Opinion Dynamics under Voter and Majority Rule Models with Biased and Stubborn Agents

We study binary opinion dynamics in a network of social agents interacting among themselves according to one of the following rules: (1) Voter rule: An updating agent simply copies the opinion of another randomly sampled agent; (2) Majority rule: An updating agent samples multiple agents and adopts the majority opinion in the selected group. While opinion dynamics have been studied extensively under these rules, no prior work considers the effect of individual biases and stubbornness of the agents. We study opinion diffusion under the above rules both when the agents are biased and when different agents have different degrees of stubbornness. We show that if the agents are biased then under the voter rule the mean consensus time is logarithmic in the network size as opposed to being linear as seen in the case of with unbiased agents. Furthermore, the probability to reach consensus on the preferred opinion increases exponentially with the network size. Under the majority rule model, we observe a phase transition phenomenon, where consensus on the preferred opinion is achieved with high probability only when the initial fraction of agents having the preferred opinion is above a certain threshold. For the majority rule model with stubborn agents we find the stationary distribution of opinions in the network in the large system limit using mean field techniques. A. Mukhopadhyay is with the Department of Computer Science, University of Warwick, Coventry UK. (email: arpan.mukhopadhyay@warwick.ac.uk). R. R. Mazumdar is with the department of Electrical and Computer Eng., University of Waterloo, Canada (email: mazum@uwaterloo.ca). R. Roy is with the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi, India (email: rahul@isid.ac.in).