Mathematical Formulation of Multilayer Networks

By Manlio De Domenico, Albert Solé-Ribalta, Emanuele Cozzo, Mikko Kivelä, Yamir Moreno, M. A. Porter, Sergio Gómez, Alex Arenas. 2013

1,066 citations recorded by the source corpus.

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.

Subfields: Biomedical Engineering, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4c12bd56cf34a8ffb6f019d1714c2a81f0942b9f

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Citations in the corpus

Multilayer Analysis and Visualization of Networks (2014, methodology): “by using multilayer generalizations of centrality measures [41,43,44].”
Multilayer networks (2013, methodology): “, order2(d + 1)] adjacency tensor9 [103] A ∈ {0, 1}|V |×|V |×|L1|×|L1|×···×|Ld|×|Ld|.”
When Social Influence Meets Item Inference (2015, background): “per, therefore, we propose a new framework for jointly sampling multiple OSNs. Our ﬁrst goal is to provide statisti cal guarantees on the difference between the crawled (and then merged or integrated [12]) real datasets and the ground truth. The ground truth in this paper is deﬁned as the perfect in”
Human mobility: Models and applications (2017, methodology): “The more intricate topology of multilayer networks necessarily affects the spreading processes taking place on the system (diffusion and navigation) [205, 206, 189] that are modeled using the methods discussed in Sec.”
The multilayer nature of ecological networks (2015, background): “biquitous and have a long history in subjects like sociology [25,26] and engineering [27], although an explicit framework for the investigation of multilayer networks was developed only very recently [23,28]. Amidst this rapid development, there have been investigations of an increasingly diverse se”
Ranking in interconnected multilayer networks reveals versatile nodes (2013, background): “Consequently, a classical single-layer network represented by a rank-2 mixed adjacency tensor W β [7] can be simply indicated by W i j , where the “abuse of notation” consists in interpreting the indices i and j as nodes and W i j would indicate intensity of the relationship between them.”
The structure and dynamics of multilayer networks (2014, background): “There have been some attempts in the literature for modeling multilayer networks properly by using the concept of tensors [21,22,47].”
Unsupervised Attributed Multiplex Network Embedding (2019, background): “…most existing network embedding methods assume that only a single type of relation exists between nodes (Veličković et al. 2017; Veličković et al. 2019; Kipf and Welling 2016), whereas in reality networks are multiplex (De Domenico et al. 2013) in nature, i.e., with multiple types of relations.”