Abstract
Stress in the global financial system propagates through multiple channels, including interbank credit and funding, overlapping portfolios, and bank–firm exposures. Multilayer networks represent these channels as interdependent layers, enabling analysis of how shocks transmit and amplify within and across channels. Yet many studies collapse these channels into a single layer, and stress tests often examine channels in isolation, in both cases removing cross-channel feedback and understating systemic vulnerability. This can lead to inadequate capital and liquidity buffers, overly optimistic risk assessments, and poorly targeted interventions. The synthesis in this review shows how multilayer financial networks are constructed and summarises reported benchmarks for how much single-layer proxies can underestimate systemic vulnerability, distilling the evidence into a practical Map–Monitor–Test–Intervene workflow that serves researchers designing multilayer propagation studies, practitioners assessing portfolio and counterparty risks, and supervisors calibrating stress tests. We synthesise 112 studies (2015–2025) across six domains: Bank–firm Networks, Corporate Networks, Global Trade and Supply-Chain Networks, Interbank Networks, Financial Markets, and Global Systemic Risks. Reported benchmarks from a subset of studies that make explicit multilayer-versus-baseline comparisons include up to 90% understatement of systemic risk when a single exposure layer is analysed in isolation, up to 50% of systemic risk missed when overlapping portfolios are omitted, and critical leverage thresholds overstated by 45% to 300% when cross-channel interactions are ignored. Although these figures come from a small number of partly overlapping calibrations and are not pooled estimates across the full review corpus, they provide upper-end evidence that reduced single-layer or single-channel representations can materially understate risk in settings where cross-layer coupling is empirically relevant. Three findings recur: layers encode distinct mechanisms, so one layer is often a poor proxy for another, with overlay aggregation hiding feedback; cross-layer coupling amplifies losses and tightens stability margins; and systemic importance depends on the state of the financial system, concentrating in a small minority of institutions that bridge multiple channels. Collapsing channels is no longer a harmless simplification when stability judgements, risk management, and policy decisions depend on how financial stress propagates.
Abbreviations
- ACS:
-
Average connectedness strength
- AOD:
-
Average overlap degree
- ASEAN:
-
Association of Southeast Asian Nations
- BRICS:
-
Brazil, Russia, India, China, South Africa
- CASP:
-
Critical Appraisal Skills Programme
- CAViaR:
-
Conditional Autoregressive Value at Risk (model)
- CCP:
-
Central counterparty
- CDS:
-
Credit default swap
- CoVaR:
-
Conditional Value at Risk (systemic risk measure)
- DCC:
-
Dynamic conditional correlation
- DY:
-
Diebold-Yilmaz spillover framework
- EL:
-
Expected loss
- ES:
-
Expected shortfall
- ETF:
-
Exchange-traded fund
- EU:
-
European Union
- FEVD:
-
Forecast error variance decomposition
- FMI:
-
Financial market infrastructure
- FX:
-
Foreign exchange
- GARCH:
-
Generalised Autoregressive Conditional Heteroskedasticity
- GFC:
-
Global Financial Crisis (2008 to 2009)
- G-SIB:
-
Global Systemically Important Bank
- IFRS:
-
International Financial Reporting Standards
- IPR:
-
Inverse participation ratio
- LASSO:
-
Least Absolute Shrinkage and Selection Operator
- LPSI:
-
Label-propagation source-identification
- MREL:
-
Minimum Requirement for own funds and Eligible Liabilities
- MRIO:
-
Multi-regional input–output
- MST:
-
Minimum spanning tree
- NBFI:
-
Non-bank Financial Intermediation
- NCC:
-
Network correlation coefficient
- NCI:
-
Network-wide Contagion Intensity
- NODF:
-
Nested overlap and decreasing fill (measure)
- NVE:
-
Network-only Volatility Effect
- OTC:
-
Over-the-counter
- PMFG:
-
Planar Maximally Filtered Graph
- PRISMA:
-
Preferred reporting items for systematic reviews and meta-analyses
- QC-ISAM-ARMA:
-
Quantile-coherency inter-slice aggregation autoregressive moving-average
- QE:
-
Quantitative easing
- SIS:
-
Susceptible-infected-susceptible
- SRC:
-
Systemic risk contribution
- TCI:
-
Total connectedness index
- TLAC:
-
Total loss-absorbing capacity
- TOPSIS:
-
Technique for order preference by similarity to ideal solution
- TVP-SV-VAR:
-
Time-varying parameter vector autoregressive model with stochastic volatility
- TVP-VAR:
-
Time-varying parameter vector autoregression
- UER:
-
Unique Edge Ratio
- UK:
-
United Kingdom
- US:
-
United States of America
- VAR:
-
Vector autoregression
- VaR:
-
Value at risk
- VTN:
-
Variance tail network
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Appendices
Appendix A Supplementary methodological details
This appendix provides extended methodological details supporting the systematic review protocol summarised in Sect. 2.
A.1 Scope and time frame rationale
The definition of financial systems was deliberately broad to include both traditional financial markets and extended economic networks with financial characteristics, such as global trade, supply chains, and policy frameworks governing these systems. The time frame was selected to capture the period following the establishment of foundational multilayer network theory in the early 2010's, and to extend beyond the broad review conducted in 2014 by Kivelä et al. (2014) and by Boccaletti et al. (2014), whilst encompassing recent developments in the field.
A.2 Database-specific search strategies
The systematic search was conducted across ten databases. Table 5 presents the full search strings adapted to each database’s syntax and field restrictions.
A.3 Detailed eligibility criteria
A.3.1 Inclusion criteria (extended)
Studies were included if they met all of the following criteria:
- (a)
Primary focus on multilayer networks: The study must employ multilayer, multiplex, interdependent, or temporal network methodologies as the principal analytical framework, rather than as a peripheral or illustrative component.
- (b)
Financial systems application: The financial systems examined must fall within the scope defined in Subsection 2.2, encompassing banking and capital markets, payment and settlement systems, corporate finance linkages, and macro-financial structures (including trade, supply chains, and policy frameworks) when these directly shape financial propagation channels.
- (c)
Publication window: Published or accepted for publication (in press with a DOI) between 1 June 2015 and 31 May 2025.
- (d)
Publication type: Peer-reviewed journal articles, book chapters, or conference proceedings. Working papers were excluded unless subsequently published in peer-reviewed form, in which case the peer-reviewed version was used.
- (e)
Citation threshold (pragmatic screen): For studies published up to and including 2023, a minimum of five citations at the time of screening was required as a light-touch indicator of scholarly uptake. No citation threshold was applied to studies published in 2024 or 2025, recognising that citation counts are immature for recent work.
- (f)
Empirical or conceptual validation: The study must provide either empirical validation using real-world or simulated data, or conceptual validation through formal theoretical development.
A.3.2 Exclusion criteria (Extended)
Studies were excluded if they met any of the following criteria:
- (a)
Methodological focus without financial application: Studies developing machine learning algorithms, software tools, or network methods without substantive application to financial systems. We excluded the term ’perceptron’ to reduce false positives from neural network papers using multilayer perceptrons.
- (b)
Non-peer-reviewed formats: Preprints (unless subsequently published), retracted papers, working papers, review papers, meta-analyses, commentaries, editorials, and thesis publications.
- (c)
Insufficient validation: Studies lacking empirical validation or practical application to financial contexts.
- (d)
Language: Publications not available in English.
Central bank and international organisation reports were treated as grey literature and cited only when they provide data or policy context rather than core empirical findings.
A.4 Study selection workflow and stage-specific counts
Our selection process involved four stages. In the initial stage, the search yielded 1,124 potentially eligible papers after removing duplicates within each database. Although the initial search targeted finance-related terms, abstract screening showed many uses of multilayer terminology in non-financial contexts, for example biology and social networks, so a second stage applied finance-specific inclusion and exclusion rules to refine the pool through comprehensive abstract screening and selective full-text examination.
Inclusion and exclusion criteria, including the citation threshold rule, are reported in Appendix A.3.
The screening process involved systematic examination of abstracts and targeted full-text searches for relevant terminology. This procedure reduced the initial pool of 1,124 papers to a preliminary selection of 208 that met the specified criteria.
The third stage reconciled results across databases, further reducing the number of papers from 208 to 108 unique publications after removing 100 cross-database duplicates. Detailed full-text review then excluded two further papers, leaving 106 core publications. A supplementary route identified eight records from websites recommending similar papers and citation checks of selected papers; six of these met the inclusion criteria. This gave a final total of 112 papers. The fourth and final stage was the synthesis of all 112 selected papers.
A.5 Quality assessment framework
Quality assessment was informed by an adaptation of the Critical Appraisal Skills Programme (CASP) framework. Given the heterogeneity of study designs spanning empirical analyses, simulations, and theoretical contributions, the framework was applied flexibly rather than as a scoring instrument, and we did not compute an aggregate score. The following dimensions were assessed where available and relevant:
- 1.
Research question clarity: Is the research question or objective clearly stated?
- 2.
Methodological appropriateness: Is the multilayer network methodology appropriate for the research question?
- 3.
Data quality and transparency: Are data sources, collection methods, and any limitations clearly described?
- 4.
Network construction transparency: Are layer definitions, edge weighting rules, and any sparsification or filtering procedures clearly specified?
- 5.
Analytical rigour: Are the analytical methods applied correctly and are assumptions stated?
- 6.
Robustness and sensitivity: Are results tested for sensitivity to key parameter choices (e.g., window length, backbone thresholds)?
- 7.
Replicability: Is sufficient detail provided to allow replication?
- 8.
Limitations acknowledgement: Are limitations of the study design and data clearly acknowledged?
A.6 Data extraction template
The following information was extracted from each included study using a standardised template:
- (1)
Bibliographic details (authors, year, journal, DOI)
- (2)
Application domain (one of six domains: Bank–firm Networks, Corporate Networks, Global Trade and Supply-Chain Networks, Interbank Networks, Financial Markets, and Global Systemic Risks)
- (3)
Network model classification (multiplex, multilayer, interdependent, temporal, hybrid)
- (4)
Layer architecture (node types, edge definitions, number of layers, inter-layer coupling)
- (5)
Mathematical formalism (e.g., supra-Laplacian, tensor, supra-adjacency matrix)
- (6)
Data sources and sample characteristics (geography, time period, sample size)
- (7)
Key findings (including reported magnitudes, benchmarks, comparisons with single-layer baselines)
- (8)
Methodological quality indicators
- (9)
Limitations acknowledged by authors
- (10)
Limitations identified during review
- (11)
Future research directions suggested
A.7 Software and tools
Reference management and deduplication were conducted using Zotero (version 6.0). Search results, screening decisions, and data extraction were recorded in Microsoft Excel. No automated screening tools were used. Initial screening was conducted manually by the lead author to ensure consistent application of eligibility criteria across the interdisciplinary literature. The screening protocol, eligibility criteria, and draft final inclusion list were reviewed jointly by both authors before and after the screening process. The second author also independently re-screened a random sample of included and excluded records against the stated inclusion and exclusion criteria. No changes to the final corpus were required.
Appendix B Cross-cutting diagnostics, methodological extensions, and illustrative applications
B.1 Introduction
Three studies show how multilayer analysis extends beyond balance-sheet and price data to capture information flows, sentiment, and organisational influence. Although these three applications sit outside the six core financial categories in the main text, they illustrate the versatility of multilayer diagnostics in non-balance sheet settings.
The three studies are: a media–country multiplex linking financial news to geography, trade, and CDS correlations (Sluban et al. 2016), a bank–media multiplex linking bank co-mentions to sentiment tone (Wang et al. 2021), and a lobbying multiplex linking organisations through formal and relational ties in the European Union (EU) (Zeng and Battiston 2016).
B.2 Data and layer architecture
In the media–country setting, the nodes are 50 countries. Two news-based layers are built from more than 1.3 million financial articles aggregated via 2,503 RSS feeds from 170 sites (November 2011 to December 2013): a co-occurrence layer and a sentiment layer. The sentiment layer is separated into positive and negative snapshots. These two news-based layers are compared with three empirical layers, geography, trade, and credit default swap (CDS) correlations, forming a five-layer system. Trade and CDS links are only included when they exceed predefined statistical thresholds (Sluban et al. 2016).
The bank–media multiplex covers 214 Chinese banks, with two sentiment layers constructed from Baidu News and EastMoney (January 2018–December 2019). Edges are undirected and weighted by monthly co-mentions, with a corpus of 152,994 co-mention records; results are reported per layer and for the aggregate (union) network (Wang et al. 2021).
The lobbying multiplex contains four layers: affiliations, shareholdings, common directors (interlocking directorates), and client relationships, across 6,637 organisations in the EU Transparency Register (Zeng and Battiston 2016). The interlocking directorate layer is undirected, while the other layers are directed, with weights available for shareholding and client ties in the underlying data (Zeng and Battiston 2016).
B.3 Methods, in brief
All three papers adapt standard diagnostics to multilayer settings. The media–country study compares layer pairs using link overlap, compares central nodes using eigenvector centrality, reports precision at k (a top k retrieval accuracy measure), and applies k-core analysis across the five-layer set (Sluban et al. 2016). The bank–media study reports small-world indicators, average path length and clustering, for each layer and for the aggregate (Wang et al. 2021). It also reports node strength, participation coefficients, inter-layer strength correlation, and Jaccard similarity between tone layers. The lobbying study computes per-layer clustering, path length, assortativity, and a multiplex centrality measure incorporating feedback dynamics, then relates degree and centrality to financial metrics (Zeng and Battiston 2016). Centrality is defined as a PageRank-style feedback process in which a node’s in-centrality depends on the centrality and relative lobbying expenditure of its in-neighbours, and out-centrality is obtained by reversing edge direction. The multiplex version then computes centrality using information from multiple link types at once, so prominence can increase when an organisation is well connected in one layer even if it is less connected in another (Zeng and Battiston 2016).
B.4 Mechanisms and comparative findings
Media-derived links align most strongly with geography and trade, rather than with CDS spread correlations, which are computed over three-month windows. Co-occurrence links align most closely with geography, whilst positive sentiment aligns more closely with trade. Overlap with the CDS correlation layer is lowest. Countries that frequently appear in the overlap core between the positive sentiment and trade layers include China, Germany, the United States, the United Kingdom, Japan, Brazil, France, and Australia. Germany appears in the overlap core between the positive sentiment and geography layers in 23 of the 26 sample months (Sluban et al. 2016).
Bank attention networks built from co-mentions exhibit short average path lengths and high clustering. The mixed (aggregate) network shows an average path length of 1.624 and clustering coefficient of 0.816, with annual values of 1.804/0.789 (path length/clustering) in 2018 and 1.626/0.835 in 2019. Inter-layer similarity between positive and negative tone layers is low when measured by Jaccard overlap of edges, yet inter-layer strength correlation is high, indicating that the same banks tend to be prominent across both tones even when the specific edges connecting them differ (Wang et al. 2021).
In the lobbying multiplex, network position and degree correlate moderately with lobbying expenditure in the affiliation layer (out-degree) and the interlocking layer (degree), but these associations are weak in the shareholding layer. Pearson correlation coefficients between lobbying expenditure and out-degree are 0.46 in the affiliation layer and \(-0.15\) in the shareholding layer, while the correlation between lobbying expenditure and degree is 0.52 in the interlocking layer; the correlation between client in-degree and turnover is 0.67. For centrality measures, correlations with lobbying expenditure are 0.172 for in-centrality in affiliation and 0.074 for out-centrality in the multiplex. These correlations suggest that financial resources and network position are related, but budget alone is an unreliable proxy for degree or centrality (Zeng and Battiston 2016).
B.5 Limitations
Text-derived layers depend on corpus coverage, threshold choices, and temporal windows (Sluban et al. 2016). The bank–media multiplex proxies for attention and tone rather than balance-sheet liquidity and also spans a short window (Wang et al. 2021). The lobbying register is self-reported and heterogeneous across actor types, and some layers are weighted by construction, but are analysed using unweighted, or undirected, snapshots (Zeng and Battiston 2016). Furthermore, data sources used in Zeng and Battiston (2016) do not provide historical information, therefore only a single snapshot at a time can be studied.
Collectively, these cross-cutting applications illustrate how multilayer diagnostics transfer to text and organisational data: media layers align more with geography and trade than with CDS comovement, bank sentiment networks exhibit small-world properties with short paths and high clustering, and network position correlates only moderately with lobbying expenditure across layers. Methods and observation windows are constrained by data availability, and some layers are treated as unweighted, so magnitudes should be interpreted with caution. Within those constraints, the diagnostics are informative and replicable within their stated designs (Sluban et al. 2016; Wang et al. 2021; Zeng and Battiston 2016).
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Scattergood, J., Oléron-Evans, T. Multilayer networks in finance: a systematic review. Appl Netw Sci (2026). https://doi.org/10.1007/s41109-026-00816-0
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DOI: https://doi.org/10.1007/s41109-026-00816-0