  We delve into the dynamics of opinions within a multiplex network using
coordination games, where agents communicate either in a 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 non-positive diagonal
elements in transition probability matrices or decomposable layers, opinions
generally converge under specific conditions, leading to a consensus. 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. Our
experimental findings corroborate that the presence of leaders expedites
convergence, with mono-directional interactions exhibiting notably faster
convergence rates compared to bidirectional interactions.
