Expanders with respect to Hadamard spaces and random graphs: extended 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 [31]. {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. This extended abstract does not contain proofs. The full version of this paper can be found at arXiv:1306.5434.