The Friedkin-Johnsen (FJ) model describes how agents adjust their opinions through repeated interactions while accounting for the influence of agents who are partially stubborn. In this work, we demonstrate that the FJ model is stepwise equivalent to solving the average consensus problem via distributed gradient descent. This perspective provides a unifying framework that bridges opinion dynamics and optimization, enabling the application of well-established results from the optimization literature. To illustrate this, we examine the recently proposed FJ model with diminishing stubbornness and extend prior results that were concerned with fixed communication graphs to time-varying and jointly connected communication graphs. We derive convergence guarantees and analyze convergence rates under these relaxed assumptions. Finally, we present numerical experiments on random graphs to showcase the impact of diminishing stubbornness dynamics on convergence in both static and time-varying settings.

This letter studies the Friedkin-Johnsen (FJ) model with diminishing competition, or stubbornness. The original FJ model assumes that each agent assigns a constant competition weight to its initial opinion. In contrast, we investigate the effect of diminishing competition on the convergence point and speed of the FJ dynamics. We prove that, if the competition is uniform across agents and vanishes asymptotically, the convergence point coincides with the nominal consensus reached with no competition. However, the diminishing competition slows down convergence according to its own rate of decay. We study this phenomenon analytically and provide upper and lower bounds on the convergence rate. Further, if competition is not uniform across agents, we show that the convergence point may not coincide with the nominal consensus point. Finally, we evaluate our analytical insights numerically.