How opinions crystallise: an analysis of polarisation in the voter model

We address the phenomenon of sedimentation of opinions in networks. We investigate how agents who never change their minds ("stubborn") can influence the opinion of a social group and foster the formation of polarised communities. We study the voter model in which users are divided in two camps and repeatedly update their opinions based on others they connect with. Assuming a proportion of the agents are stubborn, the distribution of opinions reaches an equilibrium. We give novel formulas based on Markov Chain analysis to compute the distribution of opinions at any time and speed of convergence to stationary equilibrium. Theoretical results are supported by numerical experiments on synthetic data, and we discuss a strategy to mitigate the polarisation phenomenon.