Reaching consensus in heterogeneous random opinion dynamics
Reaching consensus in heterogeneous random opinion dynamics
Disciplines
Computer Sciences (5%); Mathematics (65%); Physics, Astronomy (30%)
Keywords
-
Interacting particle systems,
Heterogeneous Social Dynamics,
Scaling Limits In Random Environment,
Convergence Rates To Equilibrium,
Mixing Times,
Averaging Process
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.
Research Output
- 1 Citations
- 19 Publications
-
2022
Title On the meeting of random walks on random DFA DOI 10.48550/arxiv.2204.02827 Type Other Author Quattropani M -
2021
Title Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domains DOI 10.48550/arxiv.2112.14196 Type Preprint Author Portinale L -
2022
Title Cutoff for the Averaging process on the hypercube and complete bipartite graphs DOI 10.48550/arxiv.2212.08870 Type Other Author Caputo P -
2022
Title On the meeting of random walks on random DFA Type Other Author Quattropani -
2022
Title On the meeting of random walks on random DFA Type Other Author Quattropani -
2022
Title Cutoff for the Averaging process on the hypercube and complete bipartite graphs Type Other Author Caputo -
2022
Title Cutoff for the Averaging process on the hypercube and complete bipartite graphs Type Other Author Caputo -
2023
Title Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models Type Other Author Chiarini -
2023
Title Mixing of the Averaging process and its discrete dual on finite-dimensional geometries Type Journal Article Author Quattropani Journal Annals of Applied Probability -
2023
Title Mixing of the Averaging process and its discrete dual on finite-dimensional geometries Type Journal Article Author Quattropani Journal Annals of Applied Probability -
2023
Title Full Gamma-expansion of reversible Markov chains level two large deviations rate functionals Type Other Author Landim -
2023
Title Full Gamma-expansion of reversible Markov chains level two large deviations rate functionals Type Other Author Landim -
2023
Title Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models Type Other Author Chiarini -
2023
Title Cutoff for the averaging process on the hypercube and complete bipartite graphs DOI 10.1214/23-ejp993 Type Journal Article Author Caputo P Journal Electronic Journal of Probability -
2023
Title Mixing of the averaging process and its discrete dual on finite-dimensional geometries DOI 10.1214/22-aap1838 Type Journal Article Author Quattropani M Journal The Annals of Applied Probability -
2023
Title Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models DOI 10.48550/arxiv.2302.10156 Type Preprint Author Chiarini A -
2023
Title Full $Γ$-expansion of reversible Markov chains level two large deviations rate functionals DOI 10.48550/arxiv.2303.00671 Type Preprint Author Landim C -
2021
Title Mixing of the Averaging process and its discrete dual on finite-dimensional geometries DOI 10.48550/arxiv.2106.09552 Type Other Author Quattropani M -
2023
Title On the meeting of random walks on random DFA DOI 10.1016/j.spa.2023.104225 Type Journal Article Author Quattropani M Journal Stochastic Processes and their Applications Pages 104225 Link Publication