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Opinion dynamics on signed graphs and graphons: Beyond the piece-wise constant case (Extended version)

Raoul Prisant, Federica Garin, Paolo Frasca

TL;DR

It is shown that the graphon dynamics well approximate the dynamics on large graphs that converge to a graphon, and this result applies to large random graphs that are sampled according to a graphon.

Abstract

In this paper we make use of graphon theory to study opinion dynamics on large undirected networks. The opinion dynamics models that we take into consideration allow for negative interactions between the individuals, i.e. competing entities whose opinions can grow apart. We consider both the repelling model and the opposing model that are studied in the literature. We define the repelling and the opposing dynamics on graphons and we show that their initial value problem's solutions exist and are unique. We then show that the graphon dynamics well approximate the dynamics on large graphs that converge to a graphon. This result applies to large random graphs that are sampled according to a graphon. All these facts are illustrated in an extended numerical example.

Opinion dynamics on signed graphs and graphons: Beyond the piece-wise constant case (Extended version)

TL;DR

It is shown that the graphon dynamics well approximate the dynamics on large graphs that converge to a graphon, and this result applies to large random graphs that are sampled according to a graphon.

Abstract

In this paper we make use of graphon theory to study opinion dynamics on large undirected networks. The opinion dynamics models that we take into consideration allow for negative interactions between the individuals, i.e. competing entities whose opinions can grow apart. We consider both the repelling model and the opposing model that are studied in the literature. We define the repelling and the opposing dynamics on graphons and we show that their initial value problem's solutions exist and are unique. We then show that the graphon dynamics well approximate the dynamics on large graphs that converge to a graphon. This result applies to large random graphs that are sampled according to a graphon. All these facts are illustrated in an extended numerical example.
Paper Structure (10 sections, 6 theorems, 44 equations, 4 figures)

This paper contains 10 sections, 6 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Opinion dynamics on graphs and graphons
  3. Opinion dynamics on signed networks
  4. Graphs and graphons
  5. Sampled graphs
  6. Opinion dynamics on signed graphons
  7. Convergence of solutions
  8. Convergence of solutions for sampled graphs
  9. Simulations
  10. Conclusion

Key Result

Theorem 1

Assume that $W\in L^{\infty}(I^2)$ and $g\in L^{\infty}(I)$. Then there exists a unique solution of DynGraphonRep, $u\in C^1(\mathbb{R}^+;L^{\infty}(I))$. The same holds for DynGraphonOpp.

Figures (4)

  • Figure 1: The graphon used in the simulations.
  • Figure 2: Comparison between solutions on graphs with $n=100$ (left) and $n=1000$ (center) and on the graphon (right) for the repelling dynamics, with initial condition $g(x)=x-\frac{1}{2}$.
  • Figure 3: Comparison between solutions on graphs with $n=100$ (left) and $n=1000$ (center) and on the graphon (right) for the opposing dynamics, with initial condition $g(x)=x-\frac{1}{2}$.
  • Figure 4: Evolution in time of $\|(u_n-u)(\cdot,t)\|_2$ for the repelling (left) and the opposing dynamics (right).

Theorems & Definitions (14)

  • Definition 1: Sampled signed graphs
  • Definition 2: Latent variables
  • Theorem 1: Existence and uniqueness
  • proof
  • Lemma 1: Graph dynamics as graphon dynamics
  • proof
  • Theorem 2: Convergence error estimate
  • proof
  • Remark 1: Choice of the convergence norm
  • Proposition 1: Convergence of sampled initial conditions
  • ...and 4 more