Expanders with respect to Hadamard spaces and random graphs
Abstract
It is shown that there exists a sequence of 3-regular graphs \Gn\n=1∞ and a Hadamard space X such that \Gn\n=1∞ forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of NS11. \Gn\n=1∞ 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.
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