Duke Mathematical Journal
- Duke Math. J.
- Volume 164, Number 8 (2015), 1471-1548.
Expanders with respect to Hadamard spaces and random graphs
It is shown that there exist a sequence of -regular graphs and a Hadamard space such that forms an expander sequence with respect to , yet random regular graphs are not expanders with respect to . This answers a question of the second author and Silberman. The graphs 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.
Duke Math. J., Volume 164, Number 8 (2015), 1471-1548.
Received: 4 July 2013
Revised: 17 July 2014
First available in Project Euclid: 28 May 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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