Structural balance and opinion separation in trust-mistrust social networks (1605.08180v1)

Abstract: Structural balance theory has been developed in sociology and psychology to explain how interacting agents, e.g., countries, political parties, opinionated individuals, with mixed trust and mistrust relationships evolve into polarized camps. Recent results have shown that structural balance is necessary for polarization in networks with fixed, strongly connected neighbor relationships when the opinion dynamics are described by DeGroot-type averaging rules. We develop this line of research in this paper in two steps. First, we consider fixed, not necessarily strongly connected, neighbor relationships. It is shown that if the network includes a strongly connected subnetwork containing mistrust, which influences the rest of the network, then no opinion clustering is possible when that subnetwork is not structurally balanced; all the opinions become neutralized in the end. In contrast, it is shown that when that subnetwork is indeed structurally balanced, the agents of the subnetwork evolve into two polarized camps and the opinions of all other agents in the network spread between these two polarized opinions. Second, we consider time-varying neighbor relationships. We show that the opinion separation criteria carry over if the conditions for fixed graphs are extended to joint graphs. The results are developed for both discrete-time and continuous-time models.

• The paper demonstrates that in fixed networks, a structurally balanced subgraph drives opinion polarization while imbalance leads to convergence toward neutrality.
• It employs both discrete-time and continuous-time DeGroot-type models to simulate opinion dynamics under fixed and time-varying conditions.
• Numerical simulations confirm that even with weaker connectivity, the interplay of trust and mistrust fosters distinct opinion clusters in social networks.

Structural Balance and Opinion Separation in Trust-Mistrust Social Networks

The paper explores the intersection of structural balance theory and opinion dynamics within social networks characterized by relationships of trust and mistrust. It builds upon established principles in sociology and psychology, notably how trust-mistrust relationships shape social dynamics and contribute to polarization. Researchers Weiguo Xia, Ming Cao, and Karl Henrik Johansson present a two-part analysis to expand the understanding of opinion dynamics under both fixed and time-varying social network conditions.

Fixed Network Topologies

In investigating networks with fixed topologies, the authors consider cases where the network structure includes a subgraph that is either structurally balanced or unbalanced, as defined by graph theory. The paper emphasizes that if this subnetwork is strongly connected and structurally balanced, the network tends to polarize into two camps with opposing opinions. In contrast, if a strongly connected subnetwork is structurally unbalanced, all opinions in the network converge toward neutrality.

For networks that contain a spanning tree, even if not strongly connected, the dynamics are similar depending on the balance of trust-mistrust relationships within the subgraph. The polarization remains robust in structurally balanced subnetworks, dictating that the encompassing network's nodes will distribute their opinions along the polarized opinions of the structural subnetwork.

Time-Varying Network Topologies

The exploration of time-varying neighbor relationships extends the fixed network results, applying them to scenarios where social interactions and trust/mistrust relations evolve. The authors demonstrate that networks that maintain structural balance during dynamic changes still lead to opinion separation and polarization, provided the structural imbalance does not consistently dominate the graph's connectivity intervals. When networks frequently encounter structural imbalance, the convergence toward neutral opinions is more likely.

The paper uses both discrete-time and continuous-time models of the DeGroot-type opinion dynamics as the analytical framework. The findings in this paper illustrate that opinion dynamics in networks extend beyond simple consensus, showing conditions under which opinion clustering and polarization occur, highlighting the complexity of social networks when trust and mistrust are both present.

Numerical Results and Claims

The paper employs theoretical proofs supported by numerical simulations to validate its hypotheses regarding network behavior under different conditions of balance. One critical implication of the research is that social networks characterized by a mix of trust and mistrust relationships can behave differently than traditionally assumed in models lacking such complexity. Results suggest that even under less stringent connectivity conditions, networks can form opinion clusters without necessitating complete polarization.

Implications and Future Directions

This research contributes to both practical and theoretical discussions surrounding opinion dynamics in complex networks. Social media platforms and organizational behavior analyses can benefit from appreciating how opinion clustering or polarization emerges based on network structure. Moreover, the introduction of time-varying dynamics reflects real-world complexities that fixed networks do not capture.

Future research could explore the mechanisms of opinion dynamics beyond the averaging processes, including biased assimilation or nonlinear multi-agent dynamics, to reflect realistic social psychology processes more accurately. Also, the introduction of noise, time delays, or further adaptive behaviors in trust-mistrust networks could provide a more comprehensive picture of social opinion formation.

Overall, this paper provides significant insights and extensions to the foundational understanding of trust-mistrust dynamics in social networks. It poses challenges and opportunities for further exploration in the paper of complex networks.