Multilayer network science: theory, methods, and applications

TL;DR

This review addresses how multilayer network science unifies the study of complex systems with multiple interaction types and temporal or higher-order structure. It consolidates theory and methods—from definitions and spectral metrics to temporal representations, higher-order frameworks, and ML-enabled embeddings—across a broad spectrum of applications, including infrastructure, biology, and social systems. Key contributions include a taxonomy of community-detection strategies, spectral indicators like $\lambda_2$ and $R$, and the integration of temporal and higher-order representations via hypergraphs and temporal hypergraphs, alongside practical ML approaches for embeddings and predictive tasks. The work highlights the need for standardized data formats and interoperable software to enable cross-disciplinary collaboration, and it argues that the multilayer paradigm will drive predictive, policy-relevant models of interdependent systems in the real world.

Abstract

Multilayer network science has emerged as a central framework for analysing interconnected and interdependent complex systems. Its relevance has grown substantially with the increasing availability of rich, heterogeneous data, which makes it possible to uncover and exploit the inherently multilayered organisation of many real-world networks. In this review, we summarise recent developments in the field. On the theoretical and methodological front, we outline core concepts and survey advances in community detection, dynamical processes, temporal networks, higher-order interactions, and machine-learning-based approaches. On the application side, we discuss progress across diverse domains, including interdependent infrastructures, spreading dynamics, computational social science, economic and financial systems, ecological and climate networks, science-of-science studies, network medicine, and network neuroscience. We conclude with a forward-looking perspective, emphasizing the need for standardized datasets and software, deeper integration of temporal and higher-order structures, and a transition toward genuinely predictive models of complex systems.

Figures (9)

• Figure 1: Comparison between the second smallest eigenvalues $\lambda_2$ of the different Laplacian matrices as a function of the interlayer coupling $D_x$. Image from gomez2013.
• Figure 2: Eigenratio $R = \lambda_N / \lambda_2$ as a function of the interlayer coupling strength $D_x$, adapted from sole2013. Circles represent exact numerical results, while the solid lines show the analytical approximations valid in the weak- and strong-coupling limits.
• Figure 3: Multilayer communities. Communities are indicated by the dashed contours. (a) Intralayer communities group state nodes on the same layer. (b) Members of cross-layer communities can be on both the same and on different layers. (c) Pillar communities group all the counterparts of the nodes in other layers in the same community.
• Figure 4: Temporal networks of face-to-face interactions of people on different days of museum visits. In each day, time evolution is represented as a color code. The data has been recorded by detecting physical proximity using wearable sensors developed by the SocioPatterns collaboration isella2011whatvandenbroeck2012making.
• Figure 5: Schematic representation of different networked systems. The bottom row depicts systems with pairwise interactions, while the top row contains group interactions including more than two nodes. As with classical pairwise models, it is possible to associate a type to each interaction, extending the framework of multilayer networks to multilayer hypergraphs. Image adapted from FerrazdeArruda2024Aug.