Clique Percolation in Random Networks
Imre Dere
´nyi,
1
Gergely Palla,
2
and Tama
´s Vicsek
1,2
1
Department of Biological Physics, Eo
¨tvo
¨s University, Pa
´zma
´ny P. stny. 1A, H-1117 Budapest, Hungary
2
Biological Physics Research Group of HAS, Pa
´zma
´ny P. stny. 1A, H-1117 Budapest, Hungary
(Received 10 November 2004; published 29 April 2005)
The notion of
k
-clique percolation in random graphs is introduced, where
k
is the size of the complete
subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdo
˝s-
Re
´nyi graph of
N
vertices we obtain that the percolation transition of
k
-cliques takes place when the
probability of two vertices being connected by an edge reaches the threshold
p
c
±
k
²³´
±
k
ÿ
1
²
N
µ
ÿ
1
=
±
k
ÿ
1
²
.
At the transition point the scaling of the giant component with
N
is highly nontrivial and depends on
k
.We
discuss why clique percolation is a novel and ef±cient approach to the identi±cation of overlapping
communities in large real networks.
DOI: 10.1103/PhysRevLett.94.160202
PACS numbers: 02.10.Ox, 05.70.Fh, 64.60.
2
i, 89.75.Hc
There has been a quickly growing interest in networks,
since they can represent the structure of a wide class of
complex systems occurring from the level of cells to
society. Data obtained on real networks show that the
corresponding graphs exhibit unexpected nontrivial prop-
erties, e.g., anomalous degree distributions, diameter,
spreading phenomena, clustering coef±cient, and correla-
tions [1]. Very recently great attention has been paid to the
local structural units of networks. Small and well de±ned
subgraphs have been introduced as ‘‘motifs’’ [2]. Their
distribution and clustering properties [2,3] can be used to
interpret global features as well. Somewhat larger units,
made up of vertices that are more densely connected to
each other than to the rest of the network, are often referred
to as communities [4], and have been considered to be the
essential structural units of real networks. They have no
obvious de±nition, and most of the recent methods for their
identi±cation rely on dividing the network into smaller
pieces. The biggest drawback of these methods is that
they do not allow for overlapping communities, although
overlaps are generally assumed to be crucial features of
communities. In this Letter we lay down the fundamentals
of a kind of percolation phenomenon on graphs, which can
also be used as an effective and deterministic method for
uniquely identifying overlapping communities in large real
networks [5].
Meanwhile, the various aspects of the classical Erdo
Re
´nyi (ER) uncorrelated random graph [6] remain still of
great interest since such a graph can serve both as a test bed
for checking all sorts of new ideas concerning complex
networks in general, and as a prototype to which all other
random graphs can be compared. Perhaps the most con-
spicuous early result on the ER graphs was related to the
percolation transition taking place at
p
³
p
c
¶
1
=N
,
where
p
is the probability that two vertices are connected