Optimal Feedback Control in Social Networks in a McKean-Vlasov-Friedkin-Johnsen System

This paper presents a comprehensive analytical formulation for deriving a closed-form optimal strategy for agents operating within a social network, modeled through a McKean-Vlasov stochastic differential equation (SDE). Each agent aims to minimize a personal dynamic cost functional that accounts for deviations from the collective opinions of others, their own past beliefs, and is influenced by randomness and inherent opinion rigidity, often described as stubbornness. To tackle this, we develop a novel methodology rooted in a Feynman-type path integral framework, incorporating a specially designed integrating factor to obtain explicit feedback control laws. This approach provides a tractable and insightful solution to the control problem in a setting shaped by both memory and noise. As part of our analysis, we adopt a modified form of the Friedkin-Johnsen opinion dynamics model to more accurately capture the influence of prior beliefs and social interactions, enabling the explicit derivation of the optimal strategy. Comparative simulations further illustrate the effectiveness and adaptability of our method across different network structures, highlighting its potential relevance to understanding opinion evolution and influence strategies in complex social systems.