Abstract
In this paper, we aim to study the cluster consensus of Friedkin–Johnsen (F-J) model with one or more stubborn nodes. Firstly, for F-J model with only one stubborn node, by algebra theory and graph theory, we not only establish consensus criterion for F-J model without oblivious agents but also establish some couple-cluster consensus criteria for F-J model with oblivious agents. Secondly, for F-J model with more than one stubborn individual, we present a novel network partition strategy to divide F-J model into many subnetworks, where every stubborn node and its pure-reachable nodes (a special reachable node without stubborn nodes except the starting node appearing in the walk) form a subnetwork, and all oblivious individuals of F-J model also constitute a subnetwork if they exist. By our network partition, some necessary and sufficient cluster consensus criteria are given. In addition, if there exist some oblivious individuals, our main results show that the influence of the initial opinion of stubborn root on the final opinion of non-oblivious individuals is proportional to the stubborn extent of the stubborn root to its initial opinion. Finally, some simulations are provided to illustrate the results.
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