Relations Between Adjacency and Modularity Graph Partitioning

In this paper the exact linear relation between the leading eigenvector of the unnormalized modularity matrix and the eigenvectors of the adjacency matrix is developed. Based on this analysis a method to approximate the leading eigenvector of the modularity matrix is given, and the relative error of the approximation is derived. A complete proof of the equivalence between normalized modularity clustering and normalized adjacency clustering is also given. A new metric is defined to describe the agreement of two clustering methods, and some applications and experiments are given to illustrate and corroborate the points that are made in the theoretical development.