Modularity spectra, eigen-subspaces, and structure of weighted graphs

TL;DR

This paper explores the spectral properties of the normalized modularity matrix in weighted graphs, focusing on eigenvalues and eigen-subspaces, and their implications for graph clustering and structure testing.

Contribution

It introduces methods for analyzing the modularity spectrum and establishes conditions under which structural eigenvalues and eigen-subspaces are testable.

Findings

Testability of structural eigenvalues and eigen-subspaces is linked to spectral gaps.

Spectral gaps enable testing of cluster variance sums.

Eigen-subspace analysis informs graph clustering quality.

Abstract

The role of the normalized modularity matrix in finding homogeneous cuts will be presented. We also discuss the testability of the structural eigenvalues and that of the subspace spanned by the corresponding eigenvectors of this matrix. In the presence of a spectral gap between the k-1 largest absolute value eigenvalues and the remainder of the spectrum, this in turn implies the testability of the sum of the inner variances of the k clusters that are obtained by applying the k-means algorithm for the appropriately chosen vertex representatives.