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Bounded-Confidence Models of Opinion Dynamics with Adaptive Confidence Bounds

Grace J. Li, Jiajie Luo, Mason A. Porter

TL;DR

This work introduces adaptive, edge-level confidence bounds in bounded-confidence models of opinion dynamics, generalizing the HK (synchronous) and DW (asynchronous) frameworks to incorporate time-varying per-dyad receptiveness. The authors prove that dyadic bounds converge to 0 or 1 and that effective graphs eventually align with nodes in the same limit opinion cluster, with adaptive HK showing stronger guarantees than adaptive DW due to synchronous updates. Numerical experiments across ER, SBM, and real networks reveal that adaptive BCMs reduce major opinion fragmentation and slow convergence relative to nonadaptive baselines, while enabling cases where adjacent nodes share a limit opinion despite becoming mutually unreceptive. The results illuminate how adaptive trust-like mechanisms reshaping inter-agent receptiveness can produce richer clustering structures and inform understanding of consensus formation in complex social networks.

Abstract

People's opinions change with time as they interact with each other. In a bounded-confidence model (BCM) of opinion dynamics, individuals (which are represented by the nodes of a network) have continuous-valued opinions and are influenced by neighboring nodes whose opinions are sufficiently similar to theirs (i.e., are within a confidence bound). In this paper, we formulate and analyze discrete-time BCMs with heterogeneous and adaptive confidence bounds. We introduce two new models: (1) a BCM with synchronous opinion updates that generalizes the Hegselmann--Krause (HK) model and (2) a BCM with asynchronous opinion updates that generalizes the Deffuant--Weisbuch (DW) model. We analytically and numerically explore our adaptive BCMs' limiting behaviors, including the confidence-bound dynamics, the formation of clusters of nodes with similar opinions, and the time evolution of an "effective graph", which is a time-dependent subgraph of a network with edges between nodes that {are currently receptive to each other.} For a variety of networks and a wide range of values of the parameters that control the increase and decrease of confidence bounds, we demonstrate numerically that our adaptive BCMs result in fewer major opinion clusters and longer convergence times than the baseline (i.e., nonadaptive) BCMs. We also show that our adaptive BCMs can have adjacent nodes that converge to the same opinion but are not {receptive to each other.} This qualitative behavior does not occur in the associated baseline BCMs.

Bounded-Confidence Models of Opinion Dynamics with Adaptive Confidence Bounds

TL;DR

This work introduces adaptive, edge-level confidence bounds in bounded-confidence models of opinion dynamics, generalizing the HK (synchronous) and DW (asynchronous) frameworks to incorporate time-varying per-dyad receptiveness. The authors prove that dyadic bounds converge to 0 or 1 and that effective graphs eventually align with nodes in the same limit opinion cluster, with adaptive HK showing stronger guarantees than adaptive DW due to synchronous updates. Numerical experiments across ER, SBM, and real networks reveal that adaptive BCMs reduce major opinion fragmentation and slow convergence relative to nonadaptive baselines, while enabling cases where adjacent nodes share a limit opinion despite becoming mutually unreceptive. The results illuminate how adaptive trust-like mechanisms reshaping inter-agent receptiveness can produce richer clustering structures and inform understanding of consensus formation in complex social networks.

Abstract

People's opinions change with time as they interact with each other. In a bounded-confidence model (BCM) of opinion dynamics, individuals (which are represented by the nodes of a network) have continuous-valued opinions and are influenced by neighboring nodes whose opinions are sufficiently similar to theirs (i.e., are within a confidence bound). In this paper, we formulate and analyze discrete-time BCMs with heterogeneous and adaptive confidence bounds. We introduce two new models: (1) a BCM with synchronous opinion updates that generalizes the Hegselmann--Krause (HK) model and (2) a BCM with asynchronous opinion updates that generalizes the Deffuant--Weisbuch (DW) model. We analytically and numerically explore our adaptive BCMs' limiting behaviors, including the confidence-bound dynamics, the formation of clusters of nodes with similar opinions, and the time evolution of an "effective graph", which is a time-dependent subgraph of a network with edges between nodes that {are currently receptive to each other.} For a variety of networks and a wide range of values of the parameters that control the increase and decrease of confidence bounds, we demonstrate numerically that our adaptive BCMs result in fewer major opinion clusters and longer convergence times than the baseline (i.e., nonadaptive) BCMs. We also show that our adaptive BCMs can have adjacent nodes that converge to the same opinion but are not {receptive to each other.} This qualitative behavior does not occur in the associated baseline BCMs.
Paper Structure (40 sections, 14 theorems, 66 equations, 13 figures, 5 tables)

This paper contains 40 sections, 14 theorems, 66 equations, 13 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Organization of our paper
  4. Baseline and adaptive BCMs
  5. The HK model
  6. Our HK model with adaptive confidence bounds
  7. The DW model
  8. Our DW model with adaptive confidence bounds
  9. Theoretical results
  10. Adaptive-confidence HK model
  11. Confidence-bound analysis
  12. Effective-graph analysis
  13. Adaptive-confidence DW model
  14. Confidence-bound analysis
  15. Effective-graph analysis
  16. ...and 25 more sections

Key Result

Theorem 1

\newlabelthm:lorenzthm0 Let $\{ A(t) \}_{t=0}^\infty \in \mathbb{R}_{\geq 0}^{N \times N}$ be a sequence of row-stochastic matrices. Suppose that each matrix satisfies the following properties: Given times $t_0$ and $t_1$ with $t_0 < t_1$, let If conditions (1)--(3) are satisfied, then there exists a time $t'$ and pairwise-disjoint classes $\mathcal{I}_{1} \cup \cdots \cup \mathcal{I}_{p} = \{1

Figures (13)

  • Figure 1: Examples of final effective graphs with $W(T_f) < 1$. We color the nodes by their initial opinion values.
  • Figure 1: The numbers of major clusters in simulations of our adaptive-confidence HK model on a 1000-node complete graph for various combinations of the BCM parameters $\gamma$, $\delta$, and $c_0$. In this and subsequent figures, we plot the mean value of our simulations for each set of BCM parameters. The bands around each curve indicate one standard deviation around the mean values. For clarity, in this figure and in subsequent figures, the vertical axes of different panels have different scales.
  • Figure 1: The numbers of major clusters in simulations of our adaptive-confidence HK model on $G(1000, p)$ ER random graphs with (A--E) $p = 0.1$ and (F--J) $p = 0.5$ for various combinations of the BCM parameters $\gamma$, $\delta$, and $c_0$.
  • Figure 1: The numbers of major clusters in simulations of (A) the baseline DW model and (B--J) our adaptive-confidence DW model on a 100-node complete graph for various combinations of the BCM parameters $\gamma$, $\delta$, $c_0$, and $\mu$. In this figure and subsequent figures, we do not use simulations in which we are unable to determine the final opinion clusters (see \ref{['tab:DW_bailout']}) to calculate the means and standard deviations. In (E), in which we show our simulations with $(\gamma, \delta) = (0.1, 0.5)$, we run all of our simulations to convergence (i.e., we ignore the bailout time) and use all of our simulations to calculate the mean numbers of major opinion clusters.
  • Figure 2: The weighted-average edge fraction $W(T_f)$ (see equation \ref{['eq:weighted_avg']}) in simulations of our adaptive-confidence HK model on a 1000-node complete graph for various combinations of the BCM parameters $\gamma$, $\delta$, and $c_0$.
  • ...and 8 more figures

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Lemma 3.3
  • Proof 3
  • Theorem 3
  • Proof 4
  • ...and 15 more