arXiv:2401.15857v5 Announce Type: replace Abstract: We delve into the dynamics of opinions within a multiplex network using coordination games, where agents communicate either through one-way or two-way interactions, and where a designated leader may be present. By employing graph theory and Markov chains, we illustrate that despite zero diagonal elements in transition probability matrices or decomposable layers, opinions converge under specific conditions on the layers, leading to a consensus, and we show by a counterexample that a spanning tree in the union of the layers does not suffice. We further scrutinize the convergence rates of opinion dynamics in networks with one-way versus two-way interactions. We find that in networks with a designated leader, opinions converge towards the initial opinion of the leader, whereas in networks without a designated leader, opinions converge to a convex combination of the opinions of agents. Moreover, we emphasize the crucial role of designated leaders in steering opinion convergence within the network. Paired experiments on 1,200 random multiplex networks show that a leader slows down convergence, and that two-way interactions slow it down in the presence of a leader but speed it up in its absence.