Giacomo Como,
Laboratory for Information and Decision Systems,
Massachusetts Institute of Technology

Title: Persistent disagreement and scaling limits for continuous opinion 
dynamics systems

Abstract:
Continuous opinion dynamics systems have been recently proposed to model 
the emergence of global behaviors in networks, with applications in 
biology, social sciences, and engineering. Agents, belonging to a large 
finite population, update their vector-valued opinions as a result of 
pairwise random interactions, and possibly under the influence of an 
external field. The interaction between two agents is typically 
attractive and its strength may depend both on the agents' identities 
(through an underlying network structure), as well as on their current 
opinions. While most of the literature has focused on conditions for the 
achievement of a consensus among the agents, there is a quest, 
especially in social applications, for models explaining the persistence 
of disagreement which is typically observed in practice.

In this talk, I will first discuss a simple gossip model in which a 
subset of `stubborn' agents, to be thought of as `zealots', or sources 
of influence, never change their opinion. This model shows quite a rich 
behavior, as the opinions keep on oscillating ergodically, and approach 
a stationary distribution, whose moments can be related to the hitting 
probabilities of a dual random walk. An analysis of such stationary 
opinion distribution reveals that, for most random networks of interest, 
the amplitude of oscillations and mutual disagreement does not vanish in 
the limit of large population.

I will then consider scaling limits for opinion dynamics systems, 
showing that, as the population size increases, the trajectory of the 
empirical opinion density concentrates, at an exponential rate, around 
the solution of an ordinary differential equation in the space of 
probability measures. Properties of this ODE will be discussed. Its 
solution converges in time to an asymptotic measure, which, in the 
important special case of bounded confidence models, turns out to be a 
convex combination of delta measures whose number depends on the initial 
condition, as well as on the confidence threshold.

This a joint work with Fabio Fagnani, of Politecnico di Torino, and 
Daron Acemoglu and Asuman Ozdaglar, of MIT.