Voter and Majority Dynamics with Biased and Stubborn Agents
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Voter and Majority Dynamics with Biased and Stubborn Agents Arpan Mukhopadhyay1
· Ravi R. Mazumdar2 · Rahul Roy3
Received: 29 March 2020 / Accepted: 12 August 2020 © The Author(s) 2020
Abstract We study binary opinion dynamics in a fully connected network of interacting agents. The agents are assumed to interact 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. We focus on the scenario where the agents are biased towards one of the opinions called the preferred opinion. Using suitably constructed branching processes, we show that under both rules the mean time to reach consensus is Θ(log N ), where N is the number of agents in the network. Furthermore, under the majority rule model, we show that consensus can be achieved on the preferred opinion with high probability even if it is initially the opinion of the minority. We also study the majority rule model when stubborn agents with fixed opinions are present. We find that the stationary distribution of opinions in the network in the large system limit using mean field techniques. Keywords Opinion dynamics · Consensus · Voter model · Majority rule · Mean field · Metastability · Branching processes Mathematics Subject Classification 68Q87 · 68W20 · 68W40
Communicated by Dheepak Dhar.
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Arpan Mukhopadhyay [email protected] Ravi R. Mazumdar [email protected] Rahul Roy [email protected]
1
Department of Computer Science, University of Warwick, Coventry, UK
2
Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Canada
3
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi, India
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A. Mukhopadhyay et al.
1 Introduction The social learning literature [4,12,13,29] studies how social agents, interacting under simple rules, learn the true utilities of their choices, opinions or technologies over time. In this context, the two central questions we study are: (1) Can social agents learn/adopt the better technology/opinion through simple rules of interactions and if so, how fast? and (2) What are the effects of the presence of stubborn agents (having fixed opinions) on the dynamics opinion diffusion? We consider a setting where the choices available to each agent are binary and are represented by {0} and {1} [2,4]. These are referred to as opinions of the agents. The interactions among the agents are modelled using two simple rules: the voter rule [7,10,19] and the majority rule [5,6,11,21]. In the voter rule, an agent randomly samples one of its neighbours at an instant when it decides to update its opinion. The updating agent then adopts the opinion of the sampled neighbour. This simple rule captures the tendency of an individual to mimic other individuals in the society. In the majority rule, instead of sampling a single agent, an updating agent samples 2K (K ≥ 1) neighbours and adopts t
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