# Phase Transition for the Maki–Thompson Rumour Model on a Small-World Network

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## Abstract

We consider the Maki–Thompson model for the stochastic propagation of a rumour within a population. In this model the population is made up of “spreaders”, “ignorants” and “stiflers”; any spreader attempts to pass the rumour to the other individuals via pair-wise interactions and in case the other individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. In a finite population the process will eventually reach an equilibrium situation where individuals are either stiflers or ignorants. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model, in such a way that interactions occur only between nearest-neighbours. This structure is realized starting from a *k*-regular ring and by inserting, in the average, *c* additional links in such a way that *k* and *c* are tuneable parameters for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter *c*. A quantitative estimate for the critical value of *c* is obtained via extensive numerical simulations.

## Keywords

Maki–Thompson model Phase-transition Small-world network## Mathematics Subject Classification

60K35 60K37 82B26## Notes

### Acknowledgements

PMR thanks FAPESP (Grants 2015/03868-7 and 2016/11648-0) for financial support. Part of this work was carried out during a stay of PMR at Laboratoire de Probabilités et Modèles Aléatoires, Université Paris-Diderot, and a visit at Università di Torino. He is grateful for their hospitality and support. AP thanks Università di Torino (XVIII tornata Programma di ricerca: “Problemi attuali della matematica 3”) for financial support. EA and FT thank INdAM-GNFM (Progetto Giovani 2016) and Sapienza Università di Roma (Progetto Avvio alla Ricerca 2015) for financial support. The authors are grateful to the anonymous reviewers for their interesting comments suggestions.

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