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Mathematical Formulation of Multilayer Networks

Manlio De Domenico1, Albert Solé-Ribalta1, Emanuele Cozzo2, Mikko Kivelä3, Yamir Moreno2,4,5, Mason A. Porter6, Sergio Gómez1, and Alex Arenas1

  • 1Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
  • 2Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50018, Spain
  • 3Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom
  • 4Department of Theoretical Physics, University of Zaragoza, Zaragoza 50009, Spain
  • 5Complex Networks and Systems Lagrange Lab, Institute for Scientific Interchange, Turin 10126, Italy
  • 6Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute and CABDyN Complexity Centre, University of Oxford, Oxford OX1 3LB, United Kingdom

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Phys. Rev. X 3, 041022 – Published 4 December, 2013

DOI: https://doi.org/10.1103/PhysRevX.3.041022

Abstract

A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems are very rich. Achieving a deep understanding of such systems necessitates generalizing “traditional” network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multilayer complex systems. In this paper, we introduce a tensorial framework to study multilayer networks, and we discuss the generalization of several important network descriptors and dynamical processes—including degree centrality, clustering coefficients, eigenvector centrality, modularity, von Neumann entropy, and diffusion—for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multilayer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems.

Popular Summary

Describing a social network based on a particular type of human social interaction, say, Facebook, is conceptually simple: a set of nodes representing the people involved in such a network, linked by their Facebook connections. But, what kind of network structure would one have if all modes of social interactions between the same people are taken into account and if one mode of interaction can influence another? Here, the notion of a “multiplex” network becomes necessary. Indeed, the scientific interest in multiplex networks has recently seen a surge. However, a fundamental scientific language that can be used consistently and broadly across the many disciplines that are involved in complex systems research was still missing. This absence is a major obstacle to further progress in this topical area of current interest. In this paper, we develop such a language, employing the concept of tensors that is widely used to describe a multitude of degrees of freedom associated with a single entity.

Our tensorial formalism provides a unified framework that makes it possible to describe both traditional “monoplex” (i.e., single-type links) and multiplex networks. Each type of interaction between the nodes is described by a single-layer network. The different modes of interaction are then described by different layers of networks. But, a node from one layer can be linked to another node in any other layer, leading to “cross talks” between the layers. High-dimensional tensors naturally capture such multidimensional patterns of connectivity. Having first developed a rigorous tensorial definition of such multilayer structures, we have also used it to generalize the many important diagnostic concepts previously known only to traditional monoplex networks, including degree centrality, clustering coefficients, and modularity.

We think that the conceptual simplicity and the fundamental rigor of our formalism will power the further development of our understanding of multiplex networks.

Article Text

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Phys. Rev. X 3, 041022– Published 4 December, 2013
  • Received 23 July 2013
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