Cheeger constants, structural balance, and spectral clustering analysis for signed graphs

TL;DR

This paper introduces multi-way Cheeger constants for signed graphs, unifying various graph measures, and develops spectral clustering methods to identify almost-balanced subgraphs, with theoretical bounds related to signed triangles.

Contribution

It unifies multiple graph-theoretical concepts into a single framework for signed graphs and extends Cheeger inequalities and spectral clustering techniques to this setting.

Findings

Defines multi-way Cheeger constants for signed graphs.

Establishes Cheeger and dual Cheeger inequalities for signed graphs.

Develops spectral clustering method for almost-balanced subgraphs.

Abstract

We introduce a family of multi-way Cheeger-type constants $\{h_k^{\sigma}, k=1,2,\ldots, n\}$ on a signed graph $\Gamma=(G,\sigma)$ such that $h_k^{\sigma}=0$ if and only if $\Gamma$ has $k$ balanced connected components. These constants are switching invariant and bring together in a unified viewpoint a number of important graph-theoretical concepts, including the classical Cheeger constant, those measures of bipartiteness introduced by Desai-Rao, Trevisan, Bauer-Jost, respectively, on unsigned graphs,, and the frustration index (originally called the line index of balance by Harary) on signed graphs. We further unify the (higher-order or improved) Cheeger and dual Cheeger inequalities for unsigned graphs as well as the underlying algorithmic proof techniques by establishing their corresponding versions on signed graphs. In particular, we develop a spectral clustering method for finding $k$ almost-balanced subgraphs, each defining a sparse cut. The proper metric for such a clustering is the metric on a real projective space. We also prove estimates of the extremal eigenvalues of signed Laplace matrix in terms of number of signed triangles ($3$-cycles).