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Continuous Social Networks

Julián Chitiva, Xavier Venel

TL;DR

This paper extends the classical DeGroot opinion dynamics to a continuum of agents using row-stochastic DiKernels, establishing that the continuous model arises as a limit of discrete networks under regularity and introducing the operator $\mathsf{T}(W)$ to evolve opinions. It provides a sufficient condition for consensus via $\gamma$-mixing DiKernels and links consensus to ergodic properties of an associated Markov chain, while also detailing the discrete-to-continuous and continuous-to-discrete connections and a method for dimensionality reduction. Beyond non-strategic dynamics, the authors formulate a Lobby Game in which two lobbies strategically influence the initial opinions; they prove existence of Nash equilibria under standard convexity and continuity assumptions and characterize equilibria in Uni-type DiKernel settings, with results extending to general DiKernels through $\gamma$-mixing approximations. Finally, the work demonstrates convergence and discretization results that relate continuous and finite-player games, offering practical tools to approximate large discrete networks and quantify approximation errors via $\|\cdot\|_\square$ dispersions.

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.

Continuous Social Networks

TL;DR

This paper extends the classical DeGroot opinion dynamics to a continuum of agents using row-stochastic DiKernels, establishing that the continuous model arises as a limit of discrete networks under regularity and introducing the operator $\mathsf{T}(W)$ to evolve opinions. It provides a sufficient condition for consensus via $\gamma$-mixing DiKernels and links consensus to ergodic properties of an associated Markov chain, while also detailing the discrete-to-continuous and continuous-to-discrete connections and a method for dimensionality reduction. Beyond non-strategic dynamics, the authors formulate a Lobby Game in which two lobbies strategically influence the initial opinions; they prove existence of Nash equilibria under standard convexity and continuity assumptions and characterize equilibria in Uni-type DiKernel settings, with results extending to general DiKernels through $\gamma$-mixing approximations. Finally, the work demonstrates convergence and discretization results that relate continuous and finite-player games, offering practical tools to approximate large discrete networks and quantify approximation errors via $\|\cdot\|_\square$ dispersions.

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.
Paper Structure (20 sections, 14 theorems, 34 equations, 5 figures)

This paper contains 20 sections, 14 theorems, 34 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Notation
  4. Continuous Social Dynamics
  5. Model
  6. Sufficient condition for emergence of consensus
  7. From Discrete to Continuous
  8. From Continuous to Discrete
  9. Reduction of Dimensionality
  10. Strategic Influence over Continuous Social Networks
  11. The Lobby Game
  12. Existence of Nash Equilibria
  13. Equilibrium Characterization
  14. One-player -- Uni-type DiKernel
  15. Two-players - Uni-type DiKernel
  16. ...and 5 more sections

Key Result

Proposition 1

Let $\gamma>0$ and $W$ be a DiKernel that satisfies Assumption assump:toto_mixing. Then, there exists $h:[0,1] \rightarrow \mathbb{R}^*_+.$

Figures (5)

  • Figure 1: DiKernels defined from $\widehat{W}$ and a given partition.
  • Figure 2: Dynamic of Opinions: From Discrete to Continuous.
  • Figure 3: Discretization of DiKernel W.
  • Figure 4: Diagram of the process of dimensionality reduction.
  • Figure 5: Dimensionality Reduction

Theorems & Definitions (45)

  • Definition 1: Continuous DeGroot Model
  • Remark 1
  • Proposition 1
  • Definition 2: Discrete to Continuous
  • Proposition 2
  • Example 1: Discrete to Continuous: Uniform Partition
  • Example 2: Discrete to Continuous: General Partition
  • Example 3: Opinion Dynamics: Uniform Partition
  • Example 4: Opinion Dynamics: General Partition
  • Example 5: Block-constant DiKernel to Weighted Discrete
  • ...and 35 more