The structure and dynamics of multilayer networks

S. Boccaletti; G. Bianconi; R. Criado; C. I. del Genio; J. Gómez-Gardeñes; M. Romance; I. Sendiña-Nadal; Z. Wang; M. Zanin

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The structure and dynamics of multilayer networks

S. Boccaletti, G. Bianconi, R. Criado, C. I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, M. Zanin

arXiv:1407.0742v2Jul 2, 2014physics.soc-phcs.SInlin.AO

TL;DR

Multilayer networks require models that represent diverse relationships, temporal variation, and interdependencies rather than treating all interactions equivalently. This review synthesizes structural measures and dynamics, showing that multilayer coupling changes robustness, diffusion, epidemic thresholds, and synchronization.

Problem
Existing network models lacked a consensus mathematical framework for representing diverse relationships and analyzing dynamics across multilayer systems.
Method
The paper reviews multilayer-network structure and dynamics, covering network representations, centrality measures, diffusion, spreading, percolation, and synchronization.
Results
Multilayer coupling produces distinct effects, including altered robustness, abrupt diffusion-regime transitions, changed epidemic thresholds, and synchronization unavailable in fixed single-layer configurations.
Takeaways & Limitations
Analyzing interactions by layer is important for understanding resilience, coordinated functioning, disease propagation, node importance, and synchronization in complex systems.
Takeaways & Limitations
Epidemic-threshold derivations can be problematic when the maximal eigenvector localizes on an infinitesimal fraction of nodes.

Abstract

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In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

8 Conclusions and open questions · 1.1. The multilayer network approach to nature

The report frames multilayer networks as a necessary extension of traditional network theory for representing systems with multiple interaction types, and surveys their structure, dynamics, applications, and open questions. Its concluding perspective identifies multilayer network science as an emerging interdisciplinary frontier.

1.1. The multilayer network approach to nature: Traditional network models represent constituents as nodes and interactions as links, providing mathematical tools for extracting information about complex systems.This representation is presented as a major effort of modern physics and as the basis of complex networks theory.
1.1. The multilayer network approach to nature: Treating all links equivalently can obscure real-world details and sometimes produce incorrect descriptions of phenomena.The limitation arises because traditional models generally encode each interaction as a weight while neglecting distinctions among relationships.
1.1. The multilayer network approach to nature: Social, transportation, and biological systems illustrate why single-layer representations can fail to capture their intrinsically multirelational structure.Examples include social relationships, air transportation and subway connections, and the C. elegans neural network of 281 neurons and around two thousands of connections.
1.1. The multilayer network approach to nature: Multilayer networks represent different relationship, activity, or category channels as layers, allowing the same node to have distinct interactions and neighbors across layers.This framework explicitly incorporates multiple channels of connectivity within a common representation.
1.1. The multilayer network approach to nature: The multilayer paradigm has produced relevant and unexpected results and is presented as a new frontier likely to stimulate expanding interdisciplinary research.The report notes that this paradigm has also been termed multiplex networks, networks of networks, interdependent networks, and hypergraphs.
8 Conclusions and open questions: The report aims to survey the current state of the art while providing a considered outlook on open questions for future research.This purpose combines a summary of existing knowledge with a perspective on issues that remain to be addressed.
8 Conclusions and open questions: The report concludes with remarks and perspective ideas after covering multilayer definitions, structure, models, robustness, spreading, synchronization, and applications.Its applications span social sciences, technology, economy, climatology, ecology, and biomedicine.

2. The structure of multilayer networks

Complexity science examines systems whose many interdependent components interact through diverse channels, producing self-organization and emergent phenomena relevant to global challenges.

Complexity science studies systems composed of many interdependent components.These systems may interact through many different channels.
Such systems can exhibit self-organization and emergent phenomena.
The field develops new ways to understand mechanisms and processes across physical, social, engineering, and information domains.It addresses challenges relevant to the future of the Worldwide Knowledge Society.

2.1. Definitions and notations

The section defines multilayer networks as collections of layer-specific graphs connected by intralayer and interlayer relationships, and distinguishes multiplex, temporal, interacting, multidimensional, interdependent, and multilevel cases. It also introduces projection and supra-adjacency representations while noting that terminology lacks consensus and alternative formalisms exist.

Core definitions: A multilayer network is M = (G, C), with graph layers Gα and interconnections Eαβ between nodes in different layers.Edges within Eα are intralayer connections, while edges in Eαβ are interlayer connections; C is the set of crossed layers.
Representations: The projection network and supra-adjacency matrix provide monolayer representations of multilayer systems, but flattening can produce behavior related to, yet different from, the original network.The supra-adjacency matrix is a block matrix, and a single node in the multilayer network corresponds to different nodes in the flattened monolayer graph.
Network types: A multiplex network is a multilayer network with the same node set in every layer and interlayer links only between counterpart nodes.Thus, multiplex networks represent a fixed set of nodes connected by different types of links.
Network types: Multilayer representations encompass temporal, interacting, multidimensional, interdependent, and multilevel networks through distinct layer and cross-layer constructions.Temporal networks map each time instant to a layer, while interacting networks use crossed layers for interactions; interdependent networks connect dependent nodes across layers, and multilevel networks use slices.
Notation and terminology: Terminology for networks with multiple relations lacks consensus, motivating precise mathematical definitions; the proposed notation is one framework among alternatives including tensor formalisms.The authors regard their notation as relatively immediate to understand, while tensor formalisms offer compact expressions for multiplex metrics.

2.2. Characterizing the structure of multilayer networks

Multilayer networks require extending node-centrality measures because node relevance is distributed across layers and vector-valued measures do not directly yield a unique ranking. The section develops aggregation-based, independent-layer, uniform, and heterogeneous eigenvector-like centralities, alongside reachability and spectral characterizations.

Degree centrality: The multiplex node degree is a vector containing each node’s degree in every layer, extending the monolayer degree definition.This vector-valued representation creates a ranking problem because there is no single clear ordering of nodes in the corresponding multidimensional space.
Eigenvector-like centrality: Independent-layer eigenvector centrality represents each node by a vector of its layer-specific centralities, which can then be aggregated numerically.Possible aggregations include the sum, maximum, or ℓp-norm.
Eigenvector-like centrality: The independent-layer eigenvector centrality does not fully capture multilevel interactions between layers and their influence on centrality.Uniform and heterogeneous formulations incorporate cross-layer influence, with heterogeneous versions allowing different layer importance through an influence matrix W.
Other structural measures: Multiplex reachability and spectral properties provide additional structural characterizations, including interdependence, supra-adjacency and Laplacian spectra, and irreducibility conditions for unique random-walker stationary states.Interdependence is slightly anti-correlated with overlapping degree, while irreducibility of the projection network’s adjacency matrix can imply irreducibility of the associated matrix B.

2.3. Matrices and tensor representation: Spectral properties

The section represents multilayer networks as tensor products whose basis elements encode node-layer states, then derives matrix representations whose spectral properties are invariant to basis ordering. It also distinguishes supra-adjacency matrices from multiplex representations and highlights the dual roles of nodes and layers.

Tensor representation: Tensor-product basis elements xi ⊗ℓα represent being in node i and layer ℓα.The node and layer sets form bases of vector spaces V and L, making their tensor products a basis of V ⊗L.
Matrix representation: A multilayer network is identified with a linear transformation on V ⊗L and is completely determined by its matrix in the node-layer basis.This extends the monolayer identification of a network with a linear transformation whose matrix is its adjacency matrix.
Spectral properties: Different matrices representing the same tensor are related by permutation similarity and therefore have identical spectral properties.If A and B represent the same tensor, then B = P · A · P −1 for a permutation matrix P.
Supra-adjacency matrices: Supra-adjacency matrices arise from reordering layers and nodes, with blocks given by layer adjacency matrices and potentially non-square blocks in general multilayer networks.Comparing representations reveals dual roles between layers and nodes.
Multilayer versus multiplex: A multilayer adjacency matrix has N 2M 2 entries, whereas a multiplex adjacency matrix with the same nodes and layers is determined by N 2M +M 2 elements.The distinction reflects the structural constraints imposed by multiplex networks.

2.4. Correlations in multiplex networks

Multiplex networks reveal correlations among node roles, link patterns, weights, and activity across layers that are absent from isolated-layer analyses. The section surveys measures for degree correlations, overlap, weighted multiplex structure, and node- and layer-activity patterns.

Interlayer degree correlations: Interlayer degree correlations indicate whether hubs in one layer are also hubs or low-degree nodes in another layer.Correlations may be positive when hubs coincide or negative when hubs in one layer are not hubs in another.
Interlayer degree correlations: Degree correlations can be measured using the full joint matrix P(kα, kβ), conditioned average degree, or Pearson, Spearman, and Kendall coefficients.An increasing or decreasing conditioned average degree indicates positive or negative correlation, respectively; Pearson can be dominated by high-degree nodes, while Spearman is not uniquely defined with tied ranks.
Overlap and multidegree: Significant link overlap occurs across diverse multiplex data sets, including airport, online-game, collaboration, citation, communication, trade, enmity, and attack layers.Overlap counts links present simultaneously in two layers, and examples include substantial overlap between friendship and communication, communication and trade, and enmity and attack.
Multistrengths and inverse multiparticipation ratio of weighted multiplex: Weighted multilink properties differ significantly between links without overlap and links with overlap in scientist collaboration and citation networks.The analyzed citation network is directed, and the exponents λ and β depend on multilink type.
Node and layer pairwise multiplexity: Node activity across layers is typically broad, following P(B_i) ≃ B_i^-δ with δ ∈ [1.5, 3.0], while layer activity is also broadly distributed.For δ ≤ 2 the node-layer bipartite network can be dense, whereas for δ > 2 it can be sparse; the typical number of active layers per node remains subject to unbound fluctuations.

3. Generative models

Generative models for multilayer networks fall into growing models with generalized preferential attachment and multilayer network ensembles with structural constraints. Although this modeling framework remains in its infancy, proposed models generate multiplex networks with controlled correlations, overlap, and diverse degree distributions.

Model classes: Multilayer generative models divide into growing networks, where node numbers increase under generalized preferential attachment, and network ensembles satisfying structural constraints.Growing models explain multilayer evolution from simple dynamical rules, while ensembles contain N nodes in each layer and can control degree-degree correlations and overlap.
Model classes: The multilayer modeling framework remains in its infancy, despite proposals for constructing multiplex networks with different topological features.The section summarizes the principal results and concepts of these constructions.
Growing multiplex models: Generalized preferential attachment can produce positive degree correlations between corresponding replica nodes, even when layers appear dynamically decoupled.In the case c1,1 = c2,2 = 1, identical replica ages generate degree-degree correlations across the duplex layers.
Growing multiplex models: Older multiplex nodes remain more connected in both layers, yielding a linear average degree in one layer conditioned on the degree in the other.This behavior was reported for other values of the attachment parameters as well as the decoupled case.
Growing multiplex models: Multiplex growth models can generate heterogeneous or homogeneous degree distributions, link condensation, and either positive or negative interlayer degree correlations depending on parameters.For one model, condensation occurs for α + β > 1 when β ≥0, or for α > 1 when β < 0; the Kendall coefficient distinguishes positive and negative correlation regions.
Network ensembles: Canonical and microcanonical network ensembles differ in constraint enforcement, and the cited multiplex ensemble can be generated layer by layer using configuration models without multiple edges or tadpoles.The Gibbs entropy is lower than the entropy S, and the two ensembles are not thermodynamically equivalent in the stated case.

4. Resilience and percolation … 4.3. Interdependent multiplex networks

Multilayer resilience and percolation differ sharply from single-layer behavior because interdependencies can propagate failures across layers and produce discontinuous emergence of a mutually connected giant component. In multiplexes, topology, interdependence, and cascade dynamics determine robustness, while partial interdependence can change the transition from discontinuous to continuous.

4.1. The fragility of multilayer interdependent networks: Interdependent layers can trigger cascading failures, making multilayer structures more fragile than single-layer networks and motivating strategies for more resilient infrastructure, financial, and biological systems.In the absence of correlations, interdependent scale-free networks can be more fragile than Poisson networks at constant average degree.
4.2. Percolation in interdependent networks: Percolation in interdependent networks is evaluated through the mutually connected giant component, the largest component surviving damage propagation back and forth across layers.For random networks without link overlap and with interdependence, its size emerges discontinuously at p = pc, unlike the continuous transition in single-layer networks.
4.2.1. The mutually connected giant component of a multilayer network: A node belongs to the mutually connected component only when it has a same-layer neighbor in that component and all nodes reachable through interdependent links have such support.For locally tree-like networks, message passing determines membership recursively while accounting for damage states and replica-node dependencies.
4.3. Interdependent multiplex networks: In fully interdependent multiplexes, correlated damage removes replica nodes together, and layer-specific degree distributions determine message equations and mutually connected-component size.A multiplex connects every layer to every other through replica-node dependencies, with each layer characterized by Pα(k).
4.3.1. Case of a multiplex formed by M Poisson networks with the same average degree: For every finite M ≥2, the mutually connected giant component in equal-degree Poisson multiplexes emerges discontinuously at p = pc(M).For a duplex, the critical value is y = yc = 2.45541 . . ., with Sc = xc = 1.25643 . . ., and the transition is a discontinuous hybrid transition with a square-root singularity.
4.3.2. Case of a duplex network formed by two Poisson networks with different average degree: For duplexes with different Poisson average degrees, the phase diagram depends only on (pzA, pzB), and the mutually connected component appears through a discontinuous hybrid transition.The transition points satisfy h(S) = 0 and h′(S) = 0.
4.3.3. Multiplex networks formed by layers of scale-free networks: Unlike single-layer scale-free networks, broader scale-free degree distributions make interdependent multiplexes more fragile: at fixed average degree, pc(γ) increases as γ decreases.Single-layer scale-free networks with γ ∈(2, 3] have pc →0 as N →∞, but this robustness does not carry over to interdependent multiplexes.
4.3.4. Cascading failures; 4.3.5. Partial interdependence; 4.3.6. Percolation in multiplex networks with multiple support-dependence relations: Failures iterated between layers can become systemic near p ≃pc, while partial interdependence changes transition type and produces long avalanches near the hybrid discontinuous regime.At r = 0 layers percolate independently through a continuous transition; at r = 1 the transition is discontinuous hybrid, with a tricritical threshold rc separating regimes. In multiple support-dependence networks, percolation is hybrid and discontinuous but becomes continuous as ˜a,˜b →∞.

4.4. Interdependent networks of networks … 4.7. Cascades on multilayer networks

The review shows that multilayer percolation depends strongly on interdependency structure, correlations, overlap, and spatial embedding. It also covers antagonistic, k-core, viability, weak-percolation, and cascade processes, which exhibit discontinuities, bistability, hysteresis, or altered vulnerability.

4.4.1. Case I: Networks of networks with fixed supernetwork and interdependent links allowed only between replica nodes: In replica-coupled networks of networks, a connected supernetwork follows multiplex MCGC equations, but the mutual component and critical point depend on layer count M.For M ≥2, the transition is hybrid and discontinuous; partial interdependence can make it continuous at a tricritical point.
4.4.2. Case II: Networks of networks with given superdegree of the layers: With heterogeneous layer superdegrees, networks of networks display multiple percolation transitions; higher-superdegree layers are more fragile, and transitions can be continuous or discontinuous when r < 1.The order parameter Σ determines σq and Sq for every superdegree q.
4.4.3. Case III: Networks of networks with fixed supernetwork and random permutation of the labels of the nodes: For randomly permuted interdependencies, tree supernetworks reproduce Case I, whereas random supernetworks with given superdegree distribution reproduce Case II’s multiple transitions and superdegree-dependent fragility.Thus, the supernetwork structure determines whether all layers percolate together or transitions occur across superdegrees.
4.5. Effects of correlations and embedding space on percolation properties of multilayer networks: Link overlap, degree correlations, and spatial embedding can substantially alter multilayer percolation, including transition order, threshold pc, and whether interdependencies sharpen transitions.Total overlap yields a single-layer giant component with a second-order transition, while maximally-positive degree correlations lower pc relative to maximally-negative correlations.
4.6. Other percolation problems: Beyond interdependencies, multilayer percolation includes classical connectivity, antagonistic networks, k-core structure, viability, and weak percolation, with distinct recursive or path-based conditions.Antagonistic duplexes can be bistable; multiplex k-cores are generally hybrid and discontinuous except the trivial (1, 1)-core.
4.6.2. Percolation of antagonistic networks: Antagonistic duplexes can have regions where neither layer percolates, only one layer percolates, or two bistable solutions exist depending on initial conditions.This behavior arises from requiring intralayer support while excluding neighbors of the replica node in the other layer.
4.7. Cascades on multilayer networks: Multilayer cascades include load, sandpile, threshold-adoption, and novel processes; multiplexity can leave avalanche-size distributions unchanged while altering detailed dynamics and vulnerability.High-degree nodes fail more often in multiplex sandpile dynamics, and increasing multiplexity can increase vulnerability to global cascades.

5. Spreading processes and games

This section revisits dynamical processes in multilayer networks, covering diffusion and transport, spreading and epidemic dynamics, and evolutionary games. It examines how multilayer structure and interlayer coupling affect relaxation, congestion, outbreak thresholds, control, and cooperation.

Diffusion and transport: Multilayer transport processes generalize diffusion and random walks by allowing flows to use multiple network layers or channels.The section also considers congestion when nodes have limited capacity for sending and storing incoming packets.
Spreading processes and games: The section extends epidemic-spreading methods to multilayer networks, emphasizing outbreak thresholds alongside control measures, disease–information coevolution, and impact transmission.It also examines how multilayer structure affects the emergence and survival of cooperation through evolutionary games and cooperative clustering.
Diffusion and transport: For Dx ≪1, the M −1 smallest non-trivial supra-Laplacian eigenvalues increase linearly with Dx, and diffusion time scales as τ ∝(M · Dx)−1.This describes the weak interlayer-coupling regime.
Diffusion and transport: For Dx ≫1, the N −1 smallest non-trivial eigenvalues approximate those of a network formed by averaging the multiplex layers.In this regime, the multiplex is faster than its slowest layer, and superdiffusion can occur when it relaxes faster than every composing layer.
Diffusion and transport: Diffusive properties change non-homogeneously with interlayer coupling, with a composition-dependent (Dx)c marking a structural transition from decoupled to systemic behavior.The transition is identified through the evolution of λ2(Dx) and changes in the associated eigenvector structure.
Diffusion and transport: For sufficiently large Dx, random walks on two-layer multiplexes cover the multiplex in less time than separately exploring each layer.Coverage measures the number of nodes visited at least once during a time window τ.

G21 G2 . . . G2M ... ... ... GM1 GM2 . . . GM

Multilayer structure shapes information flow, routing congestion, and epidemic onset. Coupling can enable spreading below isolated-network thresholds, with effects determined by spectral properties, interlayer links, and transmission conditions.

Information flow: Communicability captures information flowing in parallel between two nodes across different layers.This accounts for multilayer information transfer between within-layer representations of the nodes.
Routing and congestion: Shortest-path diffusion models transportation and information flows that follow paths minimizing distance or other costs.Such routing can concentrate flows on a small subset of edges, producing congestion.
Routing and congestion: An optimal coupling λ⋆ can balance flow spreading against congestion, measured by the Gene coefficient G ∈[0, 1].Small G values indicate spread flows, whereas large G values indicate concentrated flows across edges.
Epidemic spreading: Interconnected networks can exhibit epidemics at infection rates insufficient for propagation in either isolated network.For SIS dynamics, this result follows from heterogeneous mean-field, microscopic, and spectral analyses.
Epidemic spreading: The epidemic threshold of an interconnected system is smaller than those of isolated networks, with strong reductions when high-eigenvector-centrality nodes are linked.The critical point depends on the product of eigenvector centralities of connected nodes across networks.
Epidemic spreading: For multiplex SIR dynamics, strong coupling prevents the epidemic threshold from exceeding isolated-network thresholds, while spreading conditions can depend on the size of a participating layer.Different transmission layers can also represent distinct channels for different diseases, with onset conditions depending on τ1 = β1/µ1 and τ2 = β2/µ2.

6. Synchronization

The section distinguishes two multilayer synchronization scenarios: alternating connectivity layers that create time-dependent coupling, and simultaneously active layers with explicit interlayer interactions. It summarizes how traditional monolayer synchronization analyses extend to both cases.

Overview: Multilayer synchronization requires separate treatment of two scenarios.The section is divided into two main parts corresponding to these scenarios.
Time-dependent connectivity: In the first scenario, layers alternate between connectivity configurations, producing a time-dependent coupling structure.The key issue is how synchronization existence and stability change relative to time-independent monolayer coupling.
Time-dependent connectivity: The first scenario asks how synchronization existence and stability conditions are modified from the classic monolayer case.The monolayer reference has time-independent coupling between networked units.
Simultaneous multilayer coupling: In the second scenario, layers simultaneously couple the units and require explicit layer-layer interactions.Interlayer synchronization is more general than intralayer synchronization: units within each layer may remain out of synchrony while synchronizing across layers.
Analytical extensions: The section extends analytical treatments traditionally used for monolayer network synchronization to both multilayer scenarios.Each part reports on the corresponding extension of the main analytical approaches.

6.1. Synchronization in multilayer networks with alternating layers

The section examines synchronization when identical dynamical units switch among multilayer connections, without assuming a timescale for wiring variation. Commuting layers can enhance synchronization beyond any fixed layer, although commutativity is too restrictive for general multilayer networks.

Synchronization framework: Synchronization is analyzed for identical units whose connections switch among layers according to periodic or aperiodic sequences.The treatment does not impose an a priori timescale for variation in the coupling wiring.
Synchronization framework: Time-varying eigenvector bases add a term absent from classical Master Stability Function equations, requiring a generalized stability analysis.The analysis first treats cases where this term vanishes, then constructs quasi-static evolution among layers.
Commuting layers: Commuting-layer evolution can start from any initial wiring and provides a better condition for synchronous-state stability than a monolayer network.Eigenvalue-surrogate and ordered-selection methods substantially preserve the initial graph’s main topological features while constructing commuting layers.
Synchronization enhancement: A multilayer evolution can remain synchronizable even when fixed layers prevent synchronization, because Λmax (σλi(t)) need not be negative at all times.In the numerical example, Λmax (σλj) is positive for the first 80 eigenvalues in each fixed layer, yet the multilayer network synchronizes; Tsync scales almost exponentially with τ.
Scope and limitation: The synchronization criterion is general beyond periodic connectivity, but commuting layers remain too strong an assumption for real-world multilayer graphs.The framework allows wirings evolving over secular yet finite timescales, while motivating treatment of noncommutative layers.

6.2. Synchronization in multilayer networks with coexisting layers

Synchronization in multilayer networks depends on layer structure, interlayer connectivity, delays, and node organization, producing diverse regimes from complete and concurrent synchronization to breathing and targeted dynamics.

Structural conditions: Synchronizability generally cannot be reduced to a monolayer formalism, although reduction is possible for M = 2 when the associated Laplacians commute or under other specified conditions.The passage indicates that a broader class of two-layer networks also permits such reduction when one coupling layer satisfies an additional condition, but that condition is truncated.
Interlayer connectivity: Connecting the hubs of two networks is the best strategy for achieving complete synchronization, because connector nodes crucially determine the synchronous behavior of the whole system.This result combines experimental, numerical, and analytical studies of two-layer networks.
Interlayer connectivity: Increasing interlinks between identical layers produces a synchronizability phase transition whose exact location depends exclusively on the algebraic connectivity of the graph models.The transition is observed regardless of which graphs are interconnected.
Clustered networks: In clustered networks, the interplay between interconnections and intraconnections determines global synchronizability and distinct synchronization scenarios, with strongest synchronizability when their numbers match perfectly.The cited study considers two clusters interacting through random connections.
Delays and topology: Time delays and heterogeneous topologies generate varied synchronization behavior, including breathing synchronization, topology-dependent master-slave synchronization, parity-dependent stability, and discontinuous transitions in mediated oscillator systems.These studies examine delayed intra- and interlayer coupling, identical or non-identical connection topologies, ring subnetworks, and hub-mediated communication.
Concurrent synchronization: Concurrent synchronization consists of multiple groups of fully synchronized elements coexisting in distinct, possibly chaotic, dynamics, and its stability can be analyzed using invariant subspaces, master stability functions, and sufficient conditions for delayed coupled networks.Such multisynchronous behavior is linked to coordinated tasks in robotics and neuroscience, while some methods cannot guarantee stability when both intra-group and extra-group connections are present.

7. Applications

Multilayer networks extend applications of network theory by representing distinct relationship types, temporal layers, and interdependent systems that single-layer models oversimplify. Studies apply this framework across social, political, infrastructural, transportation, trade, and financial systems to reveal layer-specific structures and dynamics.

Social networks: Social-network applications distinguish relationship types such as personal ties, awareness, friendship, messages, trading, aggression, and online interactions.Examples include three-layer researcher networks, six-layer player networks, and an eleven-layer house-building community network.
Political networks: Temporal political layers enable community analysis of senators’ voting patterns and can help infer information about the global political situation.The network represents multiple years of call voting among US senators as separate layers.
Social networks: Multilayer social analyses show that interaction channels can differ substantially, including phone calls and SMS, while weak ties can support large-scale diffusion.A metropolitan dataset contained more than 63 million calls and 20 million SMS records; 30% of users used only calls and 8% only SMS.
Interdependent systems: Interdependent-system studies model cascading failures across coupled networks, showing that limited power-grid connectivity can help while excessive interconnectivity can worsen cascades.Applications include communication-linked electrical grids and interactions among the Western, Eastern, and Texas US power regions.
Transportation networks: Transportation applications combine air, rail, road-access, and airline layers to characterize networks with different sizes, link structures, and hub configurations.The Indian air-and-train model links airport-station pairs through direct road access, while airline layers contrast major-carrier hub-and-spoke structures with low-cost carriers.
Trade and financial networks: Trade and banking applications show that multilayer structure changes measured organization across commodities, countries, years, and financial-exposure types.Studies examine 162 countries across 97 yearly commodity layers, commodity-sensitive port measures, and Italian-bank exposures separated by maturity and secured status.

8. Conclusions and open questions

The review unifies multilayer-network concepts, models, structural processes, dynamics, and applications while identifying major open problems in mathematical formalization, generative mechanisms, symmetry, adaptive interactions, and biological data. It emphasizes that interlayer coupling produces properties and dynamics that differ substantially from single-layer networks.

Mathematical foundations: A comprehensive mathematical formalism is still needed to unify the diverse terminology and definitions used for multilayer networks.The review established common notation but identifies the formal framework as incomplete.
Generative models: Generative models for multilayer networks remain in their infancy, despite their importance for explaining structures observed in social, technological, and biological systems.Understanding the mechanisms behind real-world multilayer topology is described as a central priority.
Resilience and percolation: Interdependencies make resilience and percolation exhibit unexpected properties that cannot be reduced to single-layer behavior.The nature of interlinks matters: replica-linked networks can make mutually connected-component size independent of a connected supernetwork, whereas random interlinks make it dependent on supernetwork structure.
Spreading and evolutionary dynamics: Multilayer coupling reshapes diffusion, disease spreading, and evolutionary games by controlling coordinated functioning, outbreak thresholds, and collective states.Applications include fairness, species diversity, cyclical dominance, language evolution, and climate-change mitigation, where interactions among local systems can trigger cascades.
Synchronization: Synchronization can be enhanced and treated generally across multilayer systems, but the effects of multilayer symmetry remain unexplored and may differ between temporal and spatial layers.The review reports results ranging from commutative-layer sequences to treatments without constraints on layer relations, while identifying symmetry as an open problem.
Open applications and mechanisms: Future research should explain how multilayer links emerge through adaptive interactions and expand applications in biology and biomedicine despite difficulties recording data in living organisms.Adaptive interaction weights are identified as a mechanism for spontaneous multilayer-link organization, while biological and biomedical uptake has lagged because real-data collection is difficult.

Note added in proof

The note added in proof lists recent contributions related to the report’s sections and updates several cited arXiv manuscripts that had meanwhile appeared in peer-reviewed journals.

Additional references: The authors provide additional recent references, organized by the report sections to which they relate.These contributions were identified after editing the report and are offered for readers’ further consultation.
Publication updates: Some material came from the ArXiv repository, and the authors warn that those manuscripts may meanwhile have appeared in peer-reviewed journals.The authors report several publications they had learned about themselves.
Publication updates: Seven cited references are updated with publication details in PNAS, Phys. Rev. E, and PLoS ONE.The updates concern Refs.,,,,, [322], and.