Phase Transitions in Biased Opinion Dynamics with 2-choices Rule

TL;DR

This paper analyzes how bias towards a superior opinion influences the speed of reaching consensus in a network, revealing phase transitions based on bias strength and initial conditions.

Contribution

It introduces a model of opinion dynamics with bias, characterizes the critical bias threshold, and quantifies the consensus time scaling in fully connected networks.

Findings

Consensus time scales as Θ(n log n) for high bias levels.

Low bias levels require a high initial proportion of the superior opinion for fast consensus.

Otherwise, the convergence time is exponential in network size.

Abstract

We consider a model of binary opinion dynamics where one opinion is inherently 'superior' than the other and social agents exhibit a 'bias' towards the superior alternative. Specifically, it is assumed that an agent updates its choice to the superior alternative with probability $\alpha >0$ irrespective of its current opinion and the opinions of the other agents. With probability $1-\alpha$ it adopts the majority opinion among two randomly sampled neighbours and itself. We are interested in the time it takes for the network to converge to a consensus state where all the agents adopt the superior alternative. In a fully connected network of size $n$, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $\Theta(n \log n)$ when the bias parameter $\alpha$ is sufficiently high, i.e., $\alpha > \alpha_c$ where $\alpha_c$ is a threshold parameter that is uniquely characterised. When the bias is low, i.e., when $\alpha \in (0,\alpha_c]$, we show that the same rate of convergence can only be achieved if the initial proportion of agents with the superior opinion is above certain threshold $p_c(\alpha)$. If this is not the case, then we show that the network takes $\Omega(\exp(\Theta(n)))$ time on average to reach consensus. Through numerical simulations we observe similar behaviour for other classes of graphs.