We study the nonlinear evolution 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 that drives the agents toward the opposite opinion. The disruptive bias models external factors such as market innovations or social controllers aiming to challenge the status quo, while 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 rule that takes into account the opinions of its neighbors and the strength of the disruptive bias. We focus on random directed graphs with prescribed in- and out-degree sequences and prove that the dynamics exhibits a phase transition. When the disruptive bias is stronger than a certain critical threshold, the entire population rapidly converges to a consensus on the disruptive opinion. When the bias is weaker than this threshold, the system enters a metastable state in which only a fraction of the population adopts the new opinion, and this partial adoption persists for a long time. We explicitly characterize both the critical threshold and the long-term adoption fraction, showing that they depend only on few simple statistics of the degree sequences. Our analysis relies on a dual system of coalescing, branching, and dying particles, whose behavior is equivalent 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.

Abstract Consider a graph where each of thennodes is either in state $$\mathcal {R}$$ R or $$\mathcal {B}$$ B . Herein, we analyze thesynchronousk-Majoritydynamics, where in each discrete-time round nodes simultaneously samplekneighbors uniformly at random with replacement and adopt the majority state among those of the nodes in the sample (breaking ties uniformly at random). Differently from previous work, we study the robustness of thek-Majorityinmaintaining a $$\mathcal {R}$$ R majority, when the dynamics is subject to two forms ofbiastoward state $$\mathcal {B}$$ B . The bias models an external agent that attempts to subvert the initial majority by altering the communication between nodes, with a probability of successpin each round: in the first form of bias, the agent tries to alter the communication links by transmitting state $$\mathcal {B}$$ B ; in the second form of bias, the agent tries to corrupt nodes directly by making them update to $$\mathcal {B}$$ B . Our main result shows asharp phase transitionin both forms of bias. By considering initial configurations in which every node has probability $$q \in (\frac{1}{2},1]$$ q∈(12,1] of being in state $$\mathcal {R}$$ R , we prove that for every $$k\ge 3$$ k≥3 there exists a critical value $$p_{k,q}^\star $$ pk,q⋆ such that, with high probability, the external agent is able to subvert the initial majority either in $$n^{\omega (1)}$$ nω(1) rounds, if $$p p<pk,q⋆ , or inO(1) rounds, if $$p>p_{k,q}^\star $$ p>pk,q⋆ . When $$k<3$$ k<3 , instead, no phase transition phenomenon is observed and the disruption happens inO(1) rounds for $$p>0$$ p>0 .

We analyze the binary-state (either ℛ or ℬ) k-majority dynamics in a biased communication model where nodes have some fixed probability p, independent of the dynamics, of being seen in state ℬ by their neighbors. In this setting we study how p, as well as the initial unbalance between the two states, impact on the speed of convergence of the process, identifying sharp phase transitions.