It is shown that there exist a sequence of -regular graphs and a Hadamard space such that forms an expander sequence with respect to , yet random regular graphs are not expanders with respect to . This answers a question of the second author and Silberman. The graphs 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.
"Expanders with respect to Hadamard spaces and random graphs." Duke Math. J. 164 (8) 1471 - 1548, 1 June 2015. https://doi.org/10.1215/00127094-3119525