Abstract. "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.
These results were improved by Addario-Berry and Lei REF who proved that this mixing time is a.a.s.
The mixing time of the Newman--Watts small world
SODA
Mathematics Computer Science