Expanders with respect to Hadamard spaces and random graphs

Manor Mendel, Assaf Naor

Research output: Contribution to journalArticle

Abstract

It is shown that there exist a sequence of 3-regular graphs {G<inf>n</inf>}<sup>∞</sup><inf>n</inf>=1 and a Hadamard space X such that {G<inf>n</inf>}<sup>∞</sup><inf>n</inf>=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 {G<inf>n</inf>}<sup>∞</sup><inf>n</inf>=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 (US)
Pages (from-to)1471-1548
Number of pages78
JournalDuke Mathematical Journal
Volume164
Issue number8
DOIs
StatePublished - 2015

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Expander
Random Graphs
Regular Graph
Martingale Method
Geometric object
Approximation Algorithms
Euclidean
Cone
Subset
Computing
Graph in graph theory

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Expanders with respect to Hadamard spaces and random graphs. / Mendel, Manor; Naor, Assaf.

In: Duke Mathematical Journal, Vol. 164, No. 8, 2015, p. 1471-1548.

Research output: Contribution to journalArticle

Mendel, Manor ; Naor, Assaf. / Expanders with respect to Hadamard spaces and random graphs. In: Duke Mathematical Journal. 2015 ; Vol. 164, No. 8. pp. 1471-1548.
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