Expanders with respect to Hadamard spaces and random graphs

Manor Mendel, Assaf Naor

Research output: Contribution to journalArticlepeer-review


It is shown that there exist 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 the second author and Silberman. The graphs {Gn}n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublineartime 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.

Original languageEnglish
Pages (from-to)1471-1548
Number of pages78
JournalDuke Mathematical Journal
Issue number8
StatePublished - 2015

Bibliographical note

Publisher Copyright:
© 2015.


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