Achieving consensus in networks of increasingly stubborn voters

We study opinion evolution in networks of stubborn agents discussing a sequence of issues, modeled through the so called concatenated Friedkin-Johnsen (FJ) model. It is concatenated in the sense that agents’ opinions evolve for each issue, and the final opinion is then taken as a starting point for the next issue. We consider the scenario where agents also take a vote at the end of each issue and propose a feedback mechanism from the result (based on the median voter) to the agents’ stubbornness. Specifically, agents become increasingly stubborn during issue s+1 the more they disagree with the vote at the end of issue s. We analyze this model for a number of special cases and provide sufficient conditions for convergence to consensus stated in terms of permissible initial opinion and stubbornness. In the opposite scenario, where agents become less stubborn when disagreeing with the vote result, we prove that consensus is achieved, and we demonstrate the faster convergence of opinions compared to constant stubbornness.