  Community structure is an important feature of many networks. One of the most
popular ways to capture community structure is using a quantitative measure,
modularity, which can serve as both a standard benchmark comparing different
community detection algorithms, and a optimization objective for detecting
communities. Previous works on modularity mainly focus on the approximation
method for modularity maximization to detect communities, or minor
modifications to the definition.
  In this paper, we study modularity from an information-theoretical
perspective and show that modularity and mutual information in networks are
essentially the same. The main contribution is that we develop a family of
generalized modularity measure, $f$-Modularity, which includes the original
modularity as a special case. At a high level, we show the significance of
community structure is equivalent to the amount of information contained in the
network. On the one hand, $f$-Modularity has an information-theoretical
interpretation and enjoys the desired properties of mutual information measure.
On the other hand, quantifying community structure also provides an approach to
estimate the mutual information between discrete random samples with a large
value space but given only limited samples. We demonstrate the algorithm for
optimizing $f$-Modularity in a relatively general case, and validate it through
experimental results on simulated networks. We also apply $f$-Modularity to
real-world market networks. Our results bridge two important fields, complex
network and information theory, and also shed light on the design of measures
on community structure in the future.
