## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 23 (2018), paper no. 54, 12 pp.

### Non-convergence of proportions of types in a preferential attachment graph with three co-existing types

John Haslegrave and Jonathan Jordan

#### Abstract

We consider the preferential attachment model with multiple vertex types introduced by Antunović, Mossel and Rácz. We give an example with three types, based on the game of rock-paper-scissors, where the proportions of vertices of the different types almost surely do not converge to a limit, giving a counterexample to a conjecture of Antunović, Mossel and Rácz. We also consider another family of examples where we show that the conjecture does hold.

#### Article information

**Source**

Electron. Commun. Probab., Volume 23 (2018), paper no. 54, 12 pp.

**Dates**

Received: 29 May 2018

Accepted: 25 July 2018

First available in Project Euclid: 1 September 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1535767265

**Digital Object Identifier**

doi:10.1214/18-ECP157

**Mathematical Reviews number (MathSciNet)**

MR3852268

**Zentralblatt MATH identifier**

1398.05187

**Subjects**

Primary: 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30]

Secondary: 05C80: Random graphs [See also 60B20] 60C05: Combinatorial probability 90B15: Network models, stochastic

**Keywords**

preferential attachment stochastic approximation social networks competing types

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Haslegrave, John; Jordan, Jonathan. Non-convergence of proportions of types in a preferential attachment graph with three co-existing types. Electron. Commun. Probab. 23 (2018), paper no. 54, 12 pp. doi:10.1214/18-ECP157. https://projecteuclid.org/euclid.ecp/1535767265