Edge Expansion and Spectral Gap of Nonnegative Matrices

TL;DR

This paper explores the relationship between edge expansion and spectral gap in doubly stochastic and nonnegative matrices, providing new bounds and constructions that deepen understanding of their spectral properties.

Contribution

It introduces improved bounds and constructions relating edge expansion to spectral gap for non-symmetric doubly stochastic matrices and extends these results to general nonnegative matrices.

Findings

Constructed matrices with edge expansion much smaller than spectral gap suggests.

Established lower bounds linking edge expansion to spectral gap for all doubly stochastic matrices.

Extended bounds to nonnegative matrices, refining Perron-Frobenius theorem insights.

Abstract

The classic graphical Cheeger inequalities state that if $M$ is an $n\times n$ symmetric doubly stochastic matrix, then \[ \frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))} \] where $\phi(M)=\min_{S\subseteq[n],|S|\leq n/2}\left(\frac{1}{|S|}\sum_{i\in S,j\not\in S}M_{i,j}\right)$ is the edge expansion of $M$, and $\lambda_{2}(M)$ is the second largest eigenvalue of $M$. We study the relationship between $\phi(A)$ and the spectral gap $1-\text{Re}\lambda_{2}(A)$ for any doubly stochastic matrix $A$ (not necessarily symmetric), where $\lambda_{2}(A)$ is a nontrivial eigenvalue of $A$ with maximum real part. Fiedler showed that the upper bound on $\phi(A)$ is unaffected, i.e., $\phi(A)\leq\sqrt{2\cdot(1-\text{Re}\lambda_{2}(A))}$. With regards to the lower bound on $\phi(A)$, there are known constructions with \[ \phi(A)\in\Theta\left(\frac{1-\text{Re}\lambda_{2}(A)}{\log n}\right), \] indicating that at least a mild dependence on $n$ is necessary to lower bound $\phi(A)$. In our first result, we provide an exponentially better construction of $n\times n$ doubly stochastic matrices $A_{n}$, for which \[\phi(A_{n})\leq\frac{1-\text{Re}\lambda_{2}(A_{n})}{\sqrt{n}}.\] In fact, all nontrivial eigenvalues of our matrices are $0$, even though the matrices are highly nonexpanding. We further show that this bound is in the correct range (up to the exponent of $n$), by showing that for any doubly stochastic matrix $A$, \[\phi(A)\geq\frac{1-\text{Re}\lambda_{2}(A)}{35\cdot n}.\] Our second result extends these bounds to general nonnegative matrices $R$, obtaining a two-sided quantitative refinement of the Perron-Frobenius theorem in which the edge expansion $\phi(R)$ (appropriately defined), a quantitative measure of the irreducibility of $R$, controls the gap between the Perron-Frobenius eigenvalue and the next-largest real part of any eigenvalue.