Containment Control for a Social Network with State-Dependent Connectivity (1402.5644v1)

Abstract: Social interactions influence our thoughts, opinions and actions. In this paper, social interactions are studied within a group of individuals composed of influential social leaders and followers. Each person is assumed to maintain a social state, which can be an emotional state or an opinion. Followers update their social states based on the states of local neighbors, while social leaders maintain a constant desired state. Social interactions are modeled as a general directed graph where each directed edge represents an influence from one person to another. Motivated by the non-local property of fractional-order systems, the social response of individuals in the network are modeled by fractional-order dynamics whose states depend on influences from local neighbors and past experiences. A decentralized influence method is then developed to maintain existing social influence between individuals (i.e., without isolating peers in the group) and to influence the social group to a common desired state (i.e., within a convex hull spanned by social leaders). Mittag-Leffler stability methods are used to prove asymptotic stability of the networked fractional-order system.

Containment Control for a Social Network with State-Dependent Connectivity

This paper focuses on the domain of social network dynamics, particularly examining how opinions or emotional states within a network of individuals evolve under the influence of certain leaders. Leveraging fractional-order dynamics, the authors aim to model social state dynamics that incorporate both peer influence and past experiences. The research offers a structured approach to containment control for networks with state-dependent connectivity, ensuring that followers’ states converge to a desirable region determined by stable leaders.

Key Contributions

The primary contribution lies in the formulation of the social interactions as a fractional-order system. The paper expands upon traditional integer-order systems by incorporating fractional-order differential equations to account for past influences in an individual’s state, providing a more nuanced understanding of social dynamics. This reflects the non-local property of fractional systems, where the current state is a function of historical states.

Methodology and Control Schema

The social network is described as a directed graph where nodes represent individuals and edges denote influence from one person to another. The individuals in the network are divided into social leaders and followers. Leaders have immutable states, while followers update their states based on local neighbors and personal history.

The research presents a decentralized influence method utilizing a potential field-based approach. Here, each individual employs a potential function characterized by an attractive term for consensus and a repulsive term for maintaining network connectivity. This method ensures that followers’ states settle within a convex hull defined by the leaders, highlighting the importance of maintaining a directed spanning tree for overall network connectivity.

Stability and Convergence

Leveraging Mittag-Leffler stability, the authors demonstrate asymptotic stability of the networked system. They employ LaSalle's invariance principle to validate that followers will achieve consensus within the convex hull of leaders' states over time. By ensuring that the designed potential functions prevent the social difference from exceeding a given threshold, the paper maintains network connectivity, a crucial aspect for achieving containment control.

Implications and Future Work

The implications of this work are significant for the paper of opinion dynamics and social influence in networks, particularly when considering the integration of memory and historical data in modeling individual responses. Practically, the research provides insight into designing control algorithms for influencing social networks while preserving connectivity.

Future research is anticipated to explore complex network structures with heterogeneous dynamics, involving individuals with varied influence capabilities and responses. Additionally, considering state-dependent connectivity adds a layer of complexity, suggesting a pathway for deeper investigation into adaptive control strategies in dynamic networks.

Conclusion

This paper offers a valuable contribution to the paper of social networks by adopting fractional-order systems to model human interaction dynamics. By focusing on containment control, the research presents a robust framework for guiding a network's state evolution, setting the stage for further advancements in the fields of AI and social dynamics.