Overlapping community detection

Overlapping means that some vertices may belong to more than one community.

Overlapping

EAGLE algorithm is presented to uncover both the overlapping and hierarchical community structures of networks.

EAGLE algorithm has two stages:◦ 1. A dendrogram is generated.◦ 2. We choose an appropriate cut which breaks the

dendrogram into communities.

agglomerativE hierarchicAl clusterinG based on maximaL cliquE

1.Find out all maximal cliques in the network(Bron-Kerbosch algorithm).Note that not all maximal cliques are taken into account.We set a threshold k and neglecting all the maximal cliques with the size smaller than k.

The first stage

k=4 k=3

2.Select the pair of communities with the maximum similarity,incorporate them into a new one and calculate the similarity between the new community and other communities.

C1, C2 :community1,2 k:the degree of the vertexA:the adjacency matrix of the networkm:the total number of edges in the network

3.Repeat step 2 until only one community remains.

Stage 2: The task of the second stage of the

algorithm EAGLE is to cut the dendrogram.

Ov:the number of communities to which vertex v belongs.

Community Overlap PRopagation

Algorithm

Step1:every vertex is given a unique label. After few iterations the label of vertex is the set of pairs (c,b). c: community identifier b: belong coefficientEx.The label of vertex x={(1,0.2),(2,0.3),(3,0.5)}

Step2:each vertex x updates its label by replacing it by the label used by the greatest number of neighbours.

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Step3: if the algorithm satisfies the stop criterion , the algorithm stop.

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Ex.b

c

d

a

e

f

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(b,1)

(d,1)

(c,1) (a,1)

(e,1)

(f,1)

(g,1)(initialize)

(first iteration)

(c,1/3)(d,1/3)(a,1/3)

(c,1/3)(b,1/3)(a,1/3)

(d,1/4)(b,1/4)(e,1/4)(g,1/4)

(f,1/3)(g,1/3)(a,1/3)

(e,1/3)(f,1/3)(a,1/3)

Threshold:1/vV:the maximum number of communities to which any vertex can belong.Ex.threshold=1/2

(second iteration)

(third iteration)

(c,1/3)(e,1/3)(b,1/6)(d,1/6)

(c,1/3)(e,1/3)(b,1/6)(d,1/6)

(c,2/4)(f,1/4)(e,1/4)

(e,5/6)(g,1/6)

(g,1/6)(f,1/3)(e,3/6)