Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A simple consensus model for an increasing population of agents with i.i.d incoming opinions

Ioannis Markou

TL;DR

This work analyzes how population growth affects consensus in a symmetric all-to-all opinion model with i.i.d. incoming opinions having mean $m$ and variance $σ^2$. By formulating the dynamics with growth-induced jumps and a dissipation-driven variance functional, it derives a sharp criterion for mean-variance convergence in terms of the growth times $t_j$ and shows that sub-exponential growth $N(t)\sim e^{t^{α}}$ yields convergence to $m$ if and only if $α<1$, at an algebraic rate. The authors provide explicit moment formulas, bound the variance jumps, and apply a generalized Dawson integral analysis to connect growth rate to a precise consensus threshold. The results clarify when adding agents preserves consensus formation and quantify the long-time convergence under expanding populations, with implications for models of evolving social systems.

Abstract

In this short note we study what happens in a symmetric opinion model when we send the total interacting population $N(t)$ to infinity as $t \to \infty$. We assume that new population enters the system with opinions that are i.i.d random vectors with finite mean and variance. We give sharp conditions on the rate of population growth that is required for convergence to a global consensus in opinions. More particularly, we show that if the total population increases at a rate $N(t)\sim e^{t^α}$, then $α<1$ is necessary and sufficient condition for convergence to the mean of incoming opinions, and the convergence is achieved at an algebraic rate.

A simple consensus model for an increasing population of agents with i.i.d incoming opinions

TL;DR

This work analyzes how population growth affects consensus in a symmetric all-to-all opinion model with i.i.d. incoming opinions having mean and variance . By formulating the dynamics with growth-induced jumps and a dissipation-driven variance functional, it derives a sharp criterion for mean-variance convergence in terms of the growth times and shows that sub-exponential growth yields convergence to if and only if , at an algebraic rate. The authors provide explicit moment formulas, bound the variance jumps, and apply a generalized Dawson integral analysis to connect growth rate to a precise consensus threshold. The results clarify when adding agents preserves consensus formation and quantify the long-time convergence under expanding populations, with implications for models of evolving social systems.

Abstract

In this short note we study what happens in a symmetric opinion model when we send the total interacting population to infinity as . We assume that new population enters the system with opinions that are i.i.d random vectors with finite mean and variance. We give sharp conditions on the rate of population growth that is required for convergence to a global consensus in opinions. More particularly, we show that if the total population increases at a rate , then is necessary and sufficient condition for convergence to the mean of incoming opinions, and the convergence is achieved at an algebraic rate.
Paper Structure (6 sections, 6 theorems, 38 equations)

This paper contains 6 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main results
  3. Moments and Variance estimates
  4. Moments of the model
  5. The Variance functional
  6. Proof of Theorems

Key Result

Theorem 1

We consider an increasing population $N(t):[0,\infty) \to \mathbb{N}$ that follows eq:population, for some increasing, unbounded, sequence of times $\{ t_j\}_{j=1}^\infty$. We also assume a classical solution of eq:main - eq:IC and that the incoming opinions $X_k$ are i.i.d. random vectors with fini holds. Moreover, mean variance convergence fails if

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Corollary 1
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • ...and 3 more