Expanders with respect to Hadamard spaces and random graphs

Abstract

It is shown that there exist a sequence of $3$-regular graphs $\{G_n{\}}_{n=1}^{\infty }$ and a Hadamard space $X$ such that $\{G_n{\}}_{n=1}^{\infty }$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with respect to $X$. This answers a question of the second author and Silberman. The graphs $\{G_n{\}}_{n=1}^{\infty }$ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear-time constant-factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.