An information-theoretic framework for resolving community structure in complex networks

Abstract

To understand the structure of a large-scale biological, social, or technological network, it can be helpful to decompose the network into smaller subunits or modules. In this article, we develop an information-theoretic foundation for the concept of modularity in networks. We identify the modules of which the network is composed by finding an optimal compression of its topology, capitalizing on regularities in its structure. We explain the advantages of this approach and illustrate them by partitioning a number of real-world and model networks.

Fig. 1.

Basic framework for detecting communities as a communication process. A signaler knows the full network structure and wants to send as much information as possible about the network to a receiver over a channel with limited capacity. The signaler therefore encodes the network into modules in a way that maximizes the amount of information about the original network. This figure illustrates an encoder that compresses the network into three modules, i = circle, square, star, with n_i nodes and l_ii links connected by l_ij links between the modules. The receiver can then decode the message and construct a set of possible candidates for the original network. The smaller the set of candidates, the more information the signaler has managed to transfer.

Fig. 2.

The dolphin network of Lusseau et al. (14) partitioned with our cluster-based compression (solid line) and based on the modularity (dashed line). The right branch of the dashed line represents a split based on maximizing the modularity, which is different from the left branch solution based on the spectral analysis approximation presented in ref. . The edge-betweenness algorithm presented in ref. splits the network in the same way as our cluster-based compression method (15).

Fig. 3.

Partitioning into an optimal number of modules. (A) Network consisting of 40 journals as nodes from four different fields: multidisciplinary physics (squares), chemistry (circles), biology (stars), and ecology (triangles). The 189 links connect nodes if at least one article from one of the journals cites an article in the other journal during 2004 (32). We have selected the 10 journals with the highest impact factor in the four different fields but disregarded journals classified in one or more of the other fields. (B) Minimum description length for the network in A partitioned into 1–5 different modules. The optimal partitioning into four modules is illustrated by the lines in A.

Fig. 4.

Zachary's karate club network (29) partitioned into two modules based on the maximum mutual information with (A) and without (B) the link constraint. The partitioning with more links within modules than between modules in A clusters closely connected nodes together, and the unconstrained partitioning in B clusters nodes with similar roles together. The stars and circles represent the two observed groups in A, and the stars represent the five nodes with highest degree in B.