Continuous Social Networks

TL;DR

This paper extends the classical DeGroot social network model to a continuum of agents, establishing the continuum as a limit case, and explores conditions for consensus and applications like lobby competition and dimensionality reduction.

Contribution

It introduces a continuous agent model using DiKernels, linking it to discrete models, and applies it to analyze consensus and strategic lobbying in social networks.

Findings

Continuous model is the limit of discrete models under regularity conditions.

Sufficient conditions for consensus emergence are established.

A Nash Equilibrium exists in the lobby competition model.

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.