toplogo
Resources
Sign In

Opinion Dynamics on Signed Graphs and Graphons: Extending Beyond the Piece-wise Constant Case


Core Concepts
This paper studies opinion dynamics models that allow for negative interactions between individuals, such as the repelling and opposing models, and defines their counterparts on signed graphons. The authors prove the existence and uniqueness of solutions to these dynamics on signed graphons, and provide sufficient conditions for the solutions on signed graphs to converge to solutions on signed graphons as the number of individuals goes to infinity.
Abstract
The paper makes three main contributions: It proves the existence and uniqueness of solutions to the repelling and opposing opinion dynamics models on signed graphons (Theorem 1). It provides sufficient conditions for the solutions of the opinion dynamics on graphs of size n to converge, as n goes to infinity, to the solutions of the graphon dynamics (Theorem 2). This convergence result applies when the sequence of graphs converges to a graphon. It shows that the convergence conditions in Theorem 2 apply to large random graphs sampled from signed graphons, as long as the graphon is piece-wise Lipschitz continuous (Theorem 3). The paper first recalls the repelling and opposing opinion dynamics models on signed graphs, and then defines their counterparts on signed graphons. It then proves the existence and uniqueness of solutions to the graphon dynamics (Theorem 1). Next, the paper establishes a convergence result (Theorem 2), showing that the solutions on graphs converge to the solutions on graphons, provided that the initial conditions and the graphon approximation error converge appropriately. Finally, the paper demonstrates that the convergence conditions in Theorem 2 are satisfied when the graphs are sampled from a piece-wise Lipschitz signed graphon (Theorem 3), using results on the convergence of sampled graphs. The numerical example illustrates the differences between the solutions on graphs and graphons for the repelling and opposing dynamics, and confirms the convergence of the graph solutions to the graphon solutions as the graph size increases.
Stats
None.
Quotes
None.

Deeper Inquiries

How do the properties of the signed graphon, such as the strength and distribution of positive and negative interactions, affect the long-term behavior of the opinion dynamics

The properties of the signed graphon, such as the strength and distribution of positive and negative interactions, play a crucial role in determining the long-term behavior of the opinion dynamics. In the context of antagonistic interactions, the distribution of positive and negative edges within the graphon can lead to various outcomes. Strong positive interactions within a community can promote consensus and alignment of opinions, while negative interactions between different communities can lead to polarization and divergence of opinions. The strength of these interactions influences the rate at which opinions converge or diverge over time. Additionally, the structure of the graphon, including the connectivity patterns and weights assigned to edges, can impact the stability and resilience of the opinion dynamics. Overall, the properties of the signed graphon shape the dynamics of opinion evolution and the emergence of consensus or divergence within the network.

What are the implications of the convergence results for the design and control of large-scale social networks with antagonistic interactions

The convergence results presented in the paper have significant implications for the design and control of large-scale social networks with antagonistic interactions. By demonstrating that the dynamics on sampled graphs converge to the dynamics on graphons as the number of nodes increases, the study provides insights into how to model and analyze opinion dynamics on complex networks. These results offer a framework for understanding the behavior of opinion dynamics in large networks and can guide the development of strategies for managing and influencing opinions within such networks. The ability to approximate dynamics on large graphs with dynamics on graphons enables researchers and practitioners to study and predict the long-term behavior of social networks, identify key factors influencing opinion formation, and design interventions to steer the network towards desired outcomes.

Can the techniques developed in this paper be extended to study opinion dynamics on time-varying or directed signed graphs and graphons

The techniques developed in this paper can be extended to study opinion dynamics on time-varying or directed signed graphs and graphons. By incorporating time-varying interactions or directed edges into the models, researchers can explore how changes in relationships and influences over time impact opinion evolution within social networks. Time-varying dynamics can capture the evolution of opinions in response to changing circumstances, events, or external influences. Directed graphs and graphons can represent asymmetric relationships and information flow within the network, allowing for the analysis of how opinions are influenced by specific sources or individuals. Extending the techniques to these variations of graph structures can provide a more comprehensive understanding of opinion dynamics in complex social systems.
0