arXiv:0712.2716v1
[physics.socph]
17 Dec 2007
Community Structure in Graphs
Santo Fortunato
a
, Claudio Castellano
b
a
Complex Networks Lagrange Laboratory (CNLL), ISI Foundation,
Torino, Italy
b
SMC, INFMCNR and Dipartimento di Fisica, “Sapienza” Uni
versit`a di Roma, P. le A. Moro 2, 00185 Roma, Italy
Abstract
Graph vertices are often organized into groups that seem to live fairly in
dependently of the rest of the graph, with which they share but a few edges,
whereas the relationships between group members are stronger, as shown by
the large number of mutual connections.
Such groups of vertices, or commu
nities, can be considered as independent compartments of a graph. Detecting
communities is of great importance in sociology, biology and computer science,
disciplines where systems are often represented as graphs.
The task is very
hard, though, both conceptually, due to the ambiguity in the definition of com
munity and in the discrimination of different partitions and practically, because
algorithms must find “good” partitions among an exponentially large number of
them. Other complications are represented by the possible occurrence of hierar
chies, i.e. communities which are nested inside larger communities, and by the
existence of overlaps between communities, due to the presence of nodes belong
ing to more groups. All these aspects are dealt with in some detail and many
methods are described, from traditional approaches used in computer science
and sociology to recent techniques developed mostly within statistical physics.
1
Introduction
The origin of graph theory dates back to Euler’s solution [1] of the puzzle of
K¨
onigsberg’s bridges in 1736. Since then a lot has been learned about graphs and
their mathematical properties [2]. In the 20th century they have also become
extremely useful as representation of a wide variety of systems in different areas.
Biological, social, technological, and information networks can be studied as
graphs, and graph analysis has become crucial to understand the features of
these systems. For instance, social network analysis started in the 1930’s and
has become one of the most important topics in sociology [3, 4].
In recent
times, the computer revolution has provided scholars with a huge amount of
data and computational resources to process and analyse these data. The size
of real networks one can potentially handle has also grown considerably, reaching
1
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Figure 1: A simple graph with three communities, highlighted by the dashed
circles.
millions or even billions of vertices. The need to deal with such a large number
of units has produced a deep change in the way graphs are approached [5][9].
Real networks are not random graphs. The random graph, introduced by
P. Erd¨
os and A. R´
enyi [10], is the paradigm of a disordered graph: in it, the
probability of having an edge between a pair of vertices is equal for all possible
pairs. In a random graph, the distribution of edges among the vertices is highly
homogeneous.
For instance, the distribution of the number of neighbours of
a vertex, or
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 Spring '08
 UNGOR
 Graph Theory, The Land, vertices, Girvan M, Newman MEJ

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