Multilayer networks

Mikko KiveläAlexandre ArenasMarc BarthelemyJames P. GleesonYamir MorenoMason A. Porter

article2013Journal of Complex Networks3,429 citations

Establishes a unified mathematical framework and standardized terminology for studying complex systems with multiple layers of connectivity, resolving conflicting definitions across multiplex, interdependent, and interconnected network models.

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The study of complex systems requires moving beyond traditional single-layer network representations to account for multiple types of interactions, temporal evolution, and other complications that arise in natural and engineered systems. A comprehensive review addresses this need by developing a general mathematical framework for multilayer networks and unifying the rapidly expanding but terminologically fragmented literature on the topic.

This work sets out to provide a unified treatment of multilayer networks by constructing a general definition that can represent most existing concepts, creating a dictionary to relate disparate notions such as multiplex networks, interdependent networks, and networks of networks, and surveying data sets, diagnostics, models, and dynamical processes.

The authors employ a literature review combined with the construction of a tensor-based mathematical framework, where multilayer networks are defined as quadruplets consisting of node-layer tuples, edges between them, node sets, and layer sets. They classify existing structures by imposing constraints such as node alignment, diagonal couplings, and categorical versus ordinal inter-layer connections, and they map a wide range of network types to this representation using both tensor and supra-adjacency matrix formulations.

Key findings include the demonstration that a single general framework encompasses the vast majority of multilayer network concepts in the literature, with specific types arising from particular constraints on the general structure. The review identifies numerous empirical data sets that are naturally multilayer, ranging from social networks with multiple relationship types to transportation systems and temporal networks. Generalizations of monoplex diagnostics such as degree, clustering coefficients, centrality measures, and community detection are shown to reveal new phenomena when applied to multilayer settings, and dynamical processes like percolation exhibit first-order phase transitions in interdependent networks that differ markedly from single-layer behavior.

These results indicate that multilayer representations capture essential features of real systems that are lost in aggregation or single-layer approximations, with important consequences for understanding robustness, information flow, and structural organization. The framework reveals that inter-layer connections and multiplexity can fundamentally alter network properties and dynamics, necessitating new analytical tools rather than simple extensions of existing ones.

Further work is needed to develop computationally efficient methods for large multilayer networks, to collect and analyze data with explicit inter-layer coupling strengths, and to extend dynamical studies to more general multilayer structures. Limitations include the early stage of many generalizations and the challenge of obtaining high-quality multilayer empirical data, suggesting caution in applying results without careful validation against specific systems.

CEA Centre d'Analyse et de Mathématiques Sociales Departament d’Enginyeria Informàtica i Matemàtiques Department of Mathematics & Statistics Department of Theoretical Physics École des hautes études en sciences sociales Institut de Physique Théorique Institute for Biocomputation and Physics of Complex Systems (BIFI)MACSI Mathematical Institute Universidad de Zaragoza Universitat Rovira i Virgili University of Limerick University of Oxford network architecture Multilayer Network Multiplex Network Interdependent Network

arXiv: 1309.7233 mnets/pymnet

Paper: Catastrophic cascade of failures in interdependent networks, S. Havlin et al. (2009). This paper establishes foundational models and percolation thresholds for interdependent networks, which serve as direct prerequisites for the broader multilayer framework analyzed in the source.

Paper: Semi-Supervised Classification with Graph Convolutional Networks, Thomas N. Kipf et al. (2017). This work extends the general principles of multilayer network analysis and graph connectivity into practical, semi-supervised graph convolutional network architectures.
Paper: Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering, Michaël Defferrard et al. (2016). This paper builds directly upon general multilayer and graph-structured frameworks by introducing fast localized spectral filtering techniques for graph convolutional networks.

Abstract

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.

Topics

network architecture

Keywords

Multilayer Network Multiplex Network Interdependent Network Supra‑Adjacency Matrix Adjacency Tensor Community Structure Percolation Process Temporal Network Node‑Aligned Coupling (Categorical / Ordinal)

1 Introduction
2 Multilayer Networks
2.1 General Form
2.2 Tensor Representations
2.2.1 Constraints.
2.2.2 Tensor Flattening.
2.3 Supra-adjacency Representation
2.4 Node-Colored Networks, Interconnected Networks, Interdependent Networks, and Networks of Networks
2.5 Multiplex Networks and Multirelational Networks
2.6 Hypergraphs
2.7 Ordinal Couplings and Temporal Networks
2.7.1 Networks with Both Ordinal and Categorical Aspects
2.8 Other Types of Networks and Graphs
3 Empirical Multilayer Networks
4 Models, Methods, and Dynamics
4.1 Network Aggregation: From Multiplex Networks to Monoplex Networks
4.2 Diagnostics for Multilayer Networks
4.2.1 Node Degree and Neighborhood.
4.2.2 Walks, Paths, and Distances.
4.2.3 Clustering Coefficients, Transitivity, and Triangles.
4.2.4 Centrality Measures.
4.2.5 Inter-Layer Diagnostics.
4.3 Models of Multiplex Networks
4.4 Models of Interconnected Networks
4.5 Communities and Other Mesoscale Structures
4.5.1 Community Structure in Multilayer Networks
4.5.2 Methods Based on Tensor Decomposition.
4.6 Dynamical Systems on Multilayer Networks
4.6.1 Connected Components and Percolation.
4.6.2 Percolation Cascades.
4.6.3 Compartmental Spreading Models and Diffusion.
4.6.4 Coupled-Cell Networks
4.6.5 Other Types of Dynamical Systems.
4.6.6 Control and Dynamics.
5 Conclusions and Outlook
Acknowledgements
6 Appendix: Glossary and Notation
References

Knowls

Knowl 1 — Mathematical Definition of a General Multilayer Network
definition
A general multilayer network is defined as a quadruplet

$M = (V_M, E_M, V, \mathbf{L}),$

where:
- $V$ is the underlying set of physical entities or nodes.
- $\mathbf{L} = \{L_a\}_{a=1}^d$ is a sequence of $d$ sets of elementary layers, where $d \in \mathbb{N}_0$ denotes the number of aspects (or categorical/ordinal dimensions) of the layer structure. When $d = 0$ , the structure reduces to an ordinary monoplex graph $G = (V, E)$ .
- The complete layer space is the Cartesian product $\mathcal{L} = L_1 \times L_2 \times \dots \times L_d$ , where a specific layer is an ordered tuple $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_d)$ with $\alpha_a \in L_a$ .
- $V_M \subseteq V \times L_1 \times \dots \times L_d$ is the set of node-layer tuples $(u, \boldsymbol{\alpha}) = (u, \alpha_1, \dots, \alpha_d)$ that are actually present in the system, satisfying $\{u \mid (u, \boldsymbol{\alpha}) \in V_M\} = V$ .
- $E_M \subseteq V_M \times V_M$ is the edge set containing directed or undirected pairs of node-layer tuples $((u, \boldsymbol{\alpha}), (v, \boldsymbol{\beta}))$ .
The pair $G_M = (V_M, E_M)$ forms the underlying graph (or supra-graph) of the multilayer network.
Knowl 2 — Structural Constraints and Couplings in Multilayer Networks
definition
A general multilayer network $M = (V_M, E_M, V, \mathbf{L})$ can be constrained by specific structural properties:
- Node-aligned (Fully interconnected): Every node exists in every layer, such that $V_M = V \times L_1 \times \dots \times L_d$ .
- Layer-disjoint: Each node is present in at most one layer; that is, if $(u, \boldsymbol{\alpha}), (u, \boldsymbol{\beta}) \in V_M$ , then $\boldsymbol{\alpha} = \boldsymbol{\beta}$ .
- Intra-layer vs. Inter-layer edges: The edge set $E_M$ is partitioned into intra-layer edges $E_A = \{((u, \boldsymbol{\alpha}), (v, \boldsymbol{\beta})) \in E_M \mid \boldsymbol{\alpha} = \boldsymbol{\beta}\}$ and inter-layer edges $E_C = E_M \setminus E_A$ .
- Coupling edges: Inter-layer edges connecting the same entity across distinct layers, defined as $E_{\tilde{C}} = \{((u, \boldsymbol{\alpha}), (u, \boldsymbol{\beta})) \in E_C\}$ .
- Diagonal coupling: All inter-layer edges are coupling edges ( $E_C = E_{\tilde{C}}$ ), meaning inter-layer edges only exist between counterpart instances of the same node across different layers.
- Layer-coupled: A diagonal network where coupling presence and edge weights depend solely on the layers and are independent of the specific node identities: if $((u, \boldsymbol{\alpha}), (u, \boldsymbol{\beta})) \in E_C$ and $(v, \boldsymbol{\alpha}), (v, \boldsymbol{\beta}) \in V_M$ , then $((v, \boldsymbol{\alpha}), (v, \boldsymbol{\beta})) \in E_C$ with equal weights $w(((u, \boldsymbol{\alpha}), (u, \boldsymbol{\beta}))) = w(((v, \boldsymbol{\alpha}), (v, \boldsymbol{\beta})))$ .
- Categorical coupling: Diagonal coupling where each node is connected to its counterpart across all possible layer pairs: $(u, \boldsymbol{\alpha}), (u, \boldsymbol{\beta}) \in V_M \implies ((u, \boldsymbol{\alpha}), (u, \boldsymbol{\beta})) \in E_M$ .
- Ordinal coupling: Diagonal coupling where elementary layers in an aspect possess an ordered sequence (e.g., discrete time steps), and counterpart nodes are connected only between consecutive or bounded-horizon neighboring layers.
Knowl 3 — Adjacency Tensor Representation of Multilayer Networks
model/method
A node-aligned multilayer network $M = (V_M, E_M, V, \mathbf{L})$ with $d$ aspects can be represented by an adjacency tensor $\mathcal{A}$ of order (rank) $2(d+1)$ :

$\mathcal{A} \in \{0, 1\}^{|V| \times |V| \times |L_1| \times |L_1| \times \dots \times |L_d| \times |L_d|},$

with components denoted by $A_{uv\boldsymbol{\alpha}\boldsymbol{\beta}} = A_{uv\alpha_1\beta_1\dots\alpha_d\beta_d}$ , where $A_{uv\boldsymbol{\alpha}\boldsymbol{\beta}} = 1$ if and only if there is an edge directed from node-layer $(u, \boldsymbol{\alpha})$ to $(v, \boldsymbol{\beta})$ (i.e., $((u, \boldsymbol{\alpha}), (v, \boldsymbol{\beta})) \in E_M$ ), and $0$ otherwise.

For weighted multilayer networks, the weighted adjacency tensor $\mathcal{W}$ assigns the real-valued edge weight $W_{uv\boldsymbol{\alpha}\boldsymbol{\beta}} = w(((u, \boldsymbol{\alpha}), (v, \boldsymbol{\beta})))$ when an edge exists and $0$ otherwise.

For networks that are not node-aligned, the tensor representation can be applied by adding isolated "empty nodes" to layers where a node is absent, though care must be taken so that empty nodes do not distort calculations of network diagnostics such as degree distributions or clustering coefficients.
Knowl 4 — Supra-Adjacency and Supra-Laplacian Matrix Formulations
model/method
A multilayer network $M$ can be flattened into an ordinary single-layer supra-graph $G_M = (V_M, E_M)$ with an associated supra-adjacency matrix $\mathbf{A}_M$ .

The combinatorial supra-Laplacian matrix $\mathbf{L}_M$ of an unweighted or weighted multilayer network is defined as:

$\mathbf{L}_M = \mathbf{D}_M - \mathbf{A}_M,$

where $\mathbf{D}_M$ is the diagonal matrix whose entries are the generalized degrees (or strengths) of each node-layer tuple $(u, \boldsymbol{\alpha})$ :

$(\mathbf{D}_M)_{(u,\boldsymbol{\alpha}),(u,\boldsymbol{\alpha})} = \sum_{(v,\boldsymbol{\beta}) \in V_M} (\mathbf{A}_M)_{(u,\boldsymbol{\alpha}),(v,\boldsymbol{\beta})}.$

The normalized supra-Laplacian matrix is defined as:

$\hat{\mathbf{L}}_M = \mathbf{I} - \mathbf{D}_M^{-1/2} \mathbf{A}_M \mathbf{D}_M^{-1/2},$

where $\mathbf{I}$ is the identity matrix of dimension $|V_M| \times |V_M|$ .
Knowl 5 — Tensor Flattening and Path Counting via Multilayer Tensor Multiplication
model/method
Tensor flattening (unfolding or matricization) maps an adjacency tensor $\mathcal{A}$ of $d$ aspects to a lower-order tensor or matrix by combining pairs of aspects. For two aspects $i$ and $j$ , their index sets $L_i$ and $L_j$ are merged into a single composite aspect $L_h = L_i \times L_j$ of size $|L_h| = |L_i||L_j|$ via the bijective index mapping:

$\alpha_h = (\alpha_i - 1)|L_j| + \alpha_j.$

Flattening all aspects down to $d=0$ produces a linear bijection $f: \mathbb{R}^{|V| \times |V| \times |L_1| \times |L_1| \times \dots \times |L_d| \times |L_d|} \to \mathbb{R}^{N_S \times N_S}$ , where $N_S = |V| \prod_{a=1}^d |L_a|$ , that maps the adjacency tensor to the supra-adjacency matrix.

An induced tensor multiplication $\times_f$ on the tensor space is defined by:

$\mathcal{A} \times_f \mathcal{B} = f^{-1}(f(\mathcal{A}) \cdot f(\mathcal{B})),$

where $\cdot$ denotes standard matrix multiplication. The number of walks of length $N$ starting at node-layer $(u, \boldsymbol{\alpha})$ and ending at node-layer $(v, \boldsymbol{\beta})$ is computed directly as the tensor entry:

$\left(\underbrace{\mathcal{A} \times_f \mathcal{A} \times_f \dots \times_f \mathcal{A}}_{N \text{ times}}\right)_{uv\boldsymbol{\alpha}\boldsymbol{\beta}}.$
Knowl 6 — Structural Transition and Algebraic Connectivity in Multilayer Networks
theoretical result
In a node-aligned multiplex network with categorical interlayer coupling of weight $\omega$ and intra-layer networks $G_\alpha$ , the algebraic connectivity (the second-smallest eigenvalue $\lambda_2(\mathbf{L}_M)$ of the combinatorial supra-Laplacian $\mathbf{L}_M$ ) exhibits a discontinuous structural phase transition as a function of the inter-layer coupling strength $\omega$ :
1. Decoupled regime (small $\omega$ ): When the coupling $\omega$ is below a critical threshold $\omega^*$ , $\lambda_2(\mathbf{L}_M)$ is governed strictly by the inter-layer coupling topology and is independent of the intra-layer network structure.
2. Coupled regime (large $\omega$ ): When $\omega > \omega^*$ , $\lambda_2(\mathbf{L}_M)$ is bounded from above by a constant factor multiplied by the algebraic connectivity of the unweighted superposition (aggregated) network $\bar{G} = \sum_{\alpha=1}^b G_\alpha$ .
Knowl 7 — Network Aggregation and Eigenvalue Interlacing
model/method
Given a single-aspect multiplex network defined by a 3rd-order weighted adjacency tensor $\mathcal{W} \in \mathbb{R}^{n \times n \times b}$ with $b$ layers and $n$ nodes, an aggregated (or superposition) monoplex network is constructed as a linear combination of layer adjacency matrices:

$W_{uv} = \sum_{\alpha=1}^b m_\alpha W_{uv\alpha},$

where $\mathbf{m} = (m_1, \dots, m_b)^T \in \mathbb{R}^b$ is a vector of layer weights. When layer importances are equal and unweighted, $\mathbf{m} = (1, \dots, 1)^T$ , and $W_{uv}$ equals the total number of distinct edge types connecting node $u$ and node $v$ .

For node-aligned multiplex networks, the eigenvalues of the supra-adjacency matrix $\mathbf{A}_M$ and combinatorial supra-Laplacian $\mathbf{L}_M$ interlace with the eigenvalues of the weighted adjacency matrix $\frac{1}{b}\mathbf{W}$ and combinatorial Laplacian matrix $\frac{1}{b}\mathbf{L}_{agg}$ of the averaged aggregated network.
Knowl 8 — Multidegree, Multistrength, and Overlap Multiplicity
definition
For a single-aspect multiplex network with $b$ layers, an edge profile between two nodes is characterized by a binary vector $\mathbf{m} = (m_1, \dots, m_b) \in \{0, 1\}^b$ , where $m_\alpha = 1$ indicates the presence of an edge on layer $\alpha$ and $m_\alpha = 0$ indicates its absence.
- Multidegree ( $k_u^{\mathbf{m}}$ ): The number of neighbors with which node $u$ shares exactly the edge profile $\mathbf{m}$ .
- Multistrength ( $s_{u,\alpha}^{\mathbf{m}}$ ): The sum of edge weights incident to node $u$ on layer $\alpha$ across all multi-edges having profile $\mathbf{m}$ .
- Overlap Multiplicity ( $\nu(\mathbf{m})$ ): The number of active layers in the vector $\mathbf{m}$ :
$\nu(\mathbf{m}) = \sum_{\alpha=1}^b m_\alpha.$
- Degree at Overlap $\nu$ ( $k_u(\nu)$ ): The average multidegree of node $u$ across all possible multi-edge configurations with overlap $\nu \in \{0, \dots, b\}$ :
$k_u(\nu) = \binom{b}{\nu}^{-1} \sum_{\{\mathbf{m} \mid \nu(\mathbf{m}) = \nu\}} k_u^{\mathbf{m}}.$
Knowl 9 — Cascading Failures and Viable Cluster Collapse in Interdependent Networks
model/method
In a system of interdependent networks containing intra-layer connectivity edges and inter-layer dependency edges, node failures propagate through iterative cascades across layers:
1. Initial removal of a fraction $1 - p$ of nodes in a designated layer causes intra-layer fragmentation into connected components.
2. Nodes in other layers that depend on non-functional nodes, or that are disconnected from the giant connected component (GCC) of their respective intra-layer network, fail.
3. The removal of dependent nodes induces further disconnection within intra-layer networks, creating an alternating, recursive cascade across the layers until a steady state is reached.
The surviving functional subgraph is the mutually-connected giant component (or giant viable cluster), consisting of nodes that are simultaneously connected to each other via functioning intra-layer paths in all layers.

Unlike monoplex networks, which undergo a continuous (second-order) percolation transition under random failures where high degree heterogeneity enhances robustness, interdependent networks under the same conditions often exhibit an abrupt, discontinuous (first-order) phase transition and are substantially more vulnerable to failure when intra-layer degree distributions are broad or scale-free.
Knowl 10 — Mapping Between Multilayer Networks and Related Network Formalisms
model/method
Various non-standard network formalisms map into the general multilayer framework $M = (V_M, E_M, V, \mathbf{L})$ :
- Node-Colored / Interconnected Networks: A graph $G_c = (V_c, E_c, C, \chi)$ with color mapping $\chi: V_c \to C$ maps to a single-aspect ( $d=1$ ) layer-disjoint multilayer network with layer set $L = C$ , node set $V = V_c$ , $V_M = \{(u, c) \in V \times L \mid \chi(u) = c\}$ , and $E_M = \{((u, c_1), (v, c_2)) \in V_M \times V_M \mid (u, v) \in E_c\}$ .
- Edge-Colored Multigraphs: A multigraph $G_e = (V, E, C)$ with edge colors $C$ maps to a node-aligned multiplex network with $L = C$ , $V_M = V \times L$ , and intra-layer edges $E_M = \{((u, c), (v, c)) \mid (u, v, c) \in E\}$ .
- $k$ -Uniform Hypergraphs: An undirected $k$ -uniform hypergraph $H = (V_H, E_H)$ maps to a multiplex network with $k$ aspects, where each hyperedge $(u_1, \dots, u_k) \in E_H$ corresponds to a non-zero element in an adjacency tensor $\mathcal{W}^H_{u_1 \dots u_k}$ . Conversely, any node-aligned multiplex network with node set $V$ and layer set $L$ maps to a 3-uniform directed hypergraph with node set $V_H = V \cup L$ and directed hyperedges $(u, v, \alpha) \in E_H$ whenever $((u, \alpha), (v, \alpha)) \in E_M$ .

Coverage note — Omitted specific empirical dataset descriptions (Table 2) and detailed reviews of individual dynamical spreading models (specific variations of SIR/SIS, evolutionary games, Kuramoto oscillators, and Boolean networks) as they represent surveyed literature applications rather than the core mathematical framework, diagnostics, and structural theory contributed by the review.

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Citation

MLA

Kivela, M., et al. “Multilayer Networks”. Journal of Complex Networks, vol. 2, no. 3, 2014, pp. 203–71, https://doi.org/10.1093/comnet/cnu016.

APA

Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y., & Porter, M. A. (2014). Multilayer networks. Journal of Complex Networks, 2(3), 203–271. https://doi.org/10.1093/comnet/cnu016

Chicago

Kivela, M., A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter. 2014. “Multilayer Networks”. Journal of Complex Networks 2 (3): 203–71. https://doi.org/10.1093/comnet/cnu016.

Harvard

Kivela, M. et al. (2014) “Multilayer networks”, Journal of Complex Networks, 2(3), pp. 203–271. Available at: https://doi.org/10.1093/comnet/cnu016.

Vancouver

1. Kivela M, Arenas A, Barthelemy M, Gleeson JP, Moreno Y, Porter MA (2014) Multilayer networks. Journal of Complex Networks 2:203–271

BibTeX

@article{Kivela_2014, title={Multilayer networks}, volume={2}, ISSN={2051-1329}, url={http://dx.doi.org/10.1093/comnet/cnu016}, DOI={10.1093/comnet/cnu016}, number={3}, journal={Journal of Complex Networks}, publisher={Oxford University Press (OUP)}, author={Kivela, M. and Arenas, A. and Barthelemy, M. and Gleeson, J. P. and Moreno, Y. and Porter, M. A.}, year={2014}, month=July, pages={203–271} }

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Multilayer networks

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Knowls

Knowl 1 — Mathematical Definition of a General Multilayer Network

Knowl 2 — Structural Constraints and Couplings in Multilayer Networks

Knowl 3 — Adjacency Tensor Representation of Multilayer Networks

Knowl 4 — Supra-Adjacency and Supra-Laplacian Matrix Formulations

Knowl 5 — Tensor Flattening and Path Counting via Multilayer Tensor Multiplication

Knowl 6 — Structural Transition and Algebraic Connectivity in Multilayer Networks

Knowl 7 — Network Aggregation and Eigenvalue Interlacing

Knowl 8 — Multidegree, Multistrength, and Overlap Multiplicity

Knowl 9 — Cascading Failures and Viable Cluster Collapse in Interdependent Networks

Knowl 10 — Mapping Between Multilayer Networks and Related Network Formalisms

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