The mixing time of the Newman--Watts small world

TL;DR

This paper proves that the mixing time of the Newman--Watts small world network model is on the order of (log n)^2, confirming previous physics predictions and providing rigorous bounds.

Contribution

It establishes the precise order of the mixing time for the Newman--Watts small world model, bridging a gap between physics predictions and mathematical proof.

Findings

Mixing time is of order (log n)^2 for the Newman--Watts model.

Confirms a physics-based prediction about the mixing time.

Provides rigorous bounds matching previous conjectures.

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 n)^2. This confirms a prediction of Richard Durrett, who proved a lower bound of order (log n)^2 and an upper bound of order (log n)^3.