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Multilayer Networks
Mikko Kivelä, Alexandre Arenas, Marc Barthelemy, James P. Gleeson, Yamir Moreno, Mason A. Porter
TL;DR
Multilayer systems contain interacting subsystems and connection types that traditional network theory does not fully represent. This paper unifies terminology through a general framework and review, showing that multilayer structure can produce phenomena missed by single-layer analysis.
Problem
Traditional network theory inadequately represents systems with multiple connection types, layers, and temporal interactions.
Method
The paper develops a general multilayer-network framework, maps existing network concepts onto it, and reviews diagnostics, models, datasets, and dynamics.
Results
Multilayer dynamics can exhibit new behavior that cannot be explained by analyzing individual layers or aggregating them.
Takeaways & Limitations
Understanding complex systems requires tools that preserve multilayer structure rather than reducing systems to single-layer representations.
Takeaways & Limitations
Community-detection methods for multilayer networks remain underdeveloped, and computational difficulties intensify with system size.
Abstract
from arXiv · showhide
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.
1 Introduction
The introduction motivates multilayer networks as a necessary generalization of simple-graph models for complex systems with heterogeneous, temporal, and multiple interactions. It presents a unified framework intended to resolve fragmented terminology and organize existing concepts, data, methods, and applications.
- Motivation: Traditional network theory models entities with single, static, unweighted edges, but complex systems require representations of directed, weighted, bipartite, and time-dependent relationships.These extensions address increasingly realistic network structures across social, biological, physical, information, and engineering sciences.
- Motivation: Social systems often contain multiple relationship types, making single-layer pairwise representations an extremely crude approximation of social reality.Sociologists recognized decades ago that different types of social ties should be represented through multiple social networks.
- Scope and organization: The paper reviews multilayer-network data sets, diagnostics, models, community structure, tensor methods, connected components, and dynamical processes, then concludes with an outlook and glossary.Its organization includes a general formulation and terminology translation, a data-set review, and discussions of methods and applications.
- Problem: The rapid growth of multilayer-network research has produced disparate terminology and no consensus mathematical framework, complicating comparisons among related concepts.The literature may use different terms for the same object or the same term for different, closely related objects.
- Contribution: The paper defines a general multilayer-network framework, maps existing network types onto it, and classifies them according to the constraints they impose.This framework is designed to represent complex systems containing multiple networks or disparate and multiple interactions between entities.
2 Multilayer Networks · 2.1 General Form
The paper frames multilayer networks as a generalization of network theory for systems with multiple connectivity layers and defines a flexible formal structure that encompasses existing network types. This framework represents node-layer entities and their connections algebraically while supporting constraints, tensor representations, and reductions to familiar single-layer forms.
- 2 Multilayer Networks: The structure can be represented by an adjacency tensor, whose order can be reduced by restricting admissible multilayer networks or flattening the tensor into supra-adjacency matrices.These representations provide algebraic ways to encode and simplify multilayer connectivity.
- 2 Multilayer Networks: Multilayer networks extend ordinary graphs by adding layers to represent multiple levels, edge types, or related structural features.The framework is designed to recover existing multilayer and multiplex notions by imposing appropriate limitations and constraints.
- 2.1 General Form: The general framework permits nodes to belong to any subset of layers and allows edges between all possible combinations of node-layer tuples.Hyperedges connecting more than two entities are identified as a possible further generalization.
- 2.1 General Form: A multilayer network is defined as M = (V_M, E_M, V, L), where layers are formed from the Cartesian product of elementary-layer sets and V_M records node-layer presence.The number of aspects is finite in the paper’s main treatment, and elementary-layer sets are generally assumed finite.
- 2.1 General Form: The first two components define an underlying graph G_M = (V_M, E_M), allowing basic graph concepts and edge weights to generalize directly to multilayer networks.Edges are treated as connections between node-layer tuples, with adjacency and incidence defined analogously to monoplex networks.
- 2.1 General Form: When the number of aspects is zero, the formalism reduces to a monoplex network, with V_M = V and the node-layer set becoming redundant.This provides an explicit connection between the multilayer framework and ordinary single-layer network theory.
- 2.1 General Form: Edges are partitioned into intra-layer edges and inter-layer edges, with coupling edges reserved for connections between different layers representing the same entity.This distinction reflects the often different semantics assigned to edges within layers versus across layers.
- 2.1 General Form: Constraints such as node alignment, layer disjointness, equal layer size, diagonal couplings, and categorical couplings distinguish established multilayer-network variants.A node-aligned network contains all nodes in every layer, whereas a layer-disjoint network allows each node in at most one layer.
2.2 Tensor Representations
Multilayer networks are represented with adjacency tensors whose indices encode nodes and layer aspects, with constraints and flattening operations providing lower-rank or alternative representations.
- General tensor representation: A node-aligned multilayer network uses a rank-2(d + 1) adjacency tensor indexed by node pairs and layer pairs across d aspects.Each tensor element is binary, indicating whether the corresponding multilayer edge exists.
- General tensor representation: Weighted adjacency tensors extend this representation by assigning each element the corresponding edge weight, or 0 when no edge exists.
- Applicability beyond node-aligned networks: Although the tensor formulation is formally node-aligned, adding empty nodes allows many tensor-based methods to apply to other multilayer network types.These added nodes are not adjacent to any other nodes.
- Constraints and reduced representations: Diagonal and layer-coupled constraints force tensor elements to zero or make couplings node-independent, enabling reduced coupling tensors and lower-rank representations.With no or categorical couplings, the network is completely defined by the intra-layer adjacency tensor; for d = 1, this is rank-3.
- Flattening: Flattening combines two aspects while preserving tensor elements through a bijective mapping, and supra-adjacency matrices are the extreme d = 0 case.Flattening is also called unfolding or matricization and can simplify software implementations.
2.3 Supra-adjacency Representation
Supra-adjacency matrices represent multilayer networks in a form that enables matrix-based tools and theoretical results. From this representation, researchers can derive supra-Laplacians, study spectral behavior, and reproduce the calculations using adjacency tensors.
- Supra-adjacency matrices: Supra-adjacency matrices represent multilayer networks as adjacency matrices of corresponding supra-graphs, providing access to established matrix-based tools and theory.This representation generalizes the powerful adjacency-matrix approach from monoplex networks.
- Supra-Laplacian: The combinatorial supra-Laplacian is defined as L_M = D_M − A_M, with node-layer strengths on the diagonal of D_M.A_M is the supra-adjacency matrix corresponding to the multilayer graph G_M.
- Supra-Laplacian: The normalized supra-Laplacian exhibits qualitatively different eigen-spectrum behavior as the relative strengths of inter-layer and intra-layer edges vary.The observation concerns two-layer interconnected networks generated with correlations between intra-layer and inter-layer degrees.
- Tensor representation: Supra-adjacency calculations can be reproduced with adjacency tensors by using a bijective linear flattening function and tensor multiplication defined through ordinary matrix multiplication.The tensor product is defined as A ×_f B = f^−1(f(A) · f(B)).
2.4 Node-Colored Networks, Interconnected Networks, Interdependent Networks, and Networks of Networks
This section presents structural notions for node-colored, interconnected, interdependent, and network-of-networks formulations. It relates these structures to the multilayer-network framework, including explicit mappings for node-colored graphs.
- Interdependent Networks: Interdependent networks comprise two or more monoplex networks whose nodes are connected across networks by dependency edges.The electrical-grid and computer-network example illustrates mutual dependence between a router and a power station.
- Node-Colored Networks: Multilayer networks are equivalent to node-colored networks, although the frameworks were developed with different motivations.Here, color is used broadly as a node label, so nodes sharing a color may still be adjacent.
- Node-Colored Networks: A node-colored graph maps to the multilayer framework with d = 1 by treating each color as a layer and retaining only same-color node assignments.The resulting multilayer network is disjoint because each node has a single color, without additional restrictions.
- Node-Colored Networks: An alternative mapping renames nodes with integers unique within each color and records the layer identity alongside each node name.This representation uses integers from 1 through the maximum number of nodes sharing a color.
2.5 Multiplex Networks and Multirelational Networks
Multiplex and multirelational networks can be represented as sequences of graphs or edge-colored multigraphs, with layers encoding distinct edge types. The paper defines multiplex networks as diagonally coupled multilayer networks whose layers share at least one node, excluding layer-disjoint networks.
- Definitions: Multiplex and multirelational networks can be defined as sequences of graphs, typically with identical or partially overlapping node sets.The graph sequence uses layer-indexed edge sets, and layers may share all nodes or only some nodes.
- Definitions: They can alternatively be represented as edge-colored multigraphs, where edge colors label relationship types.An edge-colored multigraph is defined by nodes, edges, and a color set; edges incident to the same node may share a color.
- Representation: Sequences of graphs map to multilayer networks by assigning each graph to an intra-layer network, while each edge color maps to a layer.This mapping captures intra-layer edges; more general inter-layer couplings require higher-order tensor representations.
- Definition used in this paper: The paper defines multiplex networks as diagonally coupled multilayer networks in which every layer shares at least one node with another layer, excluding layer-disjoint networks.This definition includes multiplex networks that are not node-aligned and is broader in some respects than alternative usages.
- Examples: A canonical example is a social network whose layers represent friendship, family, and coworker relationships, with analogous applications in biological and transportation networks.Examples also include gene co-expression, protein interaction, and transportation networks.
2.6 Hypergraphs · 2.7 Ordinal Couplings and Temporal Networks
The paper extends multilayer-network analysis to hypergraphs, whose hyperedges can connect more than two nodes, and to ordinal and temporal networks represented by ordered layers or events. The framework also accommodates directed temporal progression, delays, continuous time, and networks combining categorical edge types with temporal order.
- 2.6 Hypergraphs: Hypergraphs generalize graphs by allowing each hyperedge to connect any number of nodes rather than exactly two.Hyperedges are represented as subsets of the node set, excluding self-edges unless explicitly included.
- 2.6 Hypergraphs: A k-uniform hypergraph requires every hyperedge to connect exactly k nodes and can represent weighted, directed structures with adjacency tensors.These structures arise in applications including folksonomies and protein-interaction interologs.
- 2.6 Hypergraphs: Weighted hypergraphs assign each existing hyperedge a weight and nonexistent hyperedges a value of 0, supporting spectral analysis through adjacency tensors.The tensor formulation is useful for investigating spectral theory in hypergraphs.
- 2.6 Hypergraphs: Multiplex networks can be mapped to directed 3-uniform hypergraphs by adding layer nodes and encoding intra-layer node connections as hyperedges.The passage also notes that the mapping can be defined in reverse.
- 2.7 Ordinal Couplings and Temporal Networks: Temporal networks can be represented as event sets or ordered sequences of graphs, including networks derived from similarities in coupled time series.An event-based temporal network uses events e = (u, v, t), with nodes u and v and timestamp t.
- 2.7 Ordinal Couplings and Temporal Networks: Ordinal multilayer couplings connect corresponding nodes only across adjacent layers, while directed interlayer edges encode time’s arrow.The framework can also incorporate generalized ordinal couplings.
- 2.7 Ordinal Couplings and Temporal Networks: Non-diagonal couplings can represent event delays, such as a flight encoded by its departure time and travel duration.The airline example represents a flight as (u, v, t, td), where td is the delay or duration element.
- 2.7.1 Networks with Both Ordinal and Categorical Aspects: The framework supports continuous time beyond finite-layer tensor and supra-adjacency representations and can distinguish long-duration events from consecutive shorter events.Multiplex temporal networks combine categorical and ordinal aspects; their tensor representation can use 6th-order tensors, or 4th-order lower-order representations.
2.8 Other Types of Networks and Graphs
This section surveys several network structures beyond standard multilayer networks, including k-partite, jointly node- and edge-colored, multilevel, and hierarchical multilayer networks. These structures extend the framework by constraining allowable connections, representing multiple node and edge types, or organizing physical and virtual layers.
- k-partite graphs: A k-partite network consists of k pairwise disjoint node sets representing distinct types, with edges prohibited between nodes of the same type.Bipartite networks are the 2-partite case, while tripartite and m-partite networks are also considered.
- k-partite graphs: Bipartite networks can be projected into layers corresponding to every pair of edge types, including repeated types, with adjacency induced through a shared neighboring node.This construction was used to study bond percolation on projected unipartite networks.
- Colored nodes and edges: Networks with both node and edge types can be mapped to the multilayer framework by treating node colors and edge colors as separate aspects.Related structures have been studied as heterogeneous information networks and coupled-cell networks with multiple arrows.
- Multilevel networks: Multilevel networks allow nodes to have any finite number of types and edges between nodes of the same type or adjacent types.A general framework was used to represent previous multilevel networks.
- Hierarchical multilayer networks: Hierarchical multilayer networks place a physical network at the bottom and virtual layers above it, with dependencies between successive layers.A node in a given layer depends on a node in the layer below, and nodes in a given layer cannot operate independently of lower layers.
3 Empirical Multilayer Networks
Empirical data are crucial for understanding how multilayer networks behave, but collecting and representing such data poses challenges because multilayer frameworks capture more detail than monoplex networks. Existing empirical studies are dominated by multiplex data, while inter-layer connection strengths remain largely unmeasured.
- Motivation: Empirical data are crucial for understanding the organization and behavior of real-world multilayer networks.Multilayer representations can reveal phenomena that monoplex-network frameworks cannot explain.
- Data-collection challenges: Multilayer networks provide more detailed system descriptions than monoplex networks, creating significant data-collection challenges for applications.A single edge type between people may be an unsuitable abstraction for social-network analysis.
- Empirical coverage: Most empirical multilayer-network studies have used data fitting the multiplex-network framework, with comparatively few studies examining other multilayer types.Examples of the latter include interdependent power-station and internet-server networks, coupled power grids, and coupled climate networks.
- Measurement gaps: Collected multiplex data have generally emphasized intra-layer networks while disregarding inter-layer connection strengths.Modeling information diffusion requires quantifying relative movement rates across layers versus within layers.
- Layer construction: Representing multilayer systems requires choosing layers creatively, such as by credit relation, social relation, hashtag, or communication language.Different layer definitions can encode distinct relationship types or interaction categories within the same system.
4 Models, Methods, and Dynamics
This section examines how concepts from monoplex networks are generalized to multilayer networks, including structural diagnostics, models, community structure, connectivity, and dynamics. It focuses primarily on single-aspect multilayer networks and typically treats multiplex networks as node-aligned.
- Models, Methods, and Dynamics: Research generalizes monoplex concepts to multilayer networks, including degrees, neighborhoods, walks, clustering, transitivity, centrality, communities, models, and component connectivity.These extensions are important because multilayer structures can substantially affect dynamical systems such as percolation and spreading.
- Models, Methods, and Dynamics: The section assumes a single aspect unless otherwise specified, reflecting that nearly all prior multilayer-network research has studied single-aspect networks.Multiplex networks are also generally treated as node-aligned unless explicitly noted.
4.1 Network Aggregation: From Multiplex Networks to Monoplex Networks
This section describes aggregating multiplex layers into a monoplex network, typically by combining corresponding inter-node edge weights with layer-specific importance weights. It also notes that aggregation can discard valuable structural information from systems whose layers represent distinct relationship types.
- Multiplex networks can be examined by aggregating their layers into a monoplex network and studying the resulting network.
- Aggregated edge weights can be defined as a linear combination of corresponding edge weights across layers, producing a weighted adjacency matrix.
- When layer importances are unavailable, aggregation typically assigns every layer weight 1; the result can represent a multigraph with edge layers discarded.
- Aggregation can discard valuable multiplex information, such as product-specific structures in international trade networks.
4.2 Diagnostics for Multilayer Networks
The section reviews diagnostics for single-aspect multilayer networks, mostly multiplexes, extending monoplex measures while exposing ambiguities in aggregation, walks, neighborhoods, and clustering. It also describes layer-comparison quantities such as overlap and inter-layer correlations.
- Scope: Diagnostics reviewed here are defined for single-aspect multilayer networks and mostly developed for multiplex networks.The section focuses on extending established network measures to this setting.
- Degree and neighborhood: Aggregation generalizes multiplex degree and neighborhood, while layer-subset definitions produce 2^b alternatives and exponentially many multidegree vectors as layers increase.Overlap multiplicity is introduced to help address the resulting computational and interpretive difficulties.
- Walks and paths: Generalized walks and paths support extensions of graph distance, connected components, betweenness centrality, random walks, communicability, and clustering coefficients.Definitions must specify whether changing layers costs a step and whether intra-layer steps differ across layers.
- Clustering coefficients: Multilayer clustering coefficients are especially difficult because neighborhoods, adjacencies, and walks or paths have multiple possible definitions.The complications exceed those encountered for weighted or directed clustering coefficients.
- Layer comparisons: Layer-comparison diagnostics include global overlap, global inter-clustering coefficients, and correlations between weighted adjacency-matrix elements.These quantities compare the intra-layer networks of two or more layers.
4.3 Models of Multiplex Networks
Models of multiplex networks extend monoplex generative mechanisms by combining independently generated layers with interlayer dependencies, constrained ensembles, and multilayer growth processes. ERGMs and microcanonical or canonical ensembles provide formal frameworks for specifying multiplex-network probabilities and constraints.
- Generative construction: Multiplex networks can be generated by applying monoplex models independently within layers and then adding interlayer dependencies through joint degrees or shared structures.Examples include Erdős-Rényi and configuration-model layers, with dependencies incorporated using joint degree distributions or shared components.
- Exponential random graph models: ERGMs model multiplex-network probabilities exponentially using network diagnostics, parameters, and a normalization factor.The diagnostics can include counts of triangles containing different edge types.
- Constrained ensembles: Microcanonical multiplex ensembles enforce strict constraints, whereas canonical ensembles maximize Shannon entropy subject to average constraints.Canonical ensembles generalize to exponential random graphs in single-layer networks.
- Growth models: Multiplex growth models generalize mechanisms such as preferential attachment by adding layers with partial node participation or by adding new nodes.These models represent alternative generative processes beyond independently constructed layers.
4.4 Models of Interconnected Networks
Models of interconnected networks extend monoplex generative models to multilayer settings, supporting analytical study of dynamics and accommodating diverse inter-layer connection mechanisms. These models range from random inter-layer wiring and multilayer degree distributions to ensembles incorporating degree-degree correlations.
- Modeling goals: Generative models for interconnected networks generalize monoplex models and facilitate approximate analytical results for qualitative features of multilayer dynamics.They are particularly useful for studying dynamical processes occurring on multilayer networks.
- Connection strategies: Inter-layer connections can be added uniformly at random to regular lattices, ER random graphs, configuration-model networks, and Barabási-Albert networks.This provides a straightforward way to construct node-colored interconnected networks from monoplex-network models.
- Connection strategies: Alternative random or deterministic connection processes can alter centrality ratios across layers and SIR spreading, while preferential attachment provides another inter-layer connection rule.The cited studies examine effects on mean centralities of node-layer tuples and SIR spreading.
- Degree distributions: A multilayer configuration model specifies Pα(k1, . . . , kb) for a layer-α node’s neighbors across layers, then connects edge stubs uniformly at random.This extends the usual configuration model by using multiple degree distributions.
- Degree correlations: Synthetic ensembles can incorporate intra-layer and inter-layer degree-degree correlations through Pαβ(k, k′), with further models encoding additional neighborhood correlations.The extended correlation input describes degrees of nodes joined across layers and their within-layer and cross-layer adjacencies.
4.5 Communities and Other Mesoscale Structures
Research on communities and other mesoscale structures in multilayer networks remains at an early stage, with computational scaling and null-model design posing major challenges. Existing approaches extend monoplex methods through layer-wise, aggregated, spectral, and tensor-based analyses.
- Multilayer community-detection research is still in its infancy, and mesoscale structures beyond communities require substantially more study.
- Computational difficulty increases with multilayer size because systems scale across both nodes and layers, while higher-order tensors grow faster than matrices.The challenge is especially prominent for temporal networks, which can contain very many layers.
- Multislice modularity requires appropriate multilayer null models to define which community connectivity patterns exceed random expectations.In monoplex networks, null models are typically specified through random-graph ensembles; constructing suitable multilayer counterparts is identified as a challenge.
- Methods for multilayer communities include adapting monoplex detection across individual layers or aggregated networks, spectral and hypergraph approaches, and tensor decompositions.Tensor methods such as CP, three-way DEDICOM, and Tucker decompositions can identify communities or classes of nodes with similar connection patterns.
4.6 Dynamical Systems on Multilayer Networks
Dynamical processes on multilayer networks can exhibit genuinely new behavior that cannot be explained by analyzing layers separately or aggregating them. Research spans percolation, cascading failures, viable clusters, epidemic and information spreading, while existing analyses rely largely on generating functions and spectral theory.
- General dynamical behavior: Multilayer dynamical processes can produce phenomena that neither single-layer analysis nor layer aggregation explains.The reviewed studies report behavior distinct from corresponding processes on monoplex networks.
- Analytical methods and limitations: Most analyses use generating functions and spectral theory, but a complete understanding may require methods based on tensor algebra and geometry.These methods inherit limitations known from monoplex-network analysis.
- Percolation and cascading failures: Negative degree correlations increase multiplex percolation thresholds, whereas positive correlations decrease them.This result concerns node-aligned multiplex networks with two correlated Erdős–Rényi layers.
- Percolation and cascading failures: A critical fraction of interlayer dependency edges separates continuous from discontinuous percolation transitions.The model allows a fraction 1−q of nodes in one layer to remain independent of the other layer.
- Percolation and cascading failures: Multiplex cascading failure can be formulated through mutually connected components, or viable clusters, requiring intra-layer paths in every layer.This process is equivalent to an interdependent-network cascade after merging nodes linked by interdependency edges.
- Spreading processes: Spreading models include multiplex SIR processes and coupled infection-information dynamics, with degree correlations and information-driven immunization altering epidemic outcomes.Positive versus negative degree correlations can reverse threshold behavior across settings, while information spreading can move nodes directly from susceptible to recovered.
5 Conclusions and Outlook
Multilayer networks require generalizing traditional network theory because real and engineered systems contain multiple subsystems and connectivity layers. The field has reviewed related frameworks and developed generalizations, but its theory remains immature and multilayer-induced correlations and new degrees of freedom are still poorly understood.
- Multiple subsystems and connectivity layers make generalizing traditional network theory necessary, because ignoring multilayer information can yield misleading results.
- The literature on multilayer networks has expanded rapidly across parallel frameworks, making the field conceptually messy despite substantial antecedent work.
- Research has extended single-layer diagnostics, methods, models, and dynamical systems to multilayer settings, highlighting multiplexity-induced correlations as an important theme.
- Multilayer-induced correlations can affect disease and idea transmission speeds and system robustness to failure, while multilayer structures introduce poorly understood new degrees of freedom.
6 Appendix: Glossary and Notation
The appendix defines core terminology and notation for multilayer networks, including network types, node-layer tuples, layer aspects, and supra-adjacency matrices. It also notes that intra-layer edge weights are generally easier to estimate reliably than inter-layer weights, with transportation networks currently the strongest multiplex candidates.
- In most existing data sets, intra-layer edge weights are easier to estimate reliably than inter-layer edge weights.
- A multilayer network is the general term for a network with multiple layers.
- A multiplex network is a multilayer network with diagonal couplings, whereas a monoplex network has a single layer.
- A node-layer tuple specifies both node and layer; an aspect is a dimension of layers, and a supra-adjacency matrix represents the network.