Mathematical Formulation of Multilayer Networks

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 ne...

Full description

Saved in:
Bibliographic Details
Published in:Physical review. X 2013-12, Vol.3 (4), p.041022, Article 041022
Main Authors: De Domenico, Manlio, Solé-Ribalta, Albert, Cozzo, Emanuele, Kivelä, Mikko, Moreno, Yamir, Porter, Mason A., Gómez, Sergio, Arenas, Alex
Format: Article
Language:English
Subjects:
Clustering
Complex systems
Eigenvectors
Formalism
Mathematical analysis
Modularity
Monolayers
Multilayers
Multimodal transportation systems
Multiplexing
Nodes
Representations
Route planning
Social factors
Social networks
Subsystems
Tensors
Transportation networks
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2160-3308
2160-3308
DOI:10.1103/PhysRevX.3.041022