Modularity and Mutual Information in Networks: Two Sides of the Same Coin

Abstract: Modularity, first proposed by [Newman and Girvan, 2004], is one of the most popular ways to quantify the significance of community structure in complex networks. It can serve as both a standard benchmark to compare different community detection algorithms, and an optimization objective to detect communities itself. Previous work on modularity has developed many efficient algorithms for modularity maximization. However, few of researchers considered the interpretation of the modularity function itself. 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 measures, f-modularity based on f-mutual information. f-Modularity has an information-theoretical interpretation, enjoys the desired properties of mutual information measure, and provides an approach to estimate the mutual information between discrete random variables. At a high level, we show the significance of community structure is equivalent to the amount of information contained in the network. The connection of f-modularity and f-mutual information bridges two important fields, complex network and information theory and also sheds light on the design of measures on community structure in future.