Mathematical Formulation of Multi-Layer Networks

TL;DR

This paper introduces a tensor-based mathematical framework for analyzing multi-layer networks, generalizing traditional network descriptors and dynamical processes to better understand complex systems with multiple subsystems and layers.

Contribution

It presents a novel tensorial approach to model multi-layer networks, extending key network measures and dynamics beyond single-layer representations.

Findings

Generalizes network descriptors like degree and clustering to multi-layer systems

Demonstrates how to recover known results for single-layer and multiplex networks

Provides tools for analyzing complex systems such as social and transportation networks

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 is 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 multi-layer complex systems. In this paper, we introduce a tensorial framework to study multi-layer 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 multi-layer 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.