Convergence and consensus analysis of a class of best-response opinion dynamics

TL;DR

The paper studies a parametric class of best-response opinion dynamics where each agent minimizes a social-cost $u_i(z;x)=\sum_j w_{ij}|z-x_j|^{α}$, introducing the BROD update $x(k+1)=β x(k)+(1-β) BR^α(x(k);W)$ with inertia $β$. It proves convergence for $α>1$ and identifies a graph-structure condition (a unique globally-reachable sink in the condensation digraph) under which consensus is achieved, while for $α<1$ convergence is not guaranteed and a counterexample is provided along with a sufficient weight-based condition for convergence. Theoretical results are complemented by simulations on small-world networks showing that network structure and the exponent $α$ jointly influence consensus probabilities and opinion diversity, with larger $α$ reducing consensus likelihood in moderately random networks. The work unifies and extends known models (DeGroot at $α=2$ and weighted median at $α=1$), clarifying when consensus arises and guiding design of opinion dynamics on networks.

Abstract

Opinion dynamics aims to understand how individuals' opinions evolve through local interactions. Recently, opinion dynamics have been modeled as network games, where individuals update their opinions in order to minimize the social pressure caused by disagreeing with others. In this paper, we study a class of best response opinion dynamics introduced by Mei et al., where a parameter $α> 0$ controls the marginal cost of opinion differences, bridging well-known mechanisms such as the DeGroot model ($α= 2$) and the weighted-median model ($α= 1$). We conduct theoretical analysis on how different values of $α$ affect the system's convergence and consensus behavior. For the case when $α> 1$, corresponding to increasing marginal costs, we establish the convergence of the dynamics and derive graph-theoretic conditions for consensus formation, which is proved to be similar to those in the DeGroot model. When $α< 1$, we show via a counterexample that convergence is not always guaranteed, and we provide sufficient conditions for convergence and consensus. Additionally, numerical simulations on small-world networks reveal how network structure and $α$ together affect opinion diversity.

Figures (3)

• Figure 1: The system describled in Example \ref{['example:not-converge']} oscillates at $\alpha=0.5,\beta=0.4$.
• Figure 2: Empirical analysis of simulation results for $\alpha$. Panel (a) shows the relationship between Proportion of Consensus and $\alpha$. The horizontal colored bars indicate the consensus probability, while the vertical ranges of the colored rectangles are the associated 95% confidence intervals, computed by the binomial distribution methodMB:15. Panel (b) shows the relationship between Opinion Diversity and $\alpha$. The horizontal bars indicate the mean of opinion diversity, while the vertical ranges of the rectangles indicate the standard deviation of opinion diversity. For different values of inertia coefficient $\beta$, the results are qualitatively similar.
• Figure 3: Relationship between Opinion Diversity and $\beta$. The results illustrated in this figure share a similar interpretation with those in Figure \ref{['fig:alpha']}. For different values of rewiring probability $p$, the results are qualitatively similar.

Theorems & Definitions (15)

• Definition 1: Best-Response Operator
• Definition 2: Best-Response Opinion Dynamics
• Definition 3: Positive function
• Definition 4: Type-K order-preserving
• Definition 5: Sub-homogeneous
• Lemma 1: deplano2020nonlinear,Theorem 13
• Theorem 2
• proof
• Theorem 3
• proof