On the Poisson Follower Model

TL;DR

This paper introduces a geometric opinion dynamics model inspired by social networks, analyzing how opinions evolve among leaders and followers, especially when initial opinions are randomly distributed as a Poisson process, revealing geometric phenomena and long-term behaviors.

Contribution

It presents a novel stochastic geometry model of opinion dynamics with asymmetric leader-follower interactions, providing integral geometry formulas and percolation-based proofs for the evolution of opinions.

Findings

Opinions cluster into leaders and followers with distinct long-term behaviors.

The likelihood of geometric phenomena can be expressed via integral geometry formulas.

Numerical simulations confirm the theoretical analysis of opinion evolution.

Abstract

We introduce a stochastic geometry dynamics inspired by opinion dynamics that captures the essence of modern asymmetric social networks with leaders and followers. Points in Euclidean space represent opinions, and the leader of an agent is the one with the closest opinion. In this dynamics, each follower updates its opinion by halving the distance to its leader. We demonstrate that this simple dynamics and its iterations exhibit several interesting purely geometric phenomena related to the evolution of leadership and opinion clusters, which resemble those observed in social networks. We also show that when the initial opinions are randomly distributed as a stationary Poisson point process, the likelihood of each of these phenomena can be expressed through an integral geometry formula involving semi-algebraic domains. Furthermore, we establish this property for step 0 and step 1 of the dynamics using percolation techniques. Finally, we analyze numerically the limiting behavior of this follower dynamics. In the Poisson case, the agents fall into two categories: ultimate followers, who continue updating their opinions indefinitely, and ultimate leaders, who adopt a fixed opinion after a finite time. Spatial discrete event simulations support all our findings.