Nodal domain partition and the number of communities in networks**This work is supported by the National Natural Science Foundation of China(no. 10971137), the National Basic Research Program (973) of China (no. 2006CB805900),and a grant of Science and Technology Commission of Shanghai Municipality (STCSM,no. 09XD1402500).

Abstract

It is difficult to detect and evaluate the number of communities in complex networks,especially when the situation involves an ambiguous boundary between the inner- andinter-community densities. In this paper, discrete nodal domain theory is used to provide acriterion to determine how many communities a network has and how to partition thesecommunities by means of topological structure and geometric characterization. Bycapturing the signs of the Laplacian eigenvectors, we separate the network into severalreasonable clusters. The method leads to a fast and effective algorithm with application toa variety of real network data sets.