Expanders with respect to Hadamard spaces and random graphs
Abstract
It is shown that there exists a sequence of 3regular 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 \cite{NS11}. $\{G_n\}_{n=1}^\infty$ 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2013
 arXiv:
 arXiv:1306.5434
 Bibcode:
 2013arXiv1306.5434M
 Keywords:

 Mathematics  Metric Geometry;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics;
 Mathematics  Functional Analysis
 EPrint:
 incorporated Referees' comments