Multilayer Network Analysis

• Multilayer network analysis is a mathematical framework that models systems with multiple interdependent connections by representing entities across distinct layers.
• It extends traditional network metrics—such as centrality, modularity, and motif analysis—using supra-adjacency matrices and tensor representations to capture intricate inter-layer dynamics.
• The approach supports applications across biology, social sciences, infrastructure, and economics, while leveraging simplification techniques like filtering and aggregation for scalable analysis.

Multilayer network analysis provides a unified mathematical and computational framework for modeling, exploring, and simplifying systems where entities interact through multiple types, contexts, or timescales of connections. Distinct from traditional monolayer (simplex) network models, multilayer networks (MLNs) represent complex systems with a high degree of heterogeneity and interconnectedness across layers, enabling richer forms of analysis, novel dynamics, and more accurate structural quantification. This article delineates the foundational definitions, principal methodologies, simplification techniques, core application domains, empirical insights from key studies, and emerging research frontiers in the field.

1. Mathematical Foundations and Definitions

A multilayer network is defined by a node set, a collection of layers, and sets of intra- and inter-layer edges. The formalism universally adopted is that of Kivelä et al.: a multilayer network $M = (V_M, E_M, V, L)$, where $V$ is the set of entities, $L$ is the set of layers (possibly structured as a Cartesian product of aspects), $V_M \subset V \times L$ denotes node-layer tuples (indicating an entity’s presence in a specific layer), and $E_M \subset V_M \times V_M$ encodes both intra- and inter-layer edges.

Layer types include:

• Multiplex: Same nodes across layers, each layer denoting a relationship type (e.g., friendship, work).
• Interconnected/Node-coloured: Layers may differ in node sets (e.g., species and habitats), with inter-layer links connecting different types.
• Temporal/Spatial: Layers encode time slices or locations.

Mathematical representations extend from block matrices (where diagonal blocks are intra-layer adjacency matrices $A^l$, and off-diagonal blocks $C^{ll'}$ are inter-layer adjacency submatrices) to higher-order tensors for complex interactions (Aleta et al., 2018).

Key multilayer metrics include:

• Multilayer degree vector: $k_i = \{k_i^1, ..., k_i^L\}$
• Edge overlap and entropy:

$$H_i = -\sum_{\alpha=1}^L \frac{k_i^\alpha}{o_i} \ln\left(\frac{k_i^\alpha}{o_i}\right)$$

where $o_i$ is total degree overlap

Supra-adjacency/tensorial models provide the computational substrate for generalizing classical algorithms (e.g., centrality, modularity) to the multilayer context (Domenico et al., 2014, Aleta et al., 2018).

2. Core Methodologies for Multilayer Analysis

2.1 Centrality and Structural Measures

• Degree centrality, eigenvector centrality, betweenness, and other diagnostics are extended to operate on the multilayer supra-adjacency matrix or tensors, yielding multilayer analogues such as eigenvector versatility and PageRank versatility (Finn et al., 2017, Domenico et al., 2014, Han et al., 2023).
• Community detection adopts multislice modularity [Mucha et al.] and multilayer InfoMap, seeking structures that are robust both within and across layers (Finn et al., 2017, Domenico et al., 2014).
• Novel measures such as multilayer betweenness centrality account for shortest paths traversing combinations of relationship types (Magnani et al., 2013), and layer-similarity indices (e.g., quantum Jensen-Shannon divergence) quantify redundancy and reducibility (Domenico et al., 2014).

2.2 Motif and Subgraph Analysis

Extending motif theory, multilayer motif analysis identifies recurrent multi-node, multi-relational building blocks. Motifs are categorized and quantified using density, triad-density, inter-layer correlation, and cosine-similarity metrics, revealing system-specific reinforcement, complementarity, or specialization in interaction patterns (Zhong et al., 2019).

2.3 Multilayer Embedding and Machine Learning

Multilayer embeddings project nodes (and sometimes layers) into continuous vector spaces reflecting both intra- and inter-layer proximities. Approaches include:

• Network aggregation (all edges collapsed), results aggregation (layer-wise embeddings concatenated), and layer co-analysis (random walks with inter-layer transitions) (Liu et al., 2017).
• Methods such as multi-node2vec explicitly generalize node2vec for efficient, scalable unsupervised feature learning via multilayer random walks; these embeddings underpin link prediction, clustering, and classification tasks (Wilson et al., 2018).
• Modern R packages (rMultiNet) implement tensor decomposition and generative multilayer models to reveal fine-grained structure and community patterns (Li et al., 2023).

2.4 Model-based Inference and Regression

Recent advancements enable statistical and econometric inference on multilayer networks:

• Multilayer network regression, combining eigenvector centrality and community structure (via fourth-order adjacency tensors), offers theoretically consistent estimation even in the presence of noise, and extracts interpretable explanatory variables for phenomena like industry production (Han et al., 2023).

3. Simplification and Compression Techniques

The intrinsic scale and complexity of empirical MLNs often require simplification before meaningful analysis. A unified taxonomy divides simplification strategies into three primary categories (Interdonato et al., 2020):

Category	Subtypes / Examples	Purpose
Selection	Filtering (centrality-based, node/layer relevance, model-based);	Reduce size/complexity; highlight relevant structure
	Sampling (random, exploration-based)
Aggregation	Flattening, layer aggregation, coarsening, community detection,	Compress network, support macroscale pattern detection
	positional equivalence, graph (MDL) summarization
Transformation	Projection-based (e.g., bipartite projection), embedding-based	Change representation (e.g., to vectors); support ML

Current gaps include native, reversible multilayer coarsening; blockmodeling for positional equivalence; and scalable model-based filtering. Embedding-based transformations (e.g., PMNE, MNE, MELL) are emerging to handle attributes, temporal information, and edge directionality.

Simplification not only manages memory and computation, but is foundational to scalability, interpretability, noise mitigation, and enabling ML workflows.

4. Applications and Empirical Insights

4.1 Biological and Social Systems

In animal behavior, the multilayer framework elucidates individual, group, and population-level social structures such as role consistency, cross-contextual influence, and core-periphery organization unattainable via aggregation (Finn et al., 2017). For example, grooming and spatial proximity networks in baboon populations yield different centrality rankings and community structures than their sum or individual layers.

4.2 Infrastructure, Economics, and Credit Risk

Empirical studies of transportation networks demonstrate that complementarity (as opposed to reinforcement) between modalities is quantifiable via motif analysis (Zhong et al., 2019). In credit risk, multilayer bipartite modeling (e.g., based on shared geography and business sector) and personalized multilayer PageRank propagate default risk more accurately than traditional models, improving predictive metrics by 10–12% and uncovering non-linear risk mitigation via large neighborhoods (Óskarsdóttir et al., 2020).

Multilayer network regression applied to world input-output data establishes that community-based centrality measures explain significant additional variance in economic output compared to traditional sector-level covariates alone, with strong post-crisis temporal signatures (Han et al., 2023).

4.3 Linguistics, Narrative, and Reaction Networks

Multilayer analysis of language systems reveals deep structural similarities and differences among syntactic, co-occurrence, syllabic, and graphemic layers, with preserved weighted overlap and motif significance profiles quantifying subsystem interaction (Margan et al., 2015). In movie script analysis, integrating character, location, and keyword layers reveals bridging entities and central themes otherwise inaccessible via character-only networks (Mourchid et al., 2018). Nuclear reaction networks, stratified by currency particles, demonstrate that degree difference and multiplex overlap track $\beta$-stability and enable scaling relationships between half-life and out-strength at low temperatures (Zhu et al., 2016).

5. Challenges, Caveats, and Future Directions

Multilayer network analysis confronts both statistical and practical challenges:

• Data integration and layer specification: Accurate modeling depends on meaningful definition of layers, interlayer links, and node correspondence, varying across domains.
• Computational complexity: The combinatorial explosion for metrics (e.g., motifs, betweenness) and the need for large-scale tensor computations demand algorithmic innovation (Magnani et al., 2013, Li et al., 2023).
• Methodological maturity: While packages such as muxViz, rMultiNet, and others now support many tasks, robust null models, standardization, and support for heterogeneity and temporal/attributed data remain emerging areas (Domenico et al., 2014, Li et al., 2023).
• Interpretability and evaluation: Many multilayer metrics are difficult to map directly to domain concepts; new benchmarks and interpretable embeddings are required.

Future advances are anticipated in reversible coarsening, positional equivalence methods, higher-order and temporal embeddings, scalable motif enumeration, and multisource data fusion, as well as domain-specific metrics in ecology, neuroscience, economic geography, and linguistics.

6. Concluding Perspective

Multilayer network analysis extends the representational and analytical capabilities of network science into domains where heterogeneity, context-dependency, and high-order interactions are fundamental. It enables modeling of systems at scales and granularities unaddressable by single-layer analysis, supporting inference, prediction, and mechanistic insight across the natural, social, and engineered worlds. As simplification, embedding, and statistical modeling approaches mature, the field is poised to support data-driven discovery and deeply comparative analyses, anchoring the paper of complex systems in a unified mathematical framework.