Reweighted Eigenvalues: A New Approach to Spectral Theory beyond Undirected Graphs

Abstract

We develop a concept called reweighted eigenvalues, to extend spectral graph theory beyond undirected graphs. Our main motivation is to derive Cheeger inequalities and spectral rounding algorithms for a general class of graph expansion problems, including vertex expansion and edge conductance in directed graphs and hypergraphs. The goal is to have a unified approach to achieve the best known results in all these settings.

The first main result is an optimal Cheeger inequality for undirected vertex expansion. Our result connects (i) reweighted eigenvalues, (ii) vertex expansion, and (iii) fastest mixing time [BDX04] of graphs, similar to the way the classical theory connects (i) Laplacian eigenvalues, (ii) edge conductance, and (iii) mixing time of graphs. We also obtain close analogues of several interesting generalizations of Cheeger’s inequality [Tre09, LOT12, LRTV12, KLLOT13] using higher reweighted eigenvalues, many of which were previously unknown.

The second main result is Cheeger inequalities for directed graphs. The idea of Eulerian reweighting is used to effectively reduce these directed expansion problems to the basic setting of edge conductance in undirected graphs. Our result connects (i) Eulerian reweighted eigenvalues, (ii) directed vertex expansion, and (iii) fastest mixing time of directed graphs. This provides the first combinatorial characterization of fastest mixing time of general (non-reversible) Markov chains. Another application is to use Eulerian reweighted eigenvalues to certify that a directed graph is an expander graph.

Several additional results are developed to support this theory. One class of results is to show that adding $\ell_2^2$ triangle inequalities [ARV09] to reweighted eigenvalues provides simpler semidefinite programming relaxations, that achieve or improve upon the previous best approximations for a general class of expansion problems. These include edge expansion and vertex expansion in directed graphs and hypergraphs, as well as multi-way variations of some undirected expansion problems. Another class of results is to prove upper bounds on reweighted eigenvalues for special classes of graphs, including planar, bounded genus, and minor free graphs. These provide the best known spectral partitioning algorithm for finding balanced separators, improving upon previous algorithms and analyses [ST96, BLR10, KLPT11] using ordinary Laplacian eigenvalues.