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

Abstract

We introduce a family of multi-way Cheeger-type constants{hσk,k=1,2,...,n}on asigned graphΓ=(G,σ) such thathσk=0 if and only ifΓhaskbalanced connectedcomponents. These constants are switching invariant and bring together in a unifiedviewpoint a number of important graph-theoretical concepts, including the classicalCheeger constant, those measures of bipartiteness introduced by Desai-Rao, Trevisan,Bauer–Jost, respectively, on unsigned graphs, and the frustration index (originally calledthelineindexofbalancebyHarary)onsignedgraphs.Wefurtherunifythe(higher-orderor improved) Cheeger and dual Cheeger inequalities for unsigned graphs as well as theunderlying algorithmic proof techniques by establishing their corresponding versionson signed graphs. In particular, we develop a spectral clustering method for findingkalmost-balanced subgraphs, each defining a sparse cut. The proper metric for sucha clustering is the metric on a real projective space. We also prove estimates of theextremal eigenvalues of signed Laplace matrix in terms of number of signed triangles(3-cycles).