Duke Mathematical Journal

Expanders with respect to Hadamard spaces and random graphs

Abstract

It is shown that there exist 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 the second author and Silberman. The graphs $\{G_{n}\}_{n=1}^{\infty}$ 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.

Article information

Source
Duke Math. J., Volume 164, Number 8 (2015), 1471-1548.

Dates
Revised: 17 July 2014
First available in Project Euclid: 28 May 2015

https://projecteuclid.org/euclid.dmj/1432817757

Digital Object Identifier
doi:10.1215/00127094-3119525

Mathematical Reviews number (MathSciNet)
MR3352040

Zentralblatt MATH identifier
1316.05109

Citation

Mendel, Manor; Naor, Assaf. Expanders with respect to Hadamard spaces and random graphs. Duke Math. J. 164 (2015), no. 8, 1471--1548. doi:10.1215/00127094-3119525. https://projecteuclid.org/euclid.dmj/1432817757

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