Small world phenomenon, rapidly mixing Markov chains, and average consensus algorithms

In this paper, we demonstrate the relationship between the diameter of a graph and the mixing time of a symmetric Markov chain defined on it. We use this relationship to show that graphs with the small world property have dramatically small mixing times. Based on this result, we conclude that addition of independent random edges with arbitrarily small probabilities to a cycle significantly increases the convergence speed of average consensus algorithms, meaning that small world networks reach consensus orders of magnitude faster than a cycle. Furthermore, this dramatic increase happens for any positive probability of random edges. The same argument is used to draw a similar conclusion for the case of addition of a random matching to the cycle.