We study the dynamics of diffusion processes acting on directed multiplex networks, i.e., coupled multilayer networks where at least one layer consists of a directed graph. By numerical and analytical analysis, we show a new phenomenology, which is genuinely induced by the directionality of the links: the emergence of a prime regime of coupling for which directed multiplex networks exhibit a faster diffusion at an intermediate degree of coupling than when the two layers are fully coupled. The speed of the processes at the optimal coupling can be higher than in any of the individual layers (Superdiffusion). Furthermore, directed multiplex networks can exhibit superdiffusion within the optimal regime, even in scenarios where the asymptotic regime is not superdiffusive. By the analysis of three simple multiplex, we tease out the characteristics of the directed dynamics that give rise to a regime in which an optimal coupling exists. Given the ubiquity of both directed and multilayer networks in nature, our results could have important implications for the dynamics of multilevel complex systems towards optimality.
