Opinion Dynamics in Social Networks with Hostile Camps: Consensus vs. Polarization (1508.05034v1)

Abstract: Most of the distributed protocols for multi-agent consensus assume that the agents are mutually cooperative and "trustful," and so the couplings among the agents bring the values of their states closer. Opinion dynamics in social groups, however, require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents, which are usually characterized by repulsive couplings and may lead to clustering of the opinions. A simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by C. Altafini, who examined first-order consensus algorithms over static signed graphs. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in signs. In this paper, we extend the modulus consensus model to the case where the network topology is an arbitrary time-varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph. For cut-balanced graphs, not only sufficient, but also necessary conditions for modulus consensus are given.

• The paper introduces an extended modulus consensus model addressing both cooperative and antagonistic interactions in social networks.
• It derives sufficient conditions for consensus in time-varying networks using uniform strong connectivity and cut-balanced graph analysis.
• Numerical results validate that additional constraints are needed in mixed-sign networks to control polarization and stabilization effects.

Opinion Dynamics in Social Networks with Hostile Camps: Consensus vs. Polarization

The paper "Opinion Dynamics in Social Networks with Hostile Camps: Consensus vs. Polarization" by Anton V. Proskurnikov, Alexey Matveev, and Ming Cao investigates the complex dynamics of opinion formation in social networks, where both cooperative and antagonistic interactions exist among agents. Leveraging a mathematical framework, the authors extend a model of opinion dynamics, initially proposed for networks with static and cooperative interactions, to networks where relationships may change over time—from trust and cooperation to mistrust and hostility.

Key Contributions

The paper makes several notable contributions to the field of opinion dynamics and multi-agent systems:

1. Modulus Consensus Model Extension: Building on Altafini’s model, the authors paper systems where agents interact via both attractive and repulsive couplings. These interactions model trustful and distrustful agent relationships, leading to behaviors such as clustering and potential polarization.
2. Static and Dynamic Topologies: One of the key advances of this paper is moving beyond static interaction topologies to consider time-varying networks, which more realistically depict social networks wherein the nature of relationships can evolve.
3. Sufficient Conditions for Modulus Consensus: The authors formulate conditions under which networks achieve modulus consensus—a scenario where all opinion modules align, but opinions may differ in sign. These conditions primarily rely on uniform strong connectivity.
4. Classification of Consensus Types: The paper delineates the outcomes of opinion dynamics into stabilization (converging to zero), consensus (alignment in opinion), and bipartite consensus (polarization along signed opinions).
5. Cut-Balanced Graph Analysis: Through rigorous mathematical analysis, the authors provide necessary and sufficient conditions specifically for cut-balanced graphs, a class of graphs characterized by their specific balance conditions.

Numerical Results and Claims

The authors substantiate their theoretical findings with precise numerical results derived from mathematical models, illustrating the efficacy of the derived conditions for achieving modulus consensus. The research also highlights that uniformly quasi-strongly connected (UQSC) properties, which are sufficient for consensus in purely cooperative networks, require further constraints in mixed-sign networks involving both positive and negative interactions.

Implications and Future Work

This research significantly impacts both theoretical and practical domains in network science and control theory. By addressing evolving interaction topologies, this paper opens avenues for further research into non-linear dynamics of real-world social networks and inter-agent communication systems. The findings hold practical implications for understanding polarization in social, political, and organizational networks, potentially guiding strategies for controlling opinion dynamics and mitigating polarization.

Further work could elucidate the role of dynamic graph properties on different types of consensus over nonlinear networks. Additionally, exploring implementations in real-world scenarios, via collaborations with social scientists, could validate and refine these theoretical models, enhancing their applicability in understanding and managing social networks.

Conclusion

The paper expertly extends existing models of opinion dynamics to account for antagonistic interactions and changing relationships within networks. Its contributions to consensus theory in networks with hostile camps provide a deeper understanding of the mechanisms leading to consensus, polarization, and stabilization in complex systems, laying a foundation for future explorations in dynamically evolving social constructs.