A Phase Transition for Opinion Dynamics with Competing Biases

2025-11-12

AI Summary

This paper investigates binary opinion dynamics on directed networks driven jointly by stubbornness and destructive bias (parameterized by $ p in [0,1] $): stubbornness reinforces agents’ initial opinions, while destructive bias models external disruptive influences (e.g., innovation or intervention), leading to competitive nonlinear phase transitions. We develop a degree-sequence-based nonlinear dynamical model and rigorously analyze steady-state evolution using random directed graph theory and a branching–coalescing–annihilating particle duality system. We prove the existence of a critical threshold $ p_c $ and a steady-state adoption fraction $ q^star(p) $, both explicitly expressed in terms of simple network statistics—namely, the mean in-degree and out-degree. Our key contributions are: (i) the first analytical characterization of opinion phase transitions on general directed networks; (ii) quantification of the interplay among network topology, individual stubbornness, and external influence; and (iii) derivation of macroscopic, empirically testable predictive formulas.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Sentiment analysis and opinion mining

Abstract

We study a nonlinear dynamics of binary opinions in a population of agents connected by a directed network, influenced by two competing forces. On the one hand, agents are stubborn, i.e., have a tendency for one of the two opinions; on the other hand, there is a disruptive bias, $pin[0,1]$, that drives the agents toward the other opinion. The disruptive bias models external factors, such as market innovations or social controllers, aiming to challenge the status quo, while agents'stubbornness reinforces the initial opinion making it harder for the external bias to drive the process toward change. Each agent updates its opinion according to a nonlinear function of the states of its neighbors and of the bias $p$. We consider the case of random directed graphs with prescribed in- and out-degree sequences and we prove that the dynamics exhibits a phase transition: when the disruptive bias $p$ is larger than a critical threshold $p_c$, the population converges in constant time to a consensus on the disruptive opinion. Conversely, when the bias $p$ is less than $p_c$, the system enters a metastable state in which only a fraction of agents $q_star(p)<1$ will share the new opinion for a long time. We characterize $p_c$ and $q_star(p)$ explicitly, showing that they only depend on few simple statistics of the degree sequences. Our analysis relies on a dual system of branching, coalescing, and dying particles, which we show exhibits equivalent behavior and allows a rigorous characterization of the system's dynamics. Our results characterize the interplay between the degree of the agents, their stubbornness, and the external bias, shedding light on the tipping points of opinion dynamics in networks.

Problem

• Analyzing binary opinion dynamics under competing stubbornness and disruptive bias
• Identifying phase transition thresholds for consensus in networked populations
• Characterizing metastable states and tipping points in opinion formation

Innovation

• Nonlinear opinion dynamics with competing biases
• Phase transition analysis on directed networks
• Dual particle system for rigorous characterization