## Advances in Applied Probability

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

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

Louigi Addario-Berry and Tao Lei

#### 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 log^{2}*n*. This confirms a
prediction of Richard Durrett [5, page 22], who proved a lower bound of order
log^{2}*n* and an upper bound of order log^{3}*n*.

#### 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