
Overlapping means that some vertices may
belong to more than one community.


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.

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.
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 vertex
A:the adjacency matrix of the network
m: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.
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 coefficient
Ex.
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.
bt (c, x) 
b
yN ( x )
t 1
(c, y )
N ( x)
y : the neighbor of the vertex x
c : community identifier
N ( x) : total number of neighbors of vertex x
bt (c, x) : the belong coefficien t of vertex x for community c at the t th iteration
Step3: if the algorithm satisfies the stop
criterion , the algorithm stop.
it  {c  V : x  V (bt (c, x)  0)}
it : the set of community identifier s in use at the t th itration
ct  {( c, i ) : c  V  i 
1
}
xV ,bt ( c , x )  0
ct : the number of vertices labelled with each community identifier at the t th itration
mt  {( c, i ) : pq ((c, p)  ct 1  (c, q )  ct  i  min( p, q )} if it  it 1
mt  ct otherwise
mt : the minimum number of vertices labelled with each community identifier
since the number of identifier s last reduced.
we stop the propagatio n as soon as m t  m t -1

Threshold:1/v
V:the maximum number of communities to
which any vertex can belong.
Ex.threshold=1/2
Ex.(b,1)
(e,1)
e
b
c
(c,1)
d
(a,1)
(f,1)
f
a
g
(c,1/3)
(d,1/3)
(a,1/3)
(f,1/3)
(g,1/3)
(a,1/3)
(g,1)
(d,1)
(initialize)
(c,1/3)
(b,1/3)
(a,1/3)
(d,1/4)
(e,1/3)
(b,1/4)
(f,1/3)
(e,1/4)
(g,1/4) (first iteration) (a,1/3)
(c,1/3)
(e,1/3)
(b,1/6)
(d,1/6)
(e,5/6)
(g,1/6)
(c,2/4)
(f,1/4)
(e,1/4)
(c,1/3)
(second iteration)
(e,1/3)
(b,1/6)
(d,1/6)
(g,1/6)
(f,1/3)
(e,3/6)
(third iteration)