Statistical significance of communities in networks

Nodes of real complex networks are topologically organized in local
clusters or modules. These
groups are generally named communities and intuitively defined as
sub-graphs with a density
of internal connections larger than the one of external links. In spite
of the intense research activity of this subject, a firm mathematical
definition of community is still missing. In this work,
we introduce a new measure aimed to quantify the statistical
significance of communities in networks. Extreme and Order Statistics
are used to predict the statistics associated with clusters in random
graphs, including confidence intervals. This allows us to quantify the
statistical significance of communities in networks, detecting
``artificial\'\' clusters arising as topological fluctuations in graphs
without intrinsic structure. The method is successfully applied in the
case of real-world networks for the evaluation of the significance of
their communities.