A Phase Transition for Opinion Dynamics with Competing Biases

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.