Multilayer network science: theory, methods, and applications (2511.23371v1)

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.

• The paper establishes multilayer network science as a paradigm to rigorously model heterogeneous, interdependent systems using layered representations.
• It introduces key matrix representations and spectral methods, such as the supra-adjacency matrix and supra-Laplacian, to analyze network connectivity and dynamics.
• The review highlights diverse applications—from social systems to neuroscience—and outlines future challenges in integrating temporal, higher-order, and machine learning approaches.

Multilayer Network Science: Theory, Methods, and Applications

Foundations and Theoretical Framework

The reviewed work "Multilayer network science: theory, methods, and applications" (2511.23371) presents a comprehensive synthesis of the formalization, analytical tools, and recent advances underlying the paper of multilayer networks—a framework required for rigorously modeling the multi-relational, heterogeneous, interdependent structures observed in natural and engineered systems.

At the core, multilayer networks incorporate multiple types of interactions or relations across distinct “layers,” each with possibly differentiated node sets, connectivity, and dynamics. Two principal matrix representations, the adjacency tensor and the supra-adjacency matrix, are discussed for encoding interactions across and within layers, generalizing key single-layer constructs. The corresponding supra-Laplacian, central to spectral graph theory, enables the extension of dynamical and structural analysis to the multi-layer setting. The behavior of the second smallest eigenvalue $\lambda_2$ of the Laplacian, an indicator of system connectivity and robustness, is analyzed as a function of interlayer coupling and shown to undergo nontrivial, discontinuous transitions as layers become effectively integrated or decoupled.

Figure 1: The interlayer coupling $D_x$ modulates the second smallest Laplacian eigenvalue $\lambda_2$, tuning the cohesiveness of multilayer systems.

The eigenratio $R = \lambda_N/\lambda_2$ is further explored for assessing synchronizability, with analytic predictions and numerics characterizing optimal regimes for interlayer coupling.

Figure 2: Eigenratio $R$ as a function of interlayer coupling strength, revealing nontrivial percolation in dynamical regimes.

The review details how multilayer network theory generalizes classical centrality, assortativity, modularity, and redundancy metrics. It emphasizes those unique to multilayer systems, such as the mutually largest connected component (MLCC), participation coefficient, and cross-layer similarity measures, all essential for quantifying information non-reducible to the aggregate network.

Mesoscale Organization and Dynamics

Community detection within multilayer networks is treated as both a theoretical and computational challenge, requiring approaches that address the diversity and complexity of community types—intralayer, cross-layer, and pillar communities—beyond simple flattening or partitioning methods.

Figure 3: Canonical community types in multilayer networks: intralayer, cross-layer, and pillar communities, each requiring distinct detection strategies.

Modern inference strategies leverage multilayer null models and SBM extensions, enabling simultaneous tasks of clustering, inference of layer interdependence, and model selection for hidden-layer network generation.

Dynamical processes on multilayer networks display novel emergent phenomena, including abrupt, hybrid, or discontinuous phase transitions, fundamentally altered critical points for diffusion, synchronization, and contagion, and metastable coexistence regimes mediated by interlayer coupling. Topology-driven instabilities and feedback among heterogeneous dynamical processes are highlighted as robust, generic features in the multiplex architecture, with claims substantiated across neuroscience, epidemiology, and social system modeling.

Temporal and Higher-Order Generalizations

A substantial portion addresses temporal networks, defining multilayer representations as sequences of correlated snapshots, justifying the need for temporal path-based metrics, time-respecting motifs, and centrality measures that account for order effects and multi-timescale dependencies.

Figure 4: Empirical temporal (multilayer) social networks, revealing dynamically evolving interaction patterns.

Group or higher-order interactions are formalized through multilayer hypergraphs and simplicial complexes, allowing encoding of interactions among $k>2$ nodes. The review discusses how inclusion of such higher-order structures produces genuinely new dynamical behaviors—discontinuous contagion transitions, multi-stability, and chaos—demonstrating the inadequacy of pairwise-only models for accurate process prediction.

Figure 5: Schematic demonstrating pairwise (bottom) vs. higher-order/group (top) interaction structures and their multilayer extensions.

Machine Learning and Graph Embeddings

Graph machine learning in the multilayer context is surveyed, focusing on embedding techniques that preserve both intralayer and interlayer structure, with algorithms ranging from random walk-based representation learning to attention mechanisms and deep graph neural networks. Both application-driven link prediction, node classification, and community discovery are covered.

Applications Across Domains

Social, Economic, and Infrastructural Systems

The review gives a technical overview of applications in social systems (modeling polarization, echo chambers, multi-context interactions), economic and financial contagion (multiplex systemic risk, cross-asset correlations), and interdependent infrastructure (cascading failures, robustness analysis). In infrastructure, multilayer modeling has revealed the counterintuitive fragility emerging from interdependence, and current extensions incorporate recovery dynamics and non-trivial network-of-network couplings.

Figure 6: Multilayer epidemic-economic systems, illustrating bidirectional feedback between health dynamics and economic activity.

Transportation and Urban Mobility

Complexity and multimodality in transport networks are visualized using multilayer models, clarifying the emergence of congestion, synchronization problems, and cognitive limits in navigation.

Figure 7: Multilayer model of Madrid public transportation, integrating tram, metro, and bus layers with explicit intermodal connections.

Biological, Ecological, and Climate Systems

In biology and network medicine, integration of diverse omics data and multiple interaction types is naturally expressed as multilayer networks, enabling enhanced discovery of disease modules, gene communities, and drug-disease associations.

Figure 8: Multilayer omics network, synthesizing gene, protein, and metabolite layers for integrative systems biology.

Multilayer approaches also yield improved analyses of ecological interactions, revealing temporal persistence, multi-trophic interactions, and the identification of keystone functional species. In climate science, coupling of physical (e.g., atmospheric) layers and environmental variables enables better vulnerability assessments and propagation analysis of climatic shocks.

Neuroscience

The brain’s multiscale structural and functional architectures are mapped using multilayer network theory, bridging modalities and enabling quantitative assessment of inter-individual or cross-type variability, structure-function relations, and higher-order connectomic features.

Data, Software, and Methodological Advances

The field’s growth is catalyzed by a rapidly expanding corpus of data repositories, standardized tools for multilayer, temporal, and higher-order networks, and a mature software ecosystem. However, the paper notes the ongoing challenge of data standardization, methodological interoperability, and multidisciplinary communication.

Implications, Open Challenges, and Outlook

This review underscores that multilayer network science is not an incremental extension but a paradigm shift in the formal language for complex systems modeling. Several bold claims are supported: there exist structural and dynamical phenomena in the multilayer representation that cannot be inferred, even in principle, from single-layer or aggregated models, mandating multilayer methods for accurate analysis and prediction.

Critical future challenges are outlined, including:

• Development and integration of frameworks for concurrent multilayer, temporal, and higher-order modeling.
• Inference of minimal (“irreducible”) multilayer descriptions and rigorous methods for model selection and layer aggregation.
• Machine learning-enabled inference of hidden layers, mesoscale organization, and cross-layer dependencies.
• Extension of control theory, explainability, and optimization to distinctly multilayer settings.
• Translation of advances to actionable, predictive, and policy-relevant domains with high-impact coupled processes.

The review anticipates that domain convergence (notably with AI, where network architectures are themselves multilayered) will be contingent on theoretical clarity, data standardization, and broad community adoption of interoperable analytic platforms.

Conclusion

"Multilayer network science: theory, methods, and applications" (2511.23371) establishes multilayer network theory as a foundational and rigorous framework for the analysis of complex systems where heterogeneity, interdependence, and temporal/higher-order structure are irreducible features. The reviewed work demonstrates that only by explicit multilayer modeling can one accurately infer dynamics, predict transitions, or design robust interventions in socio-technical, biological, and engineered systems. The field’s trajectory suggests increasing convergence with statistical inference, graph machine learning, and high-dimensional data science, positioning multilayer network science as a central pillar for future advances in complex systems and AI research.