## Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 2 (2017), 1235-1288.

### Evolving voter model on dense random graphs

Riddhipratim Basu and Allan Sly

#### Abstract

In this paper, we examine a variant of the voter model on a dynamically changing network where agents have the option of changing their friends rather than changing their opinions. We analyse, in the context of dense random graphs, two models considered in Durrett et al. [*Proc. Natl. Acad. Sci. USA* **109** (2012) 3682–3687]. When an edge with two agents holding different opinion is updated, with probability $\frac{\beta }{n}$, one agent performs a voter model step and changes its opinion to copy the other, and with probability $1-\frac{\beta }{n}$, the edge between them is broken and reconnected to a new agent chosen randomly from (i) the whole network (rewire-to-random model) or, (ii) the agents having the same opinion (rewire-to-same model). We rigorously establish in both the models, the time for this dynamics to terminate exhibits a phase transition in the model parameter $\beta $. For $\beta $ sufficiently small, with high probability the network rapidly splits into two disconnected communities with opposing opinions, whereas for $\beta $ large enough the dynamics runs for longer and the density of opinion changes significantly before the process stops. In the rewire-to-random model, we show that a positive fraction of both opinions survive with high probability.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 2 (2017), 1235-1288.

**Dates**

Received: September 2015

Revised: April 2016

First available in Project Euclid: 26 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1495764378

**Digital Object Identifier**

doi:10.1214/16-AAP1230

**Mathematical Reviews number (MathSciNet)**

MR3655865

**Zentralblatt MATH identifier**

1368.60095

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 05C80: Random graphs [See also 60B20] 05C81: Random walks on graphs

**Keywords**

Voter model coevolutionary network dense Erdős–Rényi graphs

#### Citation

Basu, Riddhipratim; Sly, Allan. Evolving voter model on dense random graphs. Ann. Appl. Probab. 27 (2017), no. 2, 1235--1288. doi:10.1214/16-AAP1230. https://projecteuclid.org/euclid.aoap/1495764378