Opinion dynamics for an increasing population of agents. A symmetric continuous agent model

TL;DR

This work develops a symmetric, continuous-in-agent model for opinion dynamics with a growing population described by $\dot{N}_t = b(t,N_t)N_t$, where new agents enter with prescribed boundary opinions $X(t,N_t)$. It proves well-posedness for the microscopic system with Lipschitz interaction $\psi$ and derives a kinetic, measure-valued formulation $f_t$ that satisfies a transport equation with a growth-driven source, establishing existence, uniqueness, and stability in the $W_1$ metric. The paper characterizes long-time behavior across finite and infinite limiting populations: finite $N_\infty$ yields clustering into a finite number of groups with total mass $N_\infty$, while infinite $N_\infty$ leads to convergence of the mean to $X(t,N_t)$ under a key condition (C1) or to Dirac masses around the boundary input when $\psi>0$ and growth is subexponential. Overall, the results connect microscopic dynamics, moments, and kinetic descriptions to reveal how population growth and incoming-opinion profiles drive consensus, clustering, and measure-valued convergence in evolving social systems.

Abstract

In this paper we formulate a continuous opinion model that takes into account population growth, i.e. increase with time in the number of interacting agents $N(t)$. In our setting the population growth is governed by a generic growth rate function $b(t, N(t))$. The two main components of our model are the growth rate $b(t, N(t))$, as well as the opinions of the incoming agents which are modeled in our system as boundary conditions in a free boundary problem. We give results on the well-posedness of the model and results that showcase how these two components affect the long time asymptotic behavior of our system. Moreover, we provide a kinetic (probabilistic) description of our model and give results on well-posedness and asymptotics for the kinetic model.

Theorems & Definitions (49)

• Remark 1
• Remark 2
• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• proof : Proof of Theorem \ref{['thm: Well_posedness1']}