Abstract
Multiplex networks have been intensively studied during the last few years as they offer a more realistic representation of many interdependent and multilevel complex networked systems. However, even if most real networks have some degree of directionality, the vast majority of the existent literature deals with multiplex networks where all layers are undirected. Here, we study the dynamics of diffusion processes acting on coupled multilayer networks where at least one layer consists of a directed graph; we call these directed multiplex networks. We reveal a new and unexpected signature of diffusion dynamics on directed multiplex networks, namely, that different from their undirected counterparts, they can exhibit a nonmonotonic rate of convergence to steady state as a function of the degree of coupling, resulting in a faster diffusion at an intermediate degree of coupling than when the two layers are fully coupled. We use synthetic multiplex examples and real-world topologies to illustrate the characteristics of the underlying dynamics that give rise to a regime in which an optimal coupling exists. We further provide analytical and numerical evidence that this new phenomenon is solely a property of directed multiplex, where at least one of the layers exhibits sufficient directionality quantified by a normalized metric of asymmetry in directional path lengths. Given the ubiquity of both directed and multilayer networks in nature, our results have important implications for studying the dynamics of multilevel complex systems.
- Received 13 February 2018
- Revised 17 July 2018
DOI:https://doi.org/10.1103/PhysRevX.8.031071
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Complex dynamical systems that originate in geophysical, biological, climatic, and social environments often exhibit a networked structure arising from interactions between constituent subsystems. In turn, the analysis of such structures reveals mechanisms behind the properties of the underlying natural processes. Of particular interest is a special class of multilayer networks known as multiplex, where layers consist of an identical set of corresponding nodes, through which they interact, while individually having different topological structure. Here, we initiate the study of multiplex networks with directed layers that are typical of many natural and engineered systems, including geophysical transport on river networks, food webs, gene regulation, and social dynamics.
We show that in contrast to undirected multiplex networks, where the fastest diffusion corresponds to maximal layer coupling, directed multiplex networks explain the emergence of the fastest diffusion at intermediate coupling. We attribute this new phenomenon to the anisotropy inherited by the directionality of the layers, which speeds up mixing via propagation in different directions over separate layers.
Our results have implications in transport dynamics of many social, biological, and natural networks that, more often than not, exhibit process directionality. In particular, our results raise new questions as to whether natural complex systems self-organize into configurations that enhance diffusivity, and they also suggest ways to design artificial networks to operate at an optimal degree of coupling for fastest transport.
