Reaching consensus in heterogeneous random opinion dynamics

In economics and social sciences, opinion dynamics is the study of how opinions form in a large population as the outcome of individuals interactions. The major issue in the field is that of investigating the conditions on the type of interactions and on the underlying social structure which ensure convergence to consensus. With the aim of quantitatively analyzing properties and timescales of this convergence, a rich class of mathematical models of opinion exchange has been proposed and studied since the pioneering works of French Jr. and DeGroot from the last century. In more recent years, the increasing complexity of the phenomena studied led to an intense activity around noisy opinion dynamics. In such stochastic models, typically, individuals are vertices of a graph, their opinions are values attached to the corresponding vertex, and individuals update their opinions according to some random local rules. This additional feature -- the noise -- offers a natural connection with other stochastic interacting particle systems from statistical mechanics. Nevertheless, despite the several analogies with statistical mechanics models, most social dynamics examples share the distinguishing feature of reaching equilibrium at a single absorbing state -- the consensus configuration -- rather than at a non-singular steady state. Due to this singularity, most probabilistic techniques from Markov chain mixing analysis break down. Despite the growing interest in developing new techniques for these challenging stochastic models, most recent rigorous quantitative results are mostly concerned with regular and homogeneous geometries: more realistic heterogeneous systems in which individuals, for instance, act differently and let their social connections evolve in the course of time remain largely uncovered. Building on our recent results on detailed and scaling properties of interacting systems in static and dynamic random environment enjoying a form of duality, we structure this research proposal around a family of unfair opinion formation models, generalizing Aldous averaging process. The project will have three main focuses. Firstly, we provide sharp estimates of various distances from equilibrium, and aim to prove a version of Aldous spectral gap identity in this degenerate context. Secondly, we investigate scaling limits of non-equilibrium fluctuations in random environment. Thirdly, we analyze the mixing behavior for these averaging processes on fluctuating in-time segments. The goal is to prove that convergence does not occur abruptly in the large population limit, linking it to the mixing of corresponding simpler dual stochastic processes. Ultimately, our aim is that of initiating a comprehensive treatment of rigorous results and probabilistic techniques on these and related opinion formation models in a highly heterogeneous context.

In economics and social sciences, opinion dynamics is the study of how opinions form in a large population as the outcome of individuals' interactions. The major issue in the field is that of investigating the conditions on the type of interactions and on the underlying social structure which ensure convergence to consensus. With the aim of quantitatively analyzing properties and timescales of this convergence, a rich class of mathematical models of opinion exchange has been proposed and studied since the pioneering works of French Jr. and DeGroot from the last century. In more recent years, the increasing complexity of the phenomena studied led to an intense activity around "noisy" opinion dynamics. In such stochastic models, typically, individuals are vertices of a graph, their opinions are values attached to the corresponding vertex, and individuals update their opinions according to some random local rules. This additional feature -- the noise -- offers a natural connection with other stochastic interacting particle systems from statistical mechanics. Nevertheless, despite the several analogies with statistical mechanics' models, most social dynamics' examples share the distinguishing feature of reaching equilibrium at a single absorbing state -- the consensus configuration -- rather than at a non-singular steady state. Due to this singularity, most probabilistic techniques from Markov chain mixing analysis break down. Despite the growing interest in developing new techniques for these challenging stochastic models, most recent rigorous quantitative results are mostly concerned with regular and homogeneous geometries: more realistic heterogeneous systems in which individuals, for instance, act differently and let their social connections evolve in the course of time remain largely uncovered. Building on our recent results on detailed and scaling properties of interacting systems in static and dynamic random environment enjoying a form of duality, we structure this research proposal around a family of "unfair" opinion formation models, generalizing Aldous' averaging process. The project will have three main focuses. Firstly, we provide sharp estimates of various distances from equilibrium, and aim to prove a version of Aldous' spectral gap identity in this degenerate context. Secondly, we investigate scaling limits of non-equilibrium fluctuations in random environment. Thirdly, we analyze the mixing behavior for these averaging processes on fluctuating in-time segments. The goal is to prove that convergence does not occur abruptly in the large population limit, linking it to the mixing of corresponding simpler "dual" stochastic processes. Ultimately, our aim is that of initiating a comprehensive treatment of rigorous results and probabilistic techniques on these and related opinion formation models in a highly heterogeneous context.