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Modeling of Rumor Propagation in Large Populations with Network via Graphon Games

Huaning Liu, Gokce Dayanikli

TL;DR

The SKIR model is extended and individual controls and weighted interactions with other agents to have controlled dynamics are implemented to understand how rumor propagates in large populations that are interacting on a network and how different policies affect the spread.

Abstract

In this paper, we propose a graphon game model to understand how rumor (such as fake news) propagates in large populations that are interacting on a network and how different policies affect the spread. We extend the SKIR model that is used to model rumor propagation and implement individual controls and weighted interactions with other agents to have controlled dynamics. The agents aim to minimize their own expected costs non-cooperatively. We give the finite player game model and the limiting graphon game model to approximate the Nash equilibrium in the population. We give the graphon game Nash equilibrium as a solution to a continuum of ordinary differential equations (ODEs) and give existence results. Finally, we give a numerical approach and analyze examples where we use piecewise constant graphon.

Modeling of Rumor Propagation in Large Populations with Network via Graphon Games

TL;DR

The SKIR model is extended and individual controls and weighted interactions with other agents to have controlled dynamics are implemented to understand how rumor propagates in large populations that are interacting on a network and how different policies affect the spread.

Abstract

In this paper, we propose a graphon game model to understand how rumor (such as fake news) propagates in large populations that are interacting on a network and how different policies affect the spread. We extend the SKIR model that is used to model rumor propagation and implement individual controls and weighted interactions with other agents to have controlled dynamics. The agents aim to minimize their own expected costs non-cooperatively. We give the finite player game model and the limiting graphon game model to approximate the Nash equilibrium in the population. We give the graphon game Nash equilibrium as a solution to a continuum of ordinary differential equations (ODEs) and give existence results. Finally, we give a numerical approach and analyze examples where we use piecewise constant graphon.

Paper Structure

This paper contains 8 sections, 3 theorems, 33 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model
  3. Finite Player Model
  4. Graphon Game Model
  5. Main Theoretical Results
  6. Example and Numerical Results
  7. Conclusions and Future Work
  8. Detailed proof of Theorem \ref{['existencetheorem']}

Key Result

Theorem III.2

Under Assumption assu:fbode_nash.(i), the graphon game Nash equilibrium control profile is written as $\hat{\alpha}^x_t = \hat{\alpha}^x(t,e,(z)_{\textbf{K},\textbf{I}}, u^x(t,\cdot))=:\hat{\phi}^x(t,e)$ for all $x\in I, t\in[0,T]$ where if the couple $(\boldsymbol{u},\boldsymbol{p})$ solves the following FBODE system:

Figures (5)

  • Figure 1: Diagram of the SKIR Model Transitions
  • Figure 2: Policy 0 vs. Policy 1 w.r.t ages
  • Figure 3: Policy 0 vs. Policy 2 w.r.t ages
  • Figure 4: Scheme 0 vs. Scheme 2
  • Figure 5: Scheme 0 vs. Scheme 4

Theorems & Definitions (8)

  • Definition II.1
  • Definition II.2
  • Theorem III.2
  • proof
  • Theorem III.3: Existence of Nash
  • proof
  • Lemma I.1
  • proof