Bounded-confidence opinion models with random-time interactions

TL;DR

This work enhances bounded-confidence opinion dynamics by introducing random-time interactions through renewal processes $R(t)$ with interevent-time distributions $\\psi$, bridging to classical HK and DW models while enabling both single- and multi-process interaction schemes. It develops a framework to separate temporal randomness from intrinsic opinion dynamics via an expectation decomposition $\\mathbb{E}[f](t) = \\sum_k \\\mathbb{E}_k[f] u_k(t)$ and derives approximate master equations for time-dependent opinions, with analytical connections shown for Markovian ITDs. The study uncovers that, for single-process BCMs, ITDs primarily affect transient behavior and, under certain conditions, can yield equivalent mean dynamics to deterministic-time BCMs; in contrast, multi-process BCMs exhibit ITD- and network-structure–dependent steady states and convergence properties, particularly for non-Markovian ITDs. Computationally, the authors implement a non-Markovian Gillespie algorithm to efficiently simulate complex renewal-process interactions and demonstrate rich dynamics across diverse networks, including potential polarization or fragmentation scenarios. Overall, the paper provides a rigorous, flexible toolkit for modeling realistic temporal randomness in opinion dynamics and offers practical pathways for analyzing, simulating, and extending BCMs to empirical settings.

Abstract

In models of opinion dynamics, agents interact with each other and can change their opinions as a result of those interactions. One type of opinion model is a bounded-confidence model (BCM), in which opinions take continuous values and interacting agents compromise their opinions with each other if their opinions are sufficiently similar. In studies of BCMs, researchers typically assume that interactions between agents occur at deterministic times. This assumption neglects an inherent element of randomness in social interactions, and it is desirable to account for it. In this paper, we study BCMs on networks and allow agents to interact at random times. To incorporate random-time interactions, we use renewal processes to determine social-interaction event times, which can follow arbitrary interevent-time distributions (ITDs). We establish connections between these random-time-interaction BCMs and deterministic-time-interaction BCMs. We analyze the quantitative impact of ITDs on the transient dynamics of BCMs and derive approximate master equations for the time-dependent expectations of the BCM dynamics. We find that BCMs with Markovian ITDs have consistent statistical properties (in particular, they have the same expected time-dependent opinions) when the ITDs have the same mean but that the statistical properties of BCMs with non-Markovian ITDs depend on the type of ITD even when the ITDs have the same mean. Additionally, we numerically examine the transient and steady-state dynamics of our models with various ITDs on different networks and compare their expected order-parameter values and expected convergence times.

Figures (4)

• Figure 1: In (a)--(c), we show the sample means of the order parameter $Q$ [see \ref{['eq: Q']}] of (a) $10$, (b) $100$, and (c) $1000$ simulations of the synchronous single-process BCM \ref{['eq: HK-random']} on a 100-node complete graph. For each simulation, we draw the initial opinions from the uniform distribution on $[0,1]$. The confidence bound is $c = 0.5$, and the tolerance parameter is $\texttt{tol} = 10^{-2}$. In (d), we plot the sample means of $Q(\bm x)$ from (c) for different ITDs and their approximations using \ref{['eq: HK_approx']}. In these approximations, we use $15$ as the upper bound of $k$.
• Figure 2: Sample means of the time-dependent opinions $x_i(t)$ in asynchronous single-process BCMs \ref{['eq: DW-random']} with Dirac delta (S-Dirac) and exponential (S-Exp) ITDs and the multiple-process BCM \ref{['eq: multiple-process-BCM']} with an exponential (M-Exp) ITD for (a) 1, (b) 100, (c) 1000, and (d) 2000 simulations. All simulations have the same initial opinions, which we draw uniformly at random from $[0,1]$. We generate one directed 25-node $G(N,p)$ ER graph with connection probability $p = 0.5$, and we run all simulations on this ER graph. The confidence bound is $c = 0.4$.
• Figure 3: Sample means of the order parameter $Q$ [see \ref{['eq: Q']}] versus time for single-process BCMs \ref{['eq: DW-random']} with Dirac delta (S-Dirac) and exponential (S-Exp) ITDs and for multiple-process BCMs \ref{['eq: multiple-process-BCM']} with exponential (M-Exp), gamma (M-Gam), and uniform (M-Uni) ITDs on (a,g) a complete graph, (b,h) directed $G(N,p)$ ER graphs with $p = 0.4$, (c,i) directed $G(N,p)$ ER graphs with $p = 0.1$, (d) symmetric and directed Chung--Lu graphs, and (e) directed SBM graphs with two communities. In (f), we show a magnification of (e). The ITD mean is $\mu = 0.01$. The confidence bound is $c = 0.5$ in panels (a)--(f) and is $c = 0.3$ in panels (g)--(i). We compute the mean of $Q$ using 3000 BCM simulations. We draw the initial opinions uniformly at random from $[0,1]$ for each simulation, and we generate a new random graph for each simulation.
• Figure 4: Normalized histograms of the convergence times of several of the simulations in Figure \ref{['fig: orderQ']}. The vertical lines indicate the mean convergence times of the BCMs on (a) a complete graph, (b) directed $G(N,p)$ ER graphs with $p = 0.4$, (c) directed $G(N,p)$ ER graphs with $p = 0.1$, (d) symmetric and directed Chung--Lu graphs, and (e) directed SBM graphs with two communities. In panels (d) and (e), which show results for graphs with heterogeneous degree distributions, the variances of the convergence times are larger than those in panels (a)--(c).