Consensus dynamics under asymmetric interactions in low dimensions

We consider a set \(N = \{ 1,\ldots ,n \}\), \(n\ge 2\), of interacting agents whose individual opinions \( x_{i}\), with \(i \in N \), take values in some domain \(\mathbb {D}\subseteq \mathbb {R}\). The interaction among the agents represents the degree of reciprocal influence which the agents exert upon each other and it is expressed by a general asymmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The present paper examines the asymmetric generalization of the linear consensus dynamics model discussed in previous publications by the same authors, in which symmetric interaction was assumed. We are mainly interested in determining the form of the asymptotic convergence towards the consensual opinion. In this respect, we present some general results plus the study of three particular versions of the linear consensus dynamics, depending on the relation between the interaction structure and the degrees of proneness to evaluation review of the various individual opinions. In the general asymmetric case, the analytic form of the asymptotic consensual solution \(\tilde{x}\) is highly more complex than that under symmetric interaction, and we have obtained it only in two low-dimensional cases. Nonetheless, we are able to write those complex analytic forms arranging the numerous terms in an intelligible way which might provide useful clues to the open quest for the analytic form of the asymptotic consensual solution in higher-dimensional cases.