Diffusion Dynamics and Optimal Coupling in Multiplex Networks with Directed Layers

Abstract

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. We reveal that directed multiplex networks may exhibit a faster diffusion at an intermediate degree of coupling than when the two layers are fully coupled. We use three simple multiplex examples and a real-world topology to illustrate 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. PACS numbers: 89.75.Hc, 89.20.a, 89.75.Kd Multiplex networks are coupled multilayer networks where each layer consists of the same set of nodes but possibly different topologies and layers interact with each other only via counterpart nodes in each layer [1-3]. Multiplex networks have been shown useful for the study of diverse processes including social networks [4][5][6], transportation networks [7], and biochemical networks [8,9], among others. Recent studies have shown that the coupling of the layers in a multiplex network can result in emergent structural [10] and dynamical behavior such as enhanced diffusion (superdiffusion) [11], increased resilience to random failure [12], and emergence of critical points in the dynamics of coupled spreading processes [4,6]. These richer dynamics arise as a direct consequence of the emergence of more paths between every pair of nodes brought about by layer switching via an interlayer link.Most of the theory for multiplex networks has been developed when all layers consist of undirected networks [1]. However, more often than not, real social, biological and natural networks are structurally directed. Additionally, even if the underlying topology is not directed, the functional and dynamical connectivity of undirected networks is often directed due to gradients or the directionality in the flow of mass or information. These processes include geophysical processes on tributary river networks [13][14][15][16] and river delta channel networks [17][18][19][20][21][22], food webs [23][24][25], gene regulation networks [23,26] and social dynamics [27] to name a few. In this paper, we study diffusion processes on directed multiplex networks, defined here as multiplex networks wherein the connectivity of at least one of the layers forms a directed graph. We document a non-monotonic increase in the rate of convergence to the steady state as a function of the degree of coupling between layers. We uncover that due to the directionality of (at least one of) the layers in a directed multiplex network an optimal coupling regime can emerge where transport processes are enhanced and diffusion is faster than when layers are fully coupled. Let x(t) represent the N × 1 vector of concentration associated with the N nodes of a network at time t (throughout, vectors are thought of as column vectors). The d...

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