Small worlds" are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman-Watts small world, the mixing time is of order log 2 n. This confirms a prediction of Richard Durrett, who proved a lower bound of order log 2 n and an upper bound of order log 3 n.The Erdős-Rényi random graph is unsatisfactory as a small-world model in two ways: first, the network does not satisfy full connectivity (a constant proportion of vertices lie outside of the giant component); second, the graph is locally tree-like -for any fixed k, the probability that there is a cycle of length at most k through a randomly chosen node node is o(1). In real-world networks showing small world behaviour (social or business networks, gene regulatory networks, networks for modelling infectious disease spread, scientific collaboration networks, and many others -the book [16] contains many interesting examples), full or almost-full connectivity is standard, and short cycles are plentiful. Several connected models have been proposed which in some respects capture the desired local structure as well as small-world behaviour, notably the Bollobás-Chung [1], Watts-Strogatz [20], and Newman-Watts [14,15] models. These models are closely related -all are based on adding sparse, long range connections to a connected "base network" which is essentially a cycle.

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