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Homogeneous Boltzmann-type equations on graphs: A framework for modelling networked social interactions

Andrea Tosin

Abstract

Homogeneous Boltzmann-type equations are an established tool for modelling interacting multi-agent systems in sociophysics by means of the principles of statistical mechanics and kinetic theory. A customary implicit assumption is that interactions are "all-to-all", meaning that every pair of randomly sampled agents may potentially interact. However, this legacy of classical kinetic theory, developed for collisions among gas molecules, may not be equally applicable to social interactions, which are often influenced by preferential connections between agents. In this paper, we discuss ongoing research on incorporating graph structures into homogeneous Boltzmann-type equations, thereby accounting for the "some-to-some" nature of social interactions.

Homogeneous Boltzmann-type equations on graphs: A framework for modelling networked social interactions

Abstract

Homogeneous Boltzmann-type equations are an established tool for modelling interacting multi-agent systems in sociophysics by means of the principles of statistical mechanics and kinetic theory. A customary implicit assumption is that interactions are "all-to-all", meaning that every pair of randomly sampled agents may potentially interact. However, this legacy of classical kinetic theory, developed for collisions among gas molecules, may not be equally applicable to social interactions, which are often influenced by preferential connections between agents. In this paper, we discuss ongoing research on incorporating graph structures into homogeneous Boltzmann-type equations, thereby accounting for the "some-to-some" nature of social interactions.

Paper Structure

This paper contains 8 sections, 2 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Networked multi-agent systems
  3. Outline of the basic qualitative theory
  4. Mass redistribution on the graph
  5. Trait distribution on the graph
  6. Networked interactions
  7. Formal derivation of a Boltzmann-type equation on a graph
  8. Embedding graphons in Boltzmann-type equations

Key Result

Theorem 2.1

If $\boldsymbol{\rho}_0\in\mathbb{R}^N_+$, i.e. if the initial masses of agents are non-negative in every vertex of $\mathcal{G}_N$, then $\rho_i(t)\geq 0$ for all $t>0$ and all $i=1,\,\dots,\,N$. In particular, if $\rho_{0,i}>0$, i.e. if the initial mass in a given vertex $i$ is non-zero, then $\rh

Figures (3)

  • Figure 1: Five networked multi-agent systems with binary interactions within each of them
  • Figure 2: A dense graph of interactions in a multi-agent system
  • Figure 3: A $4\times 4$ adjacency matrix and the contours of the corresponding piecewise constant function $W_4$ on the unit square $[0,\,1]^2$ (white is $0$, black is $1$)

Theorems & Definitions (2)

  • Theorem 2.1: See Loy2025
  • Theorem 2.2: See Loy2025