Networks of random cycles
Colin Cooper, Martin Dyer, Andrew J. Handley · 2011
We present a family of peer-to-peer network protocols that yield regular graph topologies having known Hamilton cycles. These topologics are equivalent, in a well-defined sense, to the random regular graph. As a consequence, we have connectivity deterministically, and logarithmic diameter and expansion properties with high probability. We study the efficacy of certain simple topology-altering operations, designed to introduce randomness. These operations enable the network to self-stabilise when damaged. They resemble the operations used by Cooper, Dyer and Greenhill (2007) for a similar purpose in the case of random regular graphs. There is a link between our protocols and certain combinatorial structures which have been studied previously, in particular discordant permutations and Latin rectangles. We give the first rigorous polynomial mixing-time bounds for natural Markov chains that sample these objects at random. We do so by developing a novel extension to the canonical path technique for bounding mixing times: routing via a random destination. This resembles a technique used by Valiant (1982) for low-congestion routing in hypercubes.