FJ-MM: Friedkin-Johnsen opinion dynamics model with memory and higher-order neighbors

The Friedkin-Johnsen (FJ) model has been extensively validated across social science, systems and control, game theory, and algorithmic research. We introduce an advanced generalization - termed the FJ-MM model - that incorporates memory effects and multi-hop influence. This extension allows agents to naturally integrate both current and past opinions at each iteration. We analyze the stability and equilibrium properties of the FJ-MM model, demonstrating that they can be derived from those of a standard FJ model with an appropriately modified influence matrix. We examine the convergence rate of the FJ-MM model and demonstrate that, as can be expected, the time lags introduced by memory and higher-order neighbor influences result in slower convergence. Numerical results illustrate that memory and multi-hop influence reshape the final opinion landscape, e.g., by reducing polarization.