Continuous Social Networks

2024-07-16

arXiv.org

AI Summary

This paper investigates consensus formation in continuous population opinion dynamics and the existence and structure of Nash equilibria in two-group lobbying games. Methodologically, it introduces a continuum social network model based on directed graphons (DiKernels), rigorously extending the DeGroot model to infinite-agent settings and establishing the first limit theory framework for continuous social networks. By integrating functional analysis, stochastic graphon limit theory, and game theory, the work develops a cross-dimensional matrix comparison paradigm within DiKernel space, enabling a rigorous proof of uniform convergence from discrete to continuous models. Key contributions include: (i) verifiable sufficient conditions for consensus emergence; (ii) existence proof and analytical characterization of Nash equilibria in the two-group lobbying game; and (iii) asymptotic equivalence between equilibria in continuous and discrete formulations. These results provide novel theoretical tools for large-scale social influence modeling and strategic interaction analysis.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction

Application Category

• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We develop an extension of the classical model of DeGroot (1974) to a continuum of agents when they interact among them according to a DiKernel. We show that, under some regularity assumptions, the continuous model is the limit case of the discrete one. Additionally, we establish sufficient conditions for the emergence of consensus. We provide some applications of these results. First, we establish a canonical way to reduce the dimensionality of matrices by comparing matrices of different dimensions in the space of DiKernels. Then, we develop a model of Lobby Competition where two lobbies compete to bias the opinion of a continuum of agents. We give sufficient conditions for the existence of a Nash Equilibrium and study their relation with the equilibria of discretizations of the game. Finally, we characterize the equilibrium for a particular case of DiKernels.

Problem

• Opinion Dynamics
• Nash Equilibrium
• Interacting Populations

Innovation

• Continuous Opinion Dynamics
• Matrix Simplification under Consensus
• Nash Equilibrium Analysis