Limiting behavior of a kindness model

Abstract

This paper is concerned with a stochastic model for the spread of kindness across a social network. Individuals are located on the vertices of a general finite connected graph, and are characterized by their kindness belief. Each individual, say $x$, interacts with each of its neighbors, say $y$, at rate one. The interactions can be kind or unkind, with kind interactions being more likely when the kindness belief of the sender $x$ is high. In addition, kind interactions increase the kindness belief of the recipient $y$, whereas unkind interactions decrease its kindness belief. The system also depends on two parameters modeling the impact of kind and unkind interactions, respectively. We prove that, when kind interactions have a larger impact than unkind interactions, the system converges to the purely kind configuration with probability tending to one exponentially fast in the large population limit.