Frisking-Johnsen Model with Diminishing Competition

This letter studies the Friedkin-Johnsen (FJ) model with diminishing competition, or stubbornness. The original FJ model assumes fixed competition that is manifested through a constant weight that each agent gives to its initial opinion in addition to its contribution through a consensus dynamic. This letter investigates the effect of diminishing competition on the convergence point and speed of the FJ dynamics. We show 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 evaluate this phenomenon analytically and provide upper and lower bounds on the convergence rate. If competition is not uniform across clients, we show that the convergence point may not coincide with the nominal consensus point. Finally, we evaluate and validate our analytical insights numerically.