Duke Mathematical Journal

Expanders with respect to Hadamard spaces and random graphs

Manor Mendel and Assaf Naor

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Abstract

It is shown that there exist a sequence of 3 -regular graphs { G n } n = 1 and a Hadamard space X such that { G n } 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 { G n } n = 1 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
Received: 4 July 2013
Revised: 17 July 2014
First available in Project Euclid: 28 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1432817757

Digital Object Identifier
doi:10.1215/00127094-3119525

Mathematical Reviews number (MathSciNet)
MR3352040

Zentralblatt MATH identifier
1316.05109

Subjects
Primary: 51F99: None of the above, but in this section
Secondary: 05C12: Distance in graphs 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science [See also 05C12, 68Rxx]

Keywords
expanding graphs CAT(0) spaces random graphs Euclidean cones bi-Lipschitz embeddings

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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