Advances in Applied Probability

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

Louigi Addario-Berry and Tao Lei

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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 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

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

Dates
First available in Project Euclid: 31 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1427814580

Digital Object Identifier
doi:10.1239/aap/1427814580

Mathematical Reviews number (MathSciNet)
MR3327314

Zentralblatt MATH identifier
1309.60002

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

Keywords
Random graph small world mixing time conductance bound

Citation

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. https://projecteuclid.org/euclid.aap/1427814580


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