Edge Expansion and Spectral Gap of Nonnegative Matrices

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.