Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix

TL;DR

This paper provides theoretical bounds and inequalities demonstrating that eigenvectors of the normalized modularity matrix can identify meaningful communities within networks, supporting spectral clustering methods.

Contribution

It establishes lower bounds for modularity of clusters identified by eigenvectors and introduces Cheeger-type inequalities linking eigenvalues to network subdivision.

Findings

Eigenvectors with highly positive eigenvalues indicate potential communities.

Lower bounds for modularity of clusters are derived.

Cheeger-type inequalities relate spectral properties to graph partitioning.

Abstract

Nodal theorems for generalized modularity matrices ensure that the cluster located by the positive entries of the leading eigenvector of various modularity matrices induces a connected subgraph. In this paper we obtain lower bounds for the modularity of that set of nodes showing that, under certain conditions, the nodal domains induced by eigenvectors corresponding to highly positive eigenvalues of the normalized modularity matrix have indeed positive modularity, that is they can be recognized as modules inside the network. Moreover we establish Cheeger-type inequalities for the cut-modularity of the graph, providing a theoretical support to the common understanding that highly positive eigenvalues of modularity matrices are related with the possibility of subdividing a network into communities.