Applied Mathematics Colloquium: Dr. Muruhan Rathinam

Title: Dynamics of opinions

We describe recent work on a mathematical model of how individuals interact by influencing and being influenced by each other. In most generality, such models include swarming behavior of birds and insects, herding behavior of land animals as well as human social interactions. When the identity of individuals known as "agents" is maintained and the number of agents is finite, these models usually are ODEs or more generally SDEs or Markov processes. In the continuum limit and with loss of identity of agents, these models take the form of evolution equations for measures or PDEs.

We focus particularly on a model of opinion dynamics known as the Hegselmann-Krause model and its generalizations. Equilibria of such models consist of "clusters" where each cluster corresponds to a subset of agents who have the same opinion. We present some recent results on the analysis of a class of such ODE models. Our results include Lyapunov stability as well as a form of robustness. The notion of robustness studied is concerned with what happens to a system with finite number of agents in equilibrium when a new agent is introduced. In particular, we are concerned with whether the "disruption" to the system changes continuously with the weight of the new agent.

This is joint work with Dr.  Serap Tay Stamoulas.