Spectral properties of modularity matrices

There is an exact relation between the spectra of modularity matrices introduced in social network analysis and the χ2χ2 statistic. We investigate a weighted graph with the main interest being when the hypothesis of independent attachment of the vertices is rejected, and we look for clusters of vertices with higher inter-cluster relations than expected under the hypothesis of independence. In this context, we give a sufficient condition for a weighted, and a sufficient and necessary condition for an unweighted graph to have at least one positive eigenvalue in its modularity or normalized modularity spectrum, which guarantees a community structure with more than one cluster. This property has important implications for the isoperimetric inequality, the symmetric maximal correlation, and the Newman–Girvan modularity.