Modularity and community structure in networks

Many networks of interest in the sciences, including a variety of social and biological networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure has attracted considerable recent attention. One of the most sensitive detection methods is optimization of the quality function known as "modularity" over the possible divisions of a network, but direct application of this method using, for instance, simulated annealing is computationally costly. Here we show that the modularity can be reformulated in terms of the eigenvectors of a new characteristic matrix for the network, which we call the modularity matrix, and that this reformulation leads to a spectral algorithm for community detection that returns results of better quality than competing methods in noticeably shorter running times. We demonstrate the algorithm with applications to several network data sets.