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In this paper we propose an extended version of the Friedkin–Johnsen (FJ) model where we analyze the effects that a homophily-based influence matrix has on the opinion formation process. In particular, we assume that the influence matrix varies over time, and its entries (that represent the appraisals that individuals have of the other individuals) update based on the comparison between their opinions and can take both positive and negative values. This leads to a system of two difference equations, where the first one corresponds to the standard FJ model, except that the influence matrix is no longer constant, and the second one models the dynamics of this matrix. We show that a necessary and sufficient condition for the convergence of this modified version of the classical FJ model is that the influence matrix becomes constant in a finite number of steps. Moreover, we provide the explicit expression for the agents' asymptotic opinions in some special cases: the purely cooperative setting, the case of a structurally balanced network and the case of a single discussion topic. Finally, the case when the topics are correlated and the influence matrix depends on the average opinion vectors is investigated.
Disarò et al. (Fri,) studied this question.