The isoperimetric number of random regular graphs
European Journal of Combinatorics
Sublinear time algorithms for metric space problems
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Peer-to-peer networks based on random transformations of connected regular undirected graphs
Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures
On Approximating the Average Distance Between Points
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Towards a calculus for non-linear spectral gaps
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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It is shown that there exists a sequence of 3-regular graphs {Gn}∞n=1 and a Hadamard space X such that {Gn}∞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 [31]. {Gn}∞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. This extended abstract does not contain proofs. The full version of this paper can be found at arXiv:1306.5434.