On the Poisson Follower Model

TL;DR

The paper introduces the Poisson Follower Model, a stochastic-geometry realization of leader–follower opinion dynamics where each follower moves toward its nearest neighbor by halving the distance, yielding rich geometric structures such as ultimate leader pairs and evolving parties. It develops integral-geometry representations for event frequencies, proves finiteness of follower parties at steps 0 and 1 via percolation and mass-transport techniques, and advances a conjectural long-term dichotomy between ultimate leaders and followers. Numerical simulations and semi-algebraic-set methods corroborate analytical findings and quantify densities (e.g., order-zero leader pairs with density ≈ 0.62), while asymptotic analysis explores stable trees and limiting relations. The work provides a framework for exact frequency computation through integral geometry and percolation arguments, with potential extensions to higher dimensions and broader initial processes.

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.

Figures (14)

• Figure 1: Example of a Poisson descending tree
• Figure 2: Representation of the forward and backward sets of an agent. The forward set of $x$ is connected to $x$ with dark gray arrows and the backward set of $x$ is with light gray ones. The agents connected by black arrows are in the follower party of $x$ but not in the backward or forward set of $x$.
• Figure 3: Left: Example of a Follower Loss/Follower Gain. The initial positions are shown in black. The step 1 is in red, with the new positions denoted with the prime '. Right: Example of formation of an ultimate leader pair of order 1. Same convention as in the left image
• Figure 4: Example of a follower inversion.
• Figure 5: Example of party fission

Theorems & Definitions (41)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Lemma 1
• Theorem 1
• proof
• Theorem 2
• proof