A Phase Transition for Opinion Dynamics with Competing Biases

TL;DR

This paper analyzes nonlinear biased opinion dynamics on directed networks with competing forces: a disruptive external bias $p$ and agent stubbornness. The authors formulate the model on the Directed Configuration Model (DCM) and exploit a dual process, COBRAD (coalescing, branching, dying), to rigorously characterize a phase transition: for $p \ge p_c(\varrho)$ the population rapidly reaches consensus on the disruptive opinion, while for $p < p_c(\varrho)$ a metastable fraction $q_\star(p,\varrho,\lambda)$ retains the old or mixed opinion. Remarkably, the critical threshold $p_c$ and the limiting red density $q_\star$ depend only on two coarse degree statistics, $\varrho$ and $\lambda$, highlighting a universality with respect to the full degree sequence. The analysis leverages a local random-tree approximation and a dual COBRAD process to link macroscopic tipping points to microscopic network heterogeneity, offering insights into how misperceptions and network asymmetry drive consensus or persistence of disagreement. These findings illuminate tipping points in information spread, technology adoption, and misinformation on asymmetric, large-scale networks.

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, $p\in[0,1]$, 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.

Figures (4)

• Figure 1: Graphical example of the update rule: vertex $u$ samples two neighbors $v,w$; the bias is applied on $v$, that is seen as blue, but not on $w$. Vertex $u$ updates to blue as it sees blue twice (effect of bias on $v$ and $w$ is blue).
• Figure 2: Graphical representation of the opinion dynamics with competing biases (left) and of COBRAD (right). On the left, in chronological order, $y$ samples $x,z$ without the effect of the bias, $v$ samples $u,x$ with the effect of the bias on $u$, and $x$ samples $v,y$ with the effect of the bias on both. On the right, in chronologically reverse order, the particle labeled $x$ dies, the particle labeled $v$ moves to vertex $x$, and the particle labeled $y$ branches to both vertices $x$ and $z$. At time $0$ in COBRAD there are no particles labeled $x$, hence vertex $x$ in the opinion dynamics with competing biases at time $T$ has color $b$.
• Figure 3: On the $x$-axis have the value of $\varrho$, while on the $y$-axis the value of $p$. The blue (resp. red) area represents couples $(\varrho,p)$ for which we have $p> p_c(\varrho)$ (resp. $p< p_c(\varrho)$).
• Figure 4: We run the biased opinion dynamics on three DCM graphs with $n=10^4$ vertices: in the first all vertices have in- and out-degrees equal to $6$ (red); in the second, half of the vertices have in-degree $10$ and out-degree $5$, and the other way around for the other half (green); in the third, half of the vertices have in-degree $10$ and out-degree $2$, and vice versa for the second half (blue). Notice that the red and the green graph share the same value of $\varrho=\frac{1}{6}$ (hence, they have the same $p_c\approx 0.477$) while the red and the blue graph have the same number of edges (but in the blue case $\rho=\frac{13}{30}>\frac{1}{6}$). Notice also that the value of $\lambda$ is $\frac{1}{6}$, $\frac{3}{20}$ and $\frac{3}{10}$, in the blue, red and green case, respectively. The value of the critical threshold in the case of the blue graph is $p_c\approx 0.429$. On the left a simulation for $p=0.3$, which is subcritical in all the three cases; on the right a simulation for $p=0.45$, which is subcritical for the red and the green graphs, but supercritical for the blue one. The dashed lines indicate the theoretical prediction of the long-term density of red opinions, i.e., $q_\star(p,\varrho,\lambda)$. Notice that in the case $p=0.3$ (left) we have $q_\star(\frac{3}{10},\frac{1}{6},\frac{1}{6})\approx 0.7796$ (red), $q_\star(\frac{3}{10},\frac{1}{6},\frac{3}{20})\approx 0.7810$ (green) and $q_\star(\frac{3}{10},\frac{1}{6},\frac{3}{10})\approx 0.6902$ (blue); while in the case $p=0.45$ (right) we have $q_\star(\frac{9}{20},\frac{1}{6},\frac{1}{6})\approx 0.1967$ (red) and $q_\star(\frac{9}{20},\frac{1}{6},\frac{3}{20})\approx 0.1975$ (green).

Theorems & Definitions (7)

• Proposition 1
• Definition 2
• Theorem 3
• Remark 4
• Conjecture 5
• proof : Proof of Eq. \ref{['eq:claim1']}
• proof : Proof of Eq. \ref{['eq:claim2']}