7270 community detection

8/18/2019 7270 Community Detection

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Community Detection and

Graph-based Clustering

Chapter 3

Of Lei Tang and HuanLiu’s oo!

"lides prepared by

#iang $ang%&"T% Hong'ong

(Chapter 3% Community Detection and )ining in "ocial )edia* Lei Tangand Huan Liu% )organ , Claypool% "eptember% .(.*


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Community

• Community/ 0t is formed by indi1iduals such that those2ithin a group interact 2ith each other morefreuently than 2ith those outside the group – a*!*a* group% cluster% cohesi1e subgroup% module in di4erent

conte5ts

• Community detection/ disco1ering groups in a net2or!2here indi1iduals’ group memberships are note5plicitly gi1en

•6hy communities in social media7 – Human beings are social – 8asy-to-use social media allo2s people to e5tend their

social life in unprecedented 2ays – Di9cult to meet friends in the physical 2orld% but much

easier to :nd friend online 2ith similar interests

– 0nteractions bet2een nodes can help determinecommunities

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Communities in "ocial)edia

• T2o types of groups in social media – 85plicit Groups/ formed by user subscriptions – 0mplicit Groups/ implicitly formed by social

interactions

• "ome social media sites allo2 people to ;oingroups% is it necessary to e5tract groups based onnet2or! topology7 –


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COMMUNITY DETECTION

@


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"ub;ecti1ity of CommunityDe:nition

8ach componentis a communityA densely-!nit

community

De:nition of a communitycan be sub;ecti1e*

unsuper1ised learningB


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Ta5onomy of CommunityCriteria

• Criteria 1ary depending on the tas!s• oughly% community detection methods can be

di1ided into ? categories not e5clusi1eB/• artition the 2hole net2or! into se1eral dis;oint sets

• Hierarchy-Centric Community – Construct a hierarchical structure of communities

E


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Complete )utuality/ Cliues

• Cliue/ a ma5imum complete subgraph in2hich all nodes are ad;acent to each other

• -hard to :nd the ma5imum cliue in anet2or!

• "traightfor2ard implementation to :ndcliues is 1ery e5pensi1e in time comple5ity


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inding the )a5imum Cliue

• 0n a cliue of si=e !% each node maintains degreeIJ !-( –


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)a5imum Cliue 85ample

• "uppose 2e sample a sub-net2or! 2ith nodes (-M and :nd a cliue (% % 3M of si=e 3

• 0n order to :nd a cliue I3% remo1e all nodes 2ithdegree KJ3-(J – emo1e nodes and

– emo1e nodes ( and 3

– emo1e node ?

((


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Cliue >ercolation )ethodC>)B

• Cliue is a 1ery strict de:nition% unstable• ) is such a method to :nd o1erlapping communities – Input

• A parameter !% and a net2or! – Procedure

• ind out all cliues of si=e ! in a gi1en net2or!• Construct a cliue graph* T2o cliues are ad;acent if

they share !-( nodes• 8ach connected component in the cliue graph forms

a community(


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C>) 85ample

Cliques of sie !"(% % 3M% (% 3% ?M%?% @% M% @% % EM%@% % FM% @% E% FM%% E% FM

Communities/

(% % 3% ?M?% @% % E% FM

(3


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eachability / !-cliue% !-club

• Any node in a group should be reachable in !hops

• !-cliue/ a ma5imal subgraph in 2hich the largestgeodesic distance bet2een any t2o nodes KJ !

• !-club/ a substructure of diameter KJ !

• A !-cliue might ha1e diameter larger than ! inthe subgraph – 8*g* (% % 3% ?% @M

•Commonly used in traditional "


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Group-Centric CommunityDetection/ Density-ased

Groups• The group-centric criterion reuires the 2holegroup to satisfy a certain condition – 8*g*% the group density IJ a gi1en threshold

• A subgraph is a uasi-cliue if

2here the denominator is the ma5imum number of

degrees*

• A similar strategy to that of cliues can be used – "ample a subgraph% and :nd a ma5imal

uasi-cliue say% of si=e B

– emo1e nodes 2ith degree less than the a1erage degree(@

,

<


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Clustering based on Nerte5"imilarity

• Apply !-means or similarity-based clustering tonodes

• Nerte5 similarity is de:ned in terms of thesimilarity of their neighborhood

• "tructural eui1alence/ t2o nodes are structurallyeui1alent i4 they are connecting to the same setof actors

•"tructural eui1alence is too strict for practicaluse*


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Nerte5 "imilarity

• Paccard "imilarity

• Cosine similarity

(F

(1) Clustering based on vertex similarity

(4) S t l l t i


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Cut

• )ost interactions are 2ithin group 2hereasinteractions bet2een groups are fe2

• community detection minimum cut problem

• Cut/ A partition of 1ertices of a graph into t2odis;oint sets

• )inimum cut problem/ :nd a graph partition suchthat the number of edges bet2een the t2o sets isminimi=ed

(4) Spectral clustering

(4) S t l l t i


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atio Cut ,


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atio Cut ,


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)odularity )a5imi=ation

• )odularity measures the strength of a communitypartition by ta!ing into account the degreedistribution

• Gi1en a net2or! 2ith m edges% the e5pectednumber of edges bet2een t2o nodes 2ithdegrees di and d j is

• "trength of a community/

• )odularity/ •

The e5pected number ofedges bet2een nodes (

and is3RS R(?B J 3S(?

E

(5) odularity maximi!ation

"iven t#e degree distribution


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Hierarchy-Centric CommunityDetection

• Goal/ build a hierarchical structure ofcommunities based on net2or! topology

• Allo2 the analysis of a net2or! at di4erentresolutions

• epresentati1e approaches/ – Di1isi1e Hierarchical Clustering top-

do2nB

– Agglomerati1e Hierarchical clustering

bottom-upB 3(


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Di1isi1e HierarchicalClustering

• Di1isi1e clustering – >artition nodes into se1eral sets

– 8ach set is further di1ided into smaller ones

–


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8dge et2eenness

• The strength of a tie can be measured by edgebet2eenness

• 8dge bet2eenness/ the number of shortest pathsthat pass along 2ith the edge

• The edge 2ith higher bet2eenness tends to bethe bridge bet2een t2o communities*

The edge bet2eenness ofe(% B is ? JS V (B% asall the shortest paths from to ?% @% % E% F% M ha1eto either pass e(% B or e%3B% and e(%B is theshortest path bet2een (and

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Di1isi1e clustering based onedge bet2eenness

After remo1e e?%@B% thebet2eenness of e?% B

becomes .% 2hich is thehighest

After remo1e e?%B% the edgeeE%B has the highest

bet2eenness 1alue ?% and

should be remo1ed*

0nitial bet2eenness 1alue

3?$dea% progressively removing edges &it# t#e #ig#est bet&eenness


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Agglomerati1e HierarchicalClustering

• 0nitiali=e each node as a community

• )erge communities successi1ely intolarger communities follo2ing a certain

criterion – 8*g*% based on modularity increase

3@

'endrogram according to gglomerative Clustering based on odularity


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"ummary of CommunityDetection

•

7270 community detection

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