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arXiv · 1306.5434

Expanders with respect to Hadamard spaces and random graphs

Manor Mendel·Assaf Naor
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Abstract

It is shown that there exists 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 \cite{NS11}. $\{G_n\}_{n=1}^\infty$ 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.

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Manor Mendel, Assaf Naor. 2014-07-18. Expanders with respect to Hadamard spaces and random graphs. https://doi.org/10.1215/00127094-3119525

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