Penalized versions of the Newman-Girvan modularity and their relation to normalized cuts and k-means clustering.

Two penalized-balanced and normalized-versions of the Newman-Girvan modularity are introduced and estimated by the non-negative eigenvalues of the modularity and normalized modularity matrix, respectively. In this way, the partition of the vertices that maximizes the modularity can be obtained by applying the k-means algorithm for the representatives of the vertices based on the eigenvectors belonging to the largest positive eigenvalues of the modularity or normalized modularity matrix. The proper dimension depends on the number of the structural eigenvalues of positive sign, while dominating negative eigenvalues indicate an anticommunity structure; the balance between the negative and the positive eigenvalues determines whether the underlying graph has a community, anticommunity, or randomlike structure.