In this paper we propose an extended version of the Friedkin-Johnsen (FJ) model that accounts for the effects of homophily mechanisms on the agents' mutual appraisals. The proposed model consists of two difference equations. The first one describes the opinions' evolution, namely how agents modify their opinions taking into account both their personal beliefs and the influences of other agents, as in the standard FJ model. Meanwhile, the second equation models how the influence matrix involved in the opinion formation process updates according to a homophily mechanism, by allowing both positive and negative appraisals. We derive necessary and sufficient conditions for the proposed time-varying version of the classical FJ model to asymptotically converge to a constant solution. In the case of a single discussion topic, asymptotic convergence is always ensured and the limit behavior of the system is derived in closed form.
Disarò et al. (2023) studied this question.