Source-linked AI summary
The multiplex structure of interbank networks
Leonardo Bargigli, Giovanni di Iasio, Luigi Infante, Fabrizio Lillo, Federico Pierobon
TL;DR
Assessing systemic risk requires mapping interbank linkages, but multiplex layers may not represent one another. Using Italian supervisory data broken down by contract type and maturity, the paper compares layer structures and random models, finding substantial layer heterogeneity and limited cross-layer representativeness.
Problem
Mapping interbank linkages is essential for assessing systemic risk, yet similarity between layers determines whether one layer represents another.
Method
The paper analyzes Italian supervisory data by contract type and maturity and applies random models with layer-specific topological and metric constraints.
Results
Different layers have distinct topological properties and persistence; the overall network largely mirrors unsecured overnight topology, while higher-order structures differ across layers.
Takeaways & Limitations
Analyses targeting a specific interbank segment should not assume that the overall network represents other layers.
Takeaways & Limitations
The analysis cautions that frameworks based on overall network features may be unsuitable when policymakers or researchers target a specific segment.
Abstract
from arXiv · showhide
The interbank market has a natural multiplex network representation. We employ a unique database of supervisory reports of Italian banks to the Banca d'Italia that includes all bilateral exposures broken down by maturity and by the secured and unsecured nature of the contract. We find that layers have different topological properties and persistence over time. The presence of a link in a layer is not a good predictor of the presence of the same link in other layers. Maximum entropy models reveal different unexpected substructures, such as network motifs, in different layers. Using the total interbank network or focusing on a specific layer as representative of the other layers provides a poor representation of interlinkages in the interbank market and could lead to biased estimation of systemic risk.
1 Introduction
The paper frames the Italian interbank market as a multiplex network whose layers may differ, and asks whether any layer represents the total network or predicts links in other layers. Using supervisory data, it compares layer properties, cross-layer and temporal similarity, and maximum-entropy network models.
- Motivation: Credit relations differ by maturity and collateral, motivating a multiplex representation rather than a single weighted graph.Each layer contains one type of credit relation, while the total network aggregates all layers.
- Research questions: The paper asks whether layers are topologically different, whether one layer leads the total network, and whether links predict counterparts across layers or time.
- Approach: Supervisory reports cover all bilateral exposures of Italian banks, broken down by maturity and secured or unsecured contract type.Exposures are consolidated at the banking-group level and intragroup lending is netted out.
- Approach: The analyses compare topological and metric properties, quantify joint link occurrence across layers and time, and test maximum-entropy random models.The framework examines both representation similarity and point-wise link similarity.
- Main results: Different layers have layer-specific topological and metric properties, while the total network closely mirrors the overnight market and is less informative about other layers.
- Implications: Layer heterogeneity may slow contagion across market participants, but analyses targeting a specific segment should not simply adopt the overall network's features.
2 Data description
The dataset combines supervisory reports on Italian banks with contract-level maturity and collateral distinctions, while consolidating exposures at the banking-group level. The resulting data show substantial intragroup lending and heterogeneous changes across market segments during 2008–2012.
- Data sources: The dataset uses supervisory reports from all institutions operating in Italy, covering locally incorporated banks and Italian branches of foreign banks.The observations are end-of-year outstanding balances over five dates beginning at the end of 2008.
- Network representation: The network is directed and weighted: links identify credit relations, direction indicates claims and liabilities, and weights measure exposure amounts.
- Layer construction: The layers distinguish secured and unsecured transactions, with unsecured contracts further divided into overnight, short-term, and long-term maturities.Secured observations cover OTC contracts, whereas most secured transactions occur on regulated and centrally cleared markets.
- Consolidation: More than 80% of the interbank network is intragroup on average, so consolidated data netting intragroup transactions are needed to represent market transactions clearly.A major 2010 intragroup-volume drop reflects the merger of several subsidiaries of a major group.
- Market evolution: After declining 25% from 2008 to 2010, consolidated exposure amounts in 2012 were close to 2008 levels.
- Market evolution: Unsecured short-term lending declined from 2010 while unsecured long-term lending increased, and secured lending fell amid incentives favoring centrally cleared repos.The maturity shift is consistent with liquidity-risk regulation, while bilateral trading remained useful for funding-source diversification.
3 The Multiplex Italian interbank market
The Italian interbank network is a multiplex whose layers differ substantially in structure, despite shared features such as fat-tailed degree distributions and disassortative mixing. The overnight layer closely resembles the aggregate network, but aggregate or overnight metrics provide only a partial picture of other layers.
- Network structure and data: From 2008 to 2012, the study compares the aggregated Italian Interbank Network with layers defined by maturity and secured or unsecured contracts.The network contains all bilateral exposures and is analyzed across the early financial crisis, the euro-area sovereign debt crisis, and subsequent ECB interventions.
- Network structure and data: The network is very sparse, with density approximately 1%, while the overnight layer is nearly connected and closely matches the aggregate network.Unsecured short-term lending also involves almost all banking groups but has much lower density; secured layers are smaller, especially secured long-term lending.
- Degree and strength: All layers exhibit fat-tailed degree distributions, with tail exponents ranging from 1.8 to 3.5 and most values near α = 2.3.Power laws generally cannot be rejected, except that lognormal distributions are preferred for secured short-term out-degree in almost all years.
- Degree and strength: Degree and strength are highly and significantly correlated across layers, although the correlation is much lower in the overnight market because nearly all banks operate there.The total network is therefore strongly influenced by the broad participation of banks in the overnight segment.
- Higher-order topology: All layers are disassortative, while reciprocity is generally highest in the overnight layer and may be overstated by settlement-related deposit accounts.The overnight segment also drives much of the aggregate clustering, whereas hubs have clustering coefficients close to zero in most non-overnight layers.
- Higher-order topology: Because clustering decreases with degree and is often driven by low-degree nodes, average clustering depends strongly on degree distributions and the authors instead analyze triangle counts.Consolidated data may also contribute to the relatively high clustering observed in the network.
4 Results from similarity analysis
Similarity analysis shows that interbank layers differ substantially in their persistence over time and in their link overlap, so the overnight layer is not representative of the others.
- Jaccard similarity is roughly 70% between successive years for the overnight layer, indicating comparatively stable topology over time.
- The unsecured short-term layer is slightly less persistent, with J ≈60%, while the secured short-term layer falls from 40–50% to 20–30% when moving from intersection to union.
- The unsecured long-term layer is highly variable, with J = 70% in some consecutive-year comparisons but J ≈40% in others.
- Similarity generally declines with time lag, and cosine similarity is consistently lower and more volatile than Jaccard similarity when weights are included.
- Across different layers in the same year, intersection-based Jaccard similarity is generally around 15–20% and never exceeds 50%.
- Similarity between the overnight unsecured layer and other layers is at most 30%, often around 15%, confirming its limited representativeness.
- These results confirm that network structure differs significantly across contract types, indicating complementarity between market segments.
5 Results from null models
Maximum-entropy null models explain some higher-order interbank properties but reveal layer-specific deviations in reciprocity, assortativity, and motifs. Preserving different node-level constraints changes which structures appear unexpected.
- The analysis tests whether null models preserving selected node-level metrics can explain higher-order properties such as reciprocity, clustering, assortativity, and triadic structures.
- The DBCM preserves each node’s in- and out-degree, the RCM additionally preserves reciprocated relations, and the DWCM preserves degree and strength sequences.
- In the overnight layer, largest components and reciprocated links exceed DBCM expectations, while undirected triangles are fewer than expected; similar results hold for unsecured layers.
- During 2008–2012, reciprocity across Italian interbank layers remained quite stable, contrasting with a reported pre-crisis decline in the Dutch market.
- Real layers are often more disassortative than DBCM expectations, although disassortative behavior itself is also present in the null model.
- Triadic motifs are layer-specific: most are under-expressed overnight, whereas unsecured long-term motifs are unstable across years and tend to under-express selected triads.
- The RCM better represents some layers than the DBCM but does not fully explain third-order properties, whose motif patterns and time stability differ across layers.
- The DWCM jointly accounts for strength–degree correlation and sparsity, allowing simulations to display core–periphery structure associated heuristically with credit concentration.
6 Conclusions
This work analyzes interbank networks as multiplex systems whose layers differ in topology, persistence, and resilience during 2008–2012. It finds that the overall network largely reflects unsecured overnight lending, while segment-specific analysis remains important for assessing interconnectedness and contagion.
- 6 Conclusions: During 2008–2012, Italian interbank activity shifted toward longer maturities while the domestic overnight money market remained resilient.
- 6 Conclusions: The overall network largely reflects the unsecured overnight segment, whose credit relationships persist more strongly over time than those in other segments.Persistence is defined as the probability of a link at time t given a link at time t −1.
- 6 Conclusions: Motif structures in some layers differ from those of a random network, indicating that sophisticated random models may be needed to represent interbank linkages.
- 6 Conclusions: Layer heterogeneity may slow contagion, but analyses targeting a specific segment should not rely automatically on the overall network’s features.The overnight unsecured market’s close resemblance to the total network is relevant to jurisdictions where it is central to monetary policy operations.
A Appendix A: Network Statistics Definition
Appendix A defines directed and weighted network representations and introduces measures for degree, reciprocity, assortativity, connectivity, distance, and clustering. It also records that interbank networks are sparse and generally contain a giant component.
- Network representation: A network is represented as G = (V, E), with nodes V, links E, and optional link weights wij; directedness distinguishes incoming from outgoing relations.The adjacency matrix has aij = 1 for links and aij = 0 otherwise.
- Node measures: Node degree counts neighboring links, while directed networks distinguish in-degree kin_i from out-degree kout_i; node strength extends these concepts to weighted links.
- Reciprocity: Reciprocity measures opposite-direction links between node pairs, including a correlation-based coefficient between the adjacency matrix A and its transpose A^T.Self-loops are excluded from the reciprocity calculation.
- Assortativity: Assortativity captures degree correlation between linked nodes; an increasing ⟨k_nn|k⟩ indicates that higher-degree nodes connect to higher-degree neighbors.The conditional probability P(k′|k) describes links from degree k to degree k′.
- Connectivity and clustering: Connectivity measures include density and shortest-path distance, while clustering coefficients characterize triangle structure in directed networks.Distances are computed on the symmetrized network for directed graphs and separately within connected components.
- Empirical properties: Interbank networks are sparse, yet most nodes are typically connected through paths in a giant component.Sparsity is expressed as |E| ≪ |V|^2.
B Appendix B: Similarity Analysis Methodology
Appendix B reviews similarity measures for valued, binary, and graph data, then selects Jaccard similarity after noting limitations of correlation and maximal-common-subgraph approaches. The choice reflects the need to compare network layers while accounting for data type and computational concerns.
- Similarity measures: Similarity measures should match the data type and representation being compared, with different tools available for numerical vectors, binary vectors, and graphs.
- Valued vectors: Cosine similarity and Pearson correlation are standard choices for valued vectors, but correlation assumes equally distributed entries that network data may violate.Cosine similarity lies in [−1, 1], or [0, 1] for nonnegative vectors.
- Binary vectors: Jaccard and Dice similarities are used for binary vectors, with Dice related to J by D = 2J/(1+J) but lacking a proper metric-derived distance.
- Graph similarity: Maximal-common-subgraph similarity compares |G*| with max(|G1|, |G2|), but finding G* is computationally expensive and can be sensitive to data errors, especially in weighted networks.
- Measure selection: The analysis therefore employs Jaccard similarity rather than the reviewed alternative measures.
C Appendix C: Null Models Methodology
Appendix C constructs maximum-entropy network ensembles by weighting observables over graph realizations and solving parameterized constraint systems. The resulting weighted construction combines independent topology and weight variables to satisfy degree and strength constraints, while earlier weighted ensembles can produce unrealistic topology.
- Ensemble formulation: Network observables in a maximum-entropy ensemble are averaged over graph realizations using their probabilities P(G).
- Parameter estimation: The model specifies a parameter-dependent probability distribution by maximizing a Lagrangian, with parameters determined from the imposed constraints.For the Boltzmann-Gibbs distribution, the constraint system provides maximum-likelihood estimates of the parameters.
- Binary null models: Degree-constrained binary networks yield independent Bernoulli links, while the corresponding degree equations are solved numerically for the parameters.
- Weighted null models: Strength-constrained weighted ensembles produce geometrically distributed weights but may have hard-to-solve systems and unrealistic topology unconstrained by degree distribution or connectivity.
- Interpretation: The maximum-entropy solution represents the most diversified configuration consistent with the specified constraints.
- Combined constraints: The alternative weighted construction treats artificial link weights as products of independent variables, combining topology probabilities pij with λij = xixj.This ensemble satisfies degree and strength distribution constraints simultaneously.