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

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 correspondi