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Séminaire des équipes de recherche

Random interacting agents: Consensus and diffusion

  • Date : 18/02/2016
  • Place : 2 rue Simone Iff, 75012 Paris (ou: 41 rue du Charolais) - Salle de Réunion Edsger Dijkstra, Room C114
  • Guest(s) : Ravi R. Mazumdar, University of Waterloo, Canada

We consider a network of interacting agents where each agent exists in one of two possible states {0,1} and the agents update their states through local interactions. We assume that agents in different states
show different propensity towards updating their states( the acceptance of influence). In particular, through such a mechanism we can model the presence of stubborn agents that are not affected by interactions. The motivation is opinion formation in social networks where individuals have the opportunity to interact with other agents drawn from a large population.
We first begin with a generalized voter model where the opinion of a sampled neighbour is accepted or not with a certain probability. For the classical model we know that consensus takes place depending on the
initial opinion distribution and the time for consensus is $O(n)$. In the generalized voter model with heterogeneous propensities for being influenced (asymmetric influence) various equilibria can be shown to
exist (consensus or mixtures) depending on the initial distributions and the propensity for change and the consensus time is now changed to O(\log n).
We then consider a random graph version of majority rule based opinion dynamics. This is modeled as follows: each node in state i in {0,1} considers updating its state with probability p_i and retains its state
with probability 1-p_i at the points of a unit rate Poisson process. In case it decides to update its state it does so by following a majority rule based on randomly sampling two or more other nodes from the network and observing their states. If there is no clear majority, then the agent decides to switch its state with probability p_f and retain it with probability 1-p_f. We analyze the evolution of the system in the
mean field limit. We show that when p_f \neq 0 the equilibrium fraction of agents in each state does not depend on the initial population. We further analyze the case where p_f=0 that leads to consensus. We show that the consensus time scales as log N, where N denotes the size of the network. We also study the situation where there are stubborn agents who do not change their opinion. This can result in metastability in that the network can oscillate between multiple stable equilibria.
The analysis is via mean field methods.        
This work is with Arpan Mukhopadhyay (Waterloo) and Rahul Roy (ISI, New Delhi).

Keywords: Séminaire Dyogene RAP