Majority Rule Based Opinion Dynamics with Biased and Stubborn Agents

In this paper, we investigate the impact of majority-rule based random interactions among agents in a large social network on the diffusion of opinions in the network. Opinion of each agent is assumed to be a binary variable taking values in the set {0, 1}. Interactions among agents are modeled using the majority rule, where each agent updates its opinion at random instants by adopting the 'majority' opinion among a group of randomly sampled agents. We investigate two scenarios that respectively incorporate `bias' of the agents towards a specific opinion and stubbornness of some of the agents in the majority rule dynamics. For the first scenario, where all the agents are assumed to be 'biased' towards one of the opinions, it is shown that the agents reach a consensus on the preferred opinion (with high probability) only if the initial fraction of agents having the preferred opinion is above a certain threshold. Furthermore, the mean time taken to reach the consensus is shown to be logarithmic in the network size. In the second scenario, where the presence of 'stubborn' agents, who never update their opinions, is assumed, we characterize the equilibrium distribution of opinions of the non-stubborn agents using mean field techniques. The mean field limit is shown to have multiple stable equilibrium points which leads to a phenomenon known as metastability.