# Phase transition for the Maki-Thompson rumour model on a small-world network

## TL;DR

This paper studies how the structure of a small-world network influences the spread of a rumor, revealing a phase transition between limited and widespread propagation based on network connectivity.

## Contribution

It extends the Maki-Thompson rumor model to small-world networks and identifies a critical point where the rumor transitions from localized to widespread spread.

## Key findings

- Existence of a phase transition at a finite network parameter c
- Quantitative estimate of the critical value of c
- Different regimes of rumor spread depending on network structure

## Abstract

We consider the Maki-Thompson model for the stochastic propagation of a rumour within a population. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model. 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 parameter 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.