Convergence analysis of the Friedkin–Johnsen model with multiple topics

Abstract

In this paper, we study the convergence and consensus of the Friedkin–Johnsen (F–J) model with $n$ individuals/agents and $m$ interdependent topics, where the interpersonal influence among $n$ individuals is represented by an interpersonal influence matrix $W$ and the interdependencies among $m$ topics are described by a set of “logic matrices” $C_1,\dots ,C_n$. Each topic is described by a topic subnetwork and the whole F–J model can be regarded as a large network with $m$ isomorphic subnetworks as well as the interactions among them. For the F–J model with homogeneous logic matrices (i.e., $C_1=\cdots =C_n$), our results show that it can achieve global topic consensus (consensus on each topic) if and only if all individuals are non-oblivious with only one stubborn individual and all topics are independent. For the F–J model with heterogeneous logic matrices (i.e., $\exists C_i\ne C_j$), we not only construct the integrated digraph $G(C_1,C_2,\dots ,C_n)$ but also establish the connections between this digraph and the logic networks $G\left(C_1\right),\dots ,G\left(C_n\right)$. Taking advantage of these connections, we establish the convergence and topic consensus criteria for the F–J model with heterogeneous logic matrices, where the heterogeneity is reflected in the structure, weight, and sign of $C_i$. Finally, some simulations are provided to illustrate the results.