Multilayer Networks in a Nutshell

Alberto Aleta; Yamir Moreno

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Multilayer Networks in a Nutshell

Alberto Aleta, Yamir Moreno

arXiv:1804.03488v1Apr 10, 2018physics.soc-phcond-mat.stat-mechcs.SI

TL;DR

Single-layer networks do not fully capture the interdependencies of many real complex systems. This review develops the multilayer-network framework, surveys representative dynamics and applications, and reports that multilayer structure can change robustness and diffusion behavior. Its scope is limited by unresolved theory for directed, adaptive, and temporal multilayer networks.

Problem
Many real systems contain multiple interacting networks and interdependencies that single-layer representations do not properly capture.
Method
The review extends graph-based network methodology to multilayer systems and surveys representative dynamical processes and applications.
Results
Multilayer structure changes dynamical behavior, including hybrid percolation transitions, reduced robustness in interdependent networks, and potentially faster diffusion than in separated layers.
Takeaways & Limitations
Multilayer models provide a framework for analyzing structural interdependencies in systems such as disease spreading, power grids, communication networks, and social contexts.
Takeaways & Limitations
Theoretical treatment remains concentrated on undirected and unweighted layers, while directed, adaptive, and temporal multilayer networks require further work.

Abstract

from arXiv · show

Complex systems are characterized by many interacting units that give rise to emergent behavior. A particularly advantageous way to study these systems is through the analysis of the networks that encode the interactions among the system's constituents. During the last two decades, network science has provided many insights in natural, social, biological and technological systems. However, real systems are more often than not interconnected, with many interdependencies that are not properly captured by single layer networks. To account for this source of complexity, a more general framework, in which different networks evolve or interact with each other, is needed. These are known as multilayer networks. Here we provide an overview of the basic methodology used to describe multilayer systems as well as of some representative dynamical processes that take place on top of them. We round off the review with a summary of several applications in diverse fields of science.

1. Introduction

Complex systems exhibit emergent phenomena that require attention to interactions among constituents, motivating network science as a tractable framework. Multilayer networks extend ordinary networks by representing multiple interaction types and their interconnections.

1. Introduction: Complex systems exhibit phenomena that cannot be explained or predicted solely from their constituent parts.This motivates a hierarchical and less reductionist view of complex systems.
1. Introduction: A holistic perspective accounts for both system constituents and the interactions that form between them.The properties of water illustrate why isolated components may not explain the behavior of the combined system.
1. Introduction: Network science combines graph-based modeling with techniques from other disciplines to study complex systems across economic, social, and biological fields.Graphs encode entities as nodes and their interactions as links.
1. Introduction: Single-network representations can omit information from the original system even though they have produced important structural and dynamical insights.The limitation is a reduction of the system at a different descriptive level.
1. Introduction: Multilayer networks classify interactions by characteristics, producing connected networks for different interaction types while preserving tractability.The resulting representation also depends on how layers connect, what nodes represent, and how interactions are encoded.

2. From simple graphs to multilayer networks

The paper builds multilayer networks from graph-theoretic representations by extending nodes, links, matrices, and structural measures across layers. It then describes multilayer system classes and shows that their spectral structure can alter diffusion and robustness-related dynamics.

2.1 Networks and graphs: Graphs represent systems as nodes and links, with directedness and link weights encoding interaction asymmetry and intensity.A graph is G = (V,E), where E contains links between pairs of nodes.
2.1 Networks and graphs: Network structure is characterized using measures such as degree, degree distributions, paths, random walks, and betweenness centrality.These measures connect local connectivity and path structure to network behavior.
2.1 Networks and graphs: The graph Laplacian L = D − A governs diffusion dynamics, whose eigenvalues characterize the diffusion process.The diffusion equation describes substance movement along links according to concentration differences.
2. From simple graphs to multilayer networks: Ignoring time dependence or multiple interaction types can distort analyses when dynamical processes depend on transmission ordering or link type.These omissions motivate extending isolated-network representations.
2.2 Multilayer networks: Multilayer networks add layers so nodes in different layers can be connected, distinguishing intra-layer and inter-layer interactions.Tensors and block matrices encode layer-specific adjacency structures and cross-layer couplings.
2.2 Multilayer networks: Multiplex networks reuse the same nodes across layers with potentially different interactions, whereas networks of networks represent distinct network systems.Online social platforms provide an example of multiplex structure.
2.2 Multilayer networks: Layer reducibility asks how many layers are needed to represent a system accurately and whether that representation differs from an aggregate network.Von Neumann entropy is used to assess distinguishability from the aggregated network.
2.2 Multilayer networks: Multilayer spectral properties can produce faster diffusion than in aggregated networks because the relaxation time is at most that of the aggregated network.The result follows from interlacing relationships involving quotient-network Laplacian eigenvalues.

3. Dynamical processes on multilayer networks

Multilayer networks alter both robustness and diffusion dynamics by encoding interdependencies among layers. The review examines percolation, cascading failures, epidemic spreading, and diffusion timescales in these systems.

3.1 Percolation and multilayer networks: Multilayer percolation extends connectivity across layers, while added layers can rapidly increase the number of nodes and create new robustness challenges.Robustness concerns preserving network structure under node or link failures and attacks.
3.1 Percolation and multilayer networks: Multiplex networks can exhibit hybrid transitions under random node removal, combining a discontinuity below the critical point with second-order critical behavior above it.This differs from the continuous transition reported for single-layer networks.
3.1 Percolation and multilayer networks: Interdependent power-grid and computer networks can be less robust than single networks because failures propagate through dependency links and trigger cascading breakdowns.A fragile node in one layer can shut down an important node in the other layer.
3.1 Percolation and multilayer networks: Extensions model resource depletion, multiple interlayer links, epidemic spreading, targeted attacks, link overlap, and robustness applications in brain, financial, and socio-ecological networks.Resource-depletion models can produce hysteresis and increase recovery costs.
3.2 Diffusion processes in multilayer networks: Diffusion timescales are controlled by the second smallest positive eigenvalue of the supra-Laplacian, and diffusion can be faster in multiplex networks than in separated layers for some parameters.Multilayer dynamics may also allow processes in different layers to enhance or inhibit one another.
3.2 Diffusion processes in multilayer networks: Epidemic spreading and information diffusion are the two principal diffusion processes studied in multilayer networks, with related equations sometimes used for both.Opinion dynamics can represent one individual holding different opinions across layers.
3.2 Diffusion processes in multilayer networks: Multilayer structure can change epidemic thresholds and infected fractions even with low partial overlap between layers.The review uses SIS dynamics and contact-probability matrices to describe disease spreading.
3.2 Diffusion processes in multilayer networks: For multiplex epidemic models, the dominant layer with the largest eigenvalue sets the critical properties of the entire multilayer system in the small interlayer-coupling regime.The largest eigenvalue depends on the ratio γ/β in this framework.

4. Other examples of multilayer networks

The review surveys multilayer-network applications across ecological, transportation, biological, brain, economic, and social systems. These examples show how representing multiple interaction types, modes, scales, or contexts can reveal structure and dynamics that single-layer aggregation may miss.

Ecology: Multilayer networks disentangle concurrent interaction types in ecological systems, including spatial or temporal layers.The review notes that separating interactions such as pollination and herbivory can provide insight into ecological behavior and evolution.
Transportation: Multilayer transport models represent modes as distinct layers, with cities or locations as nodes and interlayer links connecting the modes.Urban models may distinguish buses, metro, and rail, while transport modes can differ in velocity and carrying capacity.
Transportation: In European air transport, modeling each airline as a layer showed lower multilayer-network resilience than in the aggregated network.Other multilayer aviation studies examined Greek and Chinese air transport networks.
Human brain: Multilayer networks extend brain analysis across spatial, temporal, and topological scales, producing different central-node results from single-layer networks.They also support analysis of time-varying brain connectivity.
Economy: Economic applications use layers for distinct correlation measures or interconnected financial relationships, including markets, interbank networks, lobbying, and trading.One financial-market model used Pearson, Kendall, Tail, and Partial correlation layers to represent dependency structure.
Human cooperation: Multilayer social models can represent several contexts simultaneously, but cooperation gains require specific network conditions rather than social-context multiplicity alone.The review cautions that this research area remains in its infancy.

5. Outlook

The outlook identifies foundational theory, adaptive and temporal systems, and biological and disease-spreading applications as important directions for multilayer-network research. The authors expect continued growth in both foundational work and applications across scientific domains.

Open challenges: Theoretical descriptions remain incomplete because most studies focus on undirected and unweighted layers, while directed multilayer networks create harder spectral problems.Directed adjacency or supra-adjacency matrices are not symmetric, complicating their analysis.
Open challenges: Adaptive and temporal multilayer networks remain underdeveloped despite their relevance to brains, evolving social connections, and adaptive diffusion.Such models may require time-dependent diffusion coefficients.
New applications: Biological systems are a promising application because proteins and genes participate concurrently in signaling, transcriptional, and metabolic pathways shaped by environmental conditions.The review argues that these pathways call for multiple channels of representation.
New applications: Multilayer disease-spreading models can represent interacting diseases on shared host populations, including cross-immunization or cooperation between tuberculosis and HIV.These interdependencies arise across multiple networks of contacts.
Outlook: The field is expected to continue growing through foundational research and applications in social, biological, natural, and technological systems.The review presents its application list as partial and still expanding.