Advances in Applied Probability

The mixing time of the Newman-Watts small-world model

Louigi Addario-Berry and Tao Lei

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'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 model, the mixing time is of order log2n. This confirms a prediction of Richard Durrett [5, page 22], who proved a lower bound of order log2n and an upper bound of order log3n.

Article information

Adv. in Appl. Probab., Volume 47, Number 1 (2015), 37-56.

First available in Project Euclid: 31 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05C81: Random walks on graphs 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30]

Random graph small world mixing time conductance bound


Addario-Berry, Louigi; Lei, Tao. The mixing time of the Newman-Watts small-world model. Adv. in Appl. Probab. 47 (2015), no. 1, 37--56. doi:10.1239/aap/1427814580.

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