM
Ti LC
able to recover 2-mode communities even when edge directions DEMON
are dropped and networks are considered as undirected. To test 2000 CoDA
BigCLAM
this,weconvertthedirectednetworksofTwitterandGoogle+into CoDA-24
undirectedbyremovingtheedgedirections.ThenweapplyCoDA 0
0 100 200 300
(CoDA,undirected,thesecondtolastrowinTable2).Surprisingly,
Number of nodes (× 103)
CoDAachievessimilarperformanceevenwithoutexplicitedgedi-
rectionsinthenetwork.BasedonthisevidenceweconcludeCoDA
iscapableofaccuratelyfinding2-modecommunitieseveninundi- Figure4:Algorithmruntime.
rectednetworks.
Graclus[13].For allmethods weset thenumber of communities
4.4 Discovering Recipient Lists in Email Net-
Ktobethenumberofground-truthcommunities.
works
Table4showstheresults.Forthisexperiment wefocusonthe
Wealsodefineataskofautomaticallydiscoveringrecipientlists scorerelativetothatoftheworst-performingbaselineineachnet-
inthetheemailcommunicationnetwork.Theideaisthatsuchlists work(sothattheworst-performingbaselinehasascoreof100%).
exhibitadistinctstructuralpatterninthenetworkastherecipient Wecomputetherelativescorebecausethenetworksareonlypar-
listsmayhave2-modecommunitystructureasasetofuserswho tially labeled and the overall performance is thus artificially low
receivethesameemailmaynotnecessarilyemaileachother[32]. (asmethodsdiscover manyunlabeled communities).Wefindthat
We consider all Enron emails [22] with 20 or more recipients. CoDAoutperformsitsnearestcompetitorby8.4%onLiveJournal
This gives us a set of 4,572 unique recipient lists in the Enron and29%onYoutube.
dataset,whichwetreatasground-truthcommunities(Table1).Now
we are given an unlabeled directed Enron email communication 4.6 Scalability
network,whereanedgei j meansthatisentatleastonemail
→ Last,weevaluatethescalabilityofCoDAbymeasuringitsrun-
toj,andthegoalistodiscoveremailrecipientlists.
ningtimeonsyntheticnetworkswithincreasingsize.Wegenerate
We then apply CoDA as well as the baselines to this network
synthetic networks using the Forest fire model [27] with the for-
and in Table 3 we measure how accurately the communities de-
wardandbackwardprobabilitiessetto0.36and0.32,respectively.
tected byCoDA correspond totheseground-truth emailrecipient
SinceCoDAiseasilyparallelizableasdescribedinSection3,we
lists.WereportboththeF scoreandJaccardsimilarity(formeth-
1 alsoconsiderasingemachineparallelimplementationrunningwith
odsthatdonotscaletonetworksofthissize,wereportN/A).Ta-
24threads(CoDA-24).
ble3showsthatCoDAoutperformsothermethodsbyasignificant
Scalability results are shown in Figure 4. Link Clustering and
margin.CoDAoutperformsLinkclusteringby131%,DEMONby
Clique Percolation scale to networks of at most a few thousand
39%,andBigCLAMby36%.
nodes.DEMONisafastandscalableoverlappingcommunityde-
tection method. DMEON tends to be faster than CoDA (single-
4.5 Experiments onLargeNetworks
threadedimplementation)fornetworksupto100,000nodes,how-
Last,wealsoexaminetworeal-worldsocialnetworkswithmil- ever,oncethenetworkbecomeslarger,CoDAbecomesmuchfaster.
lionsof nodes inwhichnodes explicitlydeclaretheircommunity Whencomparingsingle-threadedimplementationswealsonote
memberships[44].WeconsidertheLiveJournalandYoutubesocial thatBigCLAMisthefastestmethodinourexperiments.However,
networks,andregarduser-createdgroupsasground-truthcommu- we note that CoDA takes only 30% more time than BigCLAM
nities.Weignoregroupscontainingfewerthan10nodes,yielding while it isalso solving a more complicated problem, namely de-
71,093communitiesinLiveJournaland2,078inYoutube. tectingcohesiveaswellas2-modecommunities.
Of the baselines previously mentioned, only BigCLAM could Last,wealsomeasureaparallelizedversionof CoDA(CoDA-
scale to both networks and DEMON could scale to the Youtube 24).Using24threadsonasinglemachine,weachievenearly24x
network. Therefore,wealsoconsider twolarge-scalegraphparti- speedup.Ultimately,CoDAtakesjust6minutestoprocessa300,000
tioning methods as baselines for this experiment: Metis [21] and nodenetwork.
Dataset Directed N E C S A C bluefish
PPI-Y2H ✗ 1,647 2,518 40 90.75 2.20 striped bass
PPI-LC ✗ 1,213 2,556 40 42.08 1.39
web-Stanford ✓ 281k 2,312k 19k 70.63 4.59 summer flounder
web-Google ✓ 875k 5,105k 39k 41.79 1.86
cit-HepTh ✓ 27k 353k 2,000 70.00 5.04
cit-HepPh ✓ 34k 422k 4,976 51.52 7.42
FloridaBay ✓ 121 1,745 6 45.33 2.25
Chesapeake ✓ 33 72 5 9.20 1.39
Table 5: Dataset statistics. Directed: Whether the network is
directedornot,N:numberofnodes,E:numberofedges,C: bay anchovy A B macoma spp.
number of detected communities, S: average size of detected
communities,A:communitymembershipspernode. mya arenaria
Figure 6: Examples of overlapping 2-Mode communities de-
tectedbyourmethodintheChesapeakebayfoodwebnetwork.
Seemaintextfortheexplanationofcommunitystructure.
prey relationships among fish: small fish (A) areeaten by bigger
fish (C). CoDA also discovers the overlap between two predator
groupswherewhiteperchandspotpreyonbothfishandclams.
CoDA also allows us to gain insights into biological PPI net-
works. Interestingly, CoDA discovers many2-mode communities
Figure5:TwodetectedcommunitiesinaFoodweb(Chesapeake
in the undirected protein-protein interaction network determined
Bay).Amongothercommunities,CoDAidentifiessetsofnodes
byyeast two-hybridscreening (PPI-Y2H).Forexample, Figure8
with similar predators (A, blue nodes) and with similar prey
displaystheinducedsubgraphoftwocommunitiesthatCoDAde-
(B,rednodes),bothofwhichhavelowinternalconnectivity.
tects. 2-mode communities detected by CoDA clearly reveal the
5. COMMUNITYDISCOVERY interactionbetweendifferentproteingroups.Forexample,proteins
ingroupC ofFigure8heavilyinteractwithproteinsingroupA,
SofarwehavedemonstratedthatCoDAcanreliablydetectboth even though these proteinsdo not interact withinthesamegroup
cohesiveand2-modecommunitiesindirectedaswellasundirected (withAorwithinC).
networks. In the following section, we shall demonstrate that 2- Tofurtheranalyzetheroleofthesecommunities,weusedgene
mode communities take an important role in networks. We shall
ontologiestoidentifyrelevantterms/functionsofproteinsinA,B,
use CoDA to perform a qualitative study of various networks in C,andDusingtheGOTermFinder[7].Theproteinsinthelarge
ordertodeterminetheextenttowhichcommunitystructuresvary groups(C,D)aregenerallyassociatedwithcatalyticactivityand
acrossreal-worldnetworksfromvariousdomains. ionbinding(p-value 10−4).
∼
Network data.Inadditiontothedatasetsalreadyintroduced, we However,theseproteinsareregulatedbydifferentproteingroups
alsoanalyzebiologicalnetworks,foodwebs,webgraphs,andcita- (A,B)whichhavedifferentfunctions.ProteinsinA(e.g.,YLR347C
tionnetworks(Table5).Forbiologicalnetworks,weconsider the and YNL189W) are protein transporters, whereas proteins in B
protein-protein interaction network of Saccharomyces cerevisiae: (e.g.,YLR291C)areregulators.Perhapsmoreinterestingly,YPL070W
yeast two-hybrid (PPI-Y2H) and literature-curated (PPI-LC) [2]. belongstobothAandBandregulatesbothCandD.However,its
WealsoconsidertheChesapeakeandFloridaBayfoodwebs[41], role is not yet known. But based on known functions of proteins
the web graph of Stanford University web pages (web-Stanford), ingroupsAandB wecanextrapolatethefunctionofYPL070W.
thewebgraphreleasedbyGooglein2002(web-Google)[28],and Thisexampleshowshownetworkanalysisandcommunitydetec-
thearXivcitationnetworksfromhigh-energyphysicsphenomenol- tioninparticularcanprovideresearchdirectionsforexperimental
ogy (cit-HepPh) and theory (cit-HepTh) [27] all available from biology[10].
http://snap.stanford.edu.
5.1 Biologicaland FoodwebCommunities 5.2 2-mode vs.CohesiveCommunities
Wefirstpresent2-modecommunitiesinfoodwebs,wherenodes SinceCoDAcandetectbothcohesiveand2-modecommunities,
representorganismsandanedgefromanodeutovmeansthatu wecanuseittomeasuretheextenttowhichrealnetworkdataex-
ispreyeduponbyv.WeapplyCoDAontheChesapeakeBayfood- hibits cohesive and 2-mode behavior. This analysis allows us to
webnetworkshowninFigure5,anddisplayaninducedsubgraph characterizethemesoscalestructureofreal-worldnetworksasthe
ofdetected2-modecommunitiesinFigure6. proportionof2-modeversuscohesivecommunitiescanbeusedto
Infoodwebnetworks,wefind2-modecommunitiesofgroupsof gainfurtherinsightsintocommunitystructureofnetworks.
predatorswhorelyonsimilargroupsofprey(Figure6).Theblue Experimentalsetup.Forthisexperiment,weconsider12networks
2-modecommunity(B-D)representspredatorsandpreyinthethe from6domainsinordertocharacterizetheirdifferentcommunity
ChesapeakeBaysands:nereis,macomaspp.,andmyaarenaria(in structures. We consider ego networks (Twitter, Google+) and so-
B) are small, sand-dwelling clams and worms that are fedon by cialnetworks(LiveJournal,Youtube)fromSection4.Wealsoin-
fish(inD).Alternately,theredcommunity(A-C)showspredator- clude 8 networks from 4 different domains: Biological networks,
fraction of cohesive versus 2-mode communities
n circles biological social web citations foodwebs
rtio 1 cohesive
o
p 2-mode
o
r
p
0
T witterGoogle + PPI-LC PPI-Y2H LiveJournalYoutube w eb-Stanfowredb-Google cit-HepThcit-HepPh FloriCdhae sBaapyeake Bay
Figure7:Fractionof2-Modecommunitiesandcohesivecommunitiesinsixdifferenttypesofnetworks.
munities and thus it seems as though most interactions that have
beenexploredtakeplacein“cohesive” communities[46].Onthe
otherhand,thePPI-Y2Hnetworkiscreatedbyanoisyautomatic
processandmorefaithfullyrepresentstheinteractionnetwork.In
thiscasemany2-modecommunitiesemerge[46].
Insocialnetworkswealsofindinterestingresults.InLiveJour-
nal, communities are more cohesive, which can be explained by
thefactthatedgesinLiveJournalindicate“friendships”(i.e.,shar-
ing private blog content). On the other hand, Youtube communi-
tiesarepredominantly 2-mode. Youtube differsfromother social
networks inoneimportant way: EdgesinYoutubeareessentially
“subscriptions” forcontent ratherthanmutualfriendships; conse-
quently,highdegreenodestendtoconnecttolowdegreenodes[33].
Web graphs are of interest because Kumar et al. [23] used the
existence of 2-mode communities as indicators or signatures for
Figure 8: Overlapping 2-Mode communities detected by our cohesivecommunities.Ourresultsnicelysuggesttheco-existence
methodinaProtein-Proteininteractionnetwork.Seemaintext ofcohesivecommunitiesand2-modecommunitiesbyshowingthat
fortheexplanationofcommunitystructure. webgraphshaveanequalproportionof2-modeandcohesivecom-
munities.
Finally, foodwebs as well as citation networks consist almost
webgraphs, foodwebs,andcitationnetworksamongresearchpa- entirely of 2-mode communities. These results are natural as re-
persfromTable5. ciprocalandcohesiverelationshipsareextremelyunlikelyinthese
Toclassifywhetheradetectedcommunityis2-modeorcohesive, networks. In foodwebs, for example, few species prey upon each
wemeasuretheJaccardsimilarityJ(c)= |O(c)∩I(c)| betweenthe other.Citationnetworksaredirectedacyclicgraphsandreciprocal
|O(c)∪I(c)|
set of member nodes with outgoing memberships O(c), and the citation is impossible by definition. Intuitively, cohesive commu-
setofmembernodeswithincomingmembershipsI(c).Inacom- nities in directed networks contain some number of bidirectional
pletely cohesive community, this Jaccard similarity is 1 because edges among their members, therefore a lack of such reciprocal
twosetsofmembersareidentical,whereasitis0inacompletely edgesnaturallyleadstothedominanceof2-modecommunities,as
2-modecommunity.Weregardacommunitycas2-modeifJ(c)is weobserveinFigure6.
lowerthansomethresholdγorascohesiveotherwise(J(c) γ).
Weuseγ =0.2asthissettinggivesthemostinterpretablere≥sults. 6. CONCLUSION
Experimentalresults.Figure7showsthefractionof2-modeand An accurate notion of a community is critical when studying
cohesive communities in 12 networks described above. Ego net- the mesoscale structure of networks. Traditional models consider
works (Twitterand Google+) exhibit arelatively high fraction of ‘communities’tobesetsofdenselyconnectednodes.Inaddition,
cohesivecommunitiesandasnotedearlierFacebookegonetworks here we also consider 2-mode communities, which are groups of
(not shown) haveaneven higher fraction(over 95%) ofcohesive nodeswhomaynotlinktoeachotherbutlinkinacoordinateway
communities. This result is in line with [11] where the authors totheothernodesinthenetwork.
showthatFacebookegonetworkscanbeeasilydividedintocohe- WehavepresentedCoDA,acommunitydetectionmethodwhich
sivecommunities.However,itisimportanttonotethatasignificant naturallydetectsbothdenselyconnectedand2-modecommunities.
fractionofTwitter(20%)andGoogle+(30%)communitiesexhibit CoDA can capture overlapping and hierarchical structure among
2-modestructure. communities,andhandlesbothdirectedandundirectednetworks.
Literature-curatedprotein-proteininteractionnetworks(PPI-LC) OurexperimentalfindingsrevealthatCoDAoutperformsthecur-
practicallyhaveonlycohesivecommunities(andno2-mode). On rentstate-of-the-artindetectingground-truthcommunities.More-
the other hand, in PPI networks generated based on yeast two- over, CoDA also reveals how 2-mode and cohesive communities
hybridscreening(PPI-Y2H)about50%ofthecommunitiesare2- co-existinrealnetworks.
mode.Thisdifferenceisinterestingandconfirmsapreviousstudy The versatility of CoDA to detect both cohesive and 2-mode
of PPI networks [46], which provided the following explanation: communitiesaccuratelyindirectedandundirectednetworksraises
EdgesofPPI-LCareextractedfromscientificpapersthatreportex- manyinterestingavenuesoffuturework.Forexample,understand-
perimentallyvalidatedinteractions.However,currentbiologicalex- ingtheinteractionbetween2-modecommunitiesandcohesivecom-
perimentshavemainlybeenguidedbyresearchoncohesivecom- munitiesisafruitfuldirection.Inferringtheroleofnodesfromtheir
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tendCoDAtofindimportantnodesineachcommunity.Thiscould schemeforirregulargraphs.J.ofParallelandDistributed
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[22] B.KlimtandY.Yang.Introducingtheenroncorpus.In
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wecoulddeterminewhichnodesaremostimportantandhavethe
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