Opinion dynamics on directed complex networks

TL;DR

This work develops a directed-network model of opinion dynamics that extends classical DeGroot and Friedkin-Johnsen frameworks by incorporating vertex attributes and stochastic media signals. It establishes stationary behavior on fixed graphs (d>0) and leverages local weak convergence with strong couplings to characterize the typical stationary opinion on large random graphs, yielding explicit tree-based representations via smoothing transforms. The paper derives precise conditions for consensus and polarization, analyzes the role of memory vs. no-memory, and discusses the impact of stubborn agents and selective exposure on the stationary distribution. The results provide a rigorous connection between stochastic recursions on graphs and branching processes, offering quantitative insights for how network structure and media influence shape collective opinions with practical implications for understanding real-world polarization phenomena.

Abstract

We propose and analyze a mathematical model for the evolution of opinions on directed complex networks. Our model generalizes the popular DeGroot and Friedkin-Johnsen models by allowing vertices to have attributes that may influence the opinion dynamics. We start by establishing sufficient conditions for the existence of a stationary opinion distribution on any fixed graph, and then provide an increasingly detailed characterization of its behavior by considering a sequence of directed random graphs having a local weak limit. Our most explicit results are obtained for graph sequences whose local weak limit is a marked Galton-Watson tree, in which case our model can be used to explain a variety of phenomena, e.g., conditions under which consensus can be achieved, mechanisms in which opinions can become polarized, and the effect of disruptive stubborn agents on the formation of opinions.

Figures (7)

• Figure 1: Empirical distribution of opinions in an Erdős-Rényi graph with $G(1000, 0.03)$ with internal opinion $Q_i \sim \text{Unif}(-1,1)$ and media signals $Z_{i}^{(k)}\sim \text{Unif}(-0.03,0.03)$, independent of the vertex attributes.
• Figure 2: Empirical distribution of opinions in an Erdős-Rényi graph $G(1000, 0.03)$ with internal opinion $Q_i \sim \text{Unif}(-1,1)$, and media signals $Z_{i}^{(k)}\sim \text{Unif}\{-1,1\}$, independent of the vertex attributes.
• Figure 3: Empirical distribution of opinions in an Erdős-Rényi graph $G(1000, 0.03)$ with internal opinion $Q_i \sim \text{Unif}(-1,1)$, and media signals $Z_{i}^{(k)}\sim -1+2\text{Beta}(1,8)$, independent of the vertex attributes.
• Figure 4: Empirical distribution of opinions in an Erdős-Rényi graph $G(1000, 0.03)$ with internal opinion $Q_i \sim \text{Unif}\{-1,1\}$. The media signals are biased towards one's internal opinion by setting $(Z_{i}^{(k)} | Q_i = +1) \sim -1+2\text{Beta}(8,1)$ and $(Z_i^{(k)}| Q_i = -1) \sim -1 + 2\text{Beta}(1,8)$.
• Figure 5: Empirical distribution of opinions in an inhomogeneous random digraph with 800 regular vertices that are connected using an Erdős-Rényi graph $G(800, 0.03)$ with internal opinions $(Q_i | S_i = 0) \sim \text{Unif}\{-1,1\}$. Their media signals are biased towards one's internal opinion by setting $(Z_i^{(k)}| Q_i = +1, S_i=0)\sim -1+2\text{Beta}(8,1)$ and $(Z_i^{(k)}| Q_i = -1, S_i=0)\sim -1+2\text{Beta}(1,8)$. In addition to the 800 regular vertices, the graph has 200 bots that have zero in-degree, internal opinions $(Q_i|S_i = 1) =1$, and media signals $(Z_i^{(k)}| S_i = 1) = 1$ for all $k \geq 0$. Bots connect to regular vertices using independent Bernoulli$(0.03)$ trials.

Theorems & Definitions (22)

• Theorem 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 2
• Proposition 3
• Remark 6
• Proposition 4
• Proposition 5
• Proposition 6