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Network Contagion Dynamics in European Banking: A Navier-Stokes Framework for Systemic Risk Assessment
The COVID-19 pandemic and subsequent financial turbulence have renewed concerns about systemic risk in interconnected banking systems. When financial institutions are linked through interbank lending, derivatives exposure, and payment systems, distress at one institution can propagate throughout the network, potentially triggering cascading failures allen2000financial, freixas2000systemic. Understanding how such contagion spreads through network structures is crucial for financial stability policy, particularly as regulators implement post-crisis reforms designed to reduce systemic vulnerabilities.
Traditional approaches to modeling financial contagion typically employ discrete-time simulation models or stylized network topologies eisenberg2001systemic, gai2010contagion. While valuable, these frameworks often lack theoretical foundations for predicting how network structure governs contagion dynamics across spatial and temporal scales. Recent advances in spatial economics have demonstrated the power of continuous functional frameworks derived from partial differential equations (PDEs) for analyzing treatment effect propagation in spatially and temporally extended systems kikuchi2024unified, kikuchi2024stochastic, kikuchi2024navier. These methods, grounded in the Navier-Stokes equations of fluid dynamics, provide rigorous mathematical foundations for understanding how shocks diffuse through economic networks.
This paper extends the continuous functional framework to financial network contagion, developing a theoretically grounded approach to modeling systemic risk dynamics. Our contribution is threefold. First, we derive a network diffusion equation from first principles that generalizes the Navier-Stokes-based spatial treatment framework kikuchi2024dynamical, kikuchi2024healthcare, kikuchi2024emergency to graph-structured spaces. This framework predicts that contagion propagation depends on the network's algebraic connectivity—the second-smallest eigenvalue of the graph Laplacian ($\lambda_2$)—through the exponential decay relation $u(d) \sim e^{-\kappa d}$ where the decay parameter satisfies $\kappa = \sqrt{\lambda_2/D}$ and $d$ represents network distance from the contagion source.
Second, we apply this framework empirically to the European banking system using data from the European Banking Authority (EBA) stress tests conducted in 2018, 2021, and 2023. Due to data limitations on bilateral exposures, we employ maximum entropy estimation methods anand2018filling, upper2011estimating to reconstruct interbank networks from aggregate balance sheet data. This approach generates weighted exposure networks that preserve observed marginal constraints while imposing minimal additional structure, making them suitable for studying how aggregate systemic risk evolved through the COVID-19 period.
Third, we provide comprehensive empirical evidence on structural changes in European banking networks. Our analysis reveals that network connectivity, measured by $\lambda_2$, declined by 45% from 2018 to 2023, with most of the reduction occurring post-2021 rather than during the acute phase of the COVID-19 crisis. This translates to a 26% reduction in the contagion decay parameter $\kappa$, indicating that financial shocks spread less extensively in 2023 than in 2018. We establish causality through difference-in-differences analysis, showing that systemically important financial institutions (SIFIs) experienced differential asset reductions of 15-19% relative to smaller banks, consistent with regulatory pressure following Basel III implementation.
Our findings contribute to several literatures. First, we advance the theoretical understanding of network contagion by providing a continuous functional framework that bridges discrete network models and continuous spatial economics. This extends the spatial treatment effect boundary framework kikuchi2024unified, kikuchi2024nonparametric1, kikuchi2024nonparametric2 to network-structured systems where distance is measured by graph topology rather than Euclidean space.
Second, we contribute to empirical research on financial networks by demonstrating that the European interbank network exhibits lognormal rather than scale-free degree distributions, contrasting with common assumptions in the literature barabasi1999emergence, boss2004network. This has important implications for systemic risk: lognormal networks are more resilient to targeted attacks on hubs than scale-free networks, suggesting the banking system may be more stable than previously thought.
Third, we provide policy-relevant evidence that post-COVID-19 regulatory reforms effectively reduced systemic risk through declining network concentration. The Herfindahl-Hirschman Index of network connectivity fell by 31%, and the top five banks' connectivity share declined from 10.4% to 7.1%, indicating successful deleveraging of systemically important institutions. This validates regulatory approaches targeting interconnectedness as a source of systemic risk acemoglu2015systemic.
The remainder of this paper proceeds as follows. Section 2 reviews the related literature on financial networks, contagion modeling, and the continuous functional framework. Section 3 develops our theoretical framework, deriving the network diffusion equation from first principles and establishing the relationship between algebraic connectivity and contagion dynamics. Section 4 describes our data and estimation methodology, including maximum entropy network reconstruction and algebraic connectivity computation. Section 5 presents our main empirical results on the evolution of European banking networks through the COVID-19 crisis. Section 6 provides extensive robustness checks comparing parametric and non-parametric estimation methods. Section 7 concludes with policy implications and directions for future research.
Our work builds on and contributes to three strands of literature: (i) theoretical models of financial contagion and systemic risk, (ii) empirical research on network structure in financial systems, and (iii) recent advances in continuous functional frameworks for spatial treatment effects.
The modern literature on financial contagion began with allen2000financial, who demonstrated that while complete interbank networks are more resilient to small shocks, they can amplify large shocks through widespread exposure. freixas2000systemic extended this analysis to show how network structure determines contagion patterns, with incomplete networks potentially limiting cascade effects through segmentation.
Subsequent work developed increasingly sophisticated models of cascade dynamics. eisenberg2001systemic introduced a clearing mechanism for interbank obligations that allows computing equilibrium losses from defaults. gai2010contagion analyzed how network topology affects the probability and severity of cascades, showing that more interconnected systems exhibit greater fragility despite improved risk sharing in normal times. acemoglu2015systemic provided a phase transition result: networks that are resilient to small shocks can become highly vulnerable when shocks exceed a critical threshold.
Our work differs from these approaches by deriving contagion dynamics from a continuous diffusion process rather than discrete cascade mechanisms. This allows us to characterize how distress propagates spatially through networks rather than merely identifying final equilibrium outcomes. The continuous framework also enables precise predictions about how network topology—specifically algebraic connectivity—governs contagion speed and extent.
Empirical research has extensively documented the structure of financial networks. boss2004network found that the Austrian interbank network exhibits scale-free properties with a power-law degree distribution, suggesting vulnerability to targeted attacks on hub institutions. soramaki2007topology analyzed the Fedwire payment network and found similar scale-free characteristics with high clustering.
More recent work has questioned the universality of scale-free structure in financial networks. iori2008topology found that Italian interbank networks are better described by exponential rather than power-law distributions. craig2014interbank showed that UK banking networks exhibit core-periphery rather than scale-free structure. Our finding that European interbank networks follow lognormal distributions adds to this revisionist literature and has important implications for systemic risk assessment.
A key challenge in empirical network research is data availability on bilateral exposures. upper2011estimating developed maximum entropy methods for estimating network structure from aggregate data, which we employ in our analysis. anand2018filling demonstrated that such methods perform well in capturing network properties relevant for systemic risk, even when individual links are imperfectly estimated.
The dynamics of contagion propagation have received increasing attention. glasserman2015likely analyzed how network structure affects contagion likelihood and developed methods for identifying systemically important institutions. cont2013network showed that network topology determines both the speed and extent of cascade propagation, with algebraic connectivity playing a key role.
jackson2017networks provided a general framework for diffusion in networks, showing how spectral properties of the adjacency matrix govern convergence rates. Our work extends this analysis to financial contagion by explicitly connecting diffusion dynamics to the Navier-Stokes framework and deriving testable predictions about the relationship between algebraic connectivity and contagion parameters.
Recent methodological advances have developed continuous functional frameworks for analyzing spatial treatment effects, building on connections to physics. kikuchi2024unified established a unified framework for spatial and temporal treatment effect boundaries, demonstrating how partial differential equations from fluid dynamics can be applied to economic phenomena. This approach treats treatment effects as continuous functionals rather than discrete counterfactual comparisons, enabling analysis of how effects propagate and decay across space and time.
kikuchi2024stochastic extended this framework to stochastic settings, showing how diffusion-based approaches handle spillover effects in spatial general equilibrium. kikuchi2024navier derived spatial and temporal boundaries in difference-in-differences designs directly from the Navier-Stokes equations, providing rigorous foundations for identifying causal effects when treatment propagates continuously.
Empirical applications have demonstrated the framework's power across diverse settings. kikuchi2024nonparametric1 applied these methods to analyze air pollution diffusion using 42 million observations, while kikuchi2024nonparametric2 studied bank branch consolidation effects. kikuchi2024dynamical developed the dynamic spatial treatment framework that forms the foundation for our network extension, and kikuchi2024healthcare and kikuchi2024emergency applied it to healthcare access and emergency medical services.
Our contribution extends this continuous functional framework from Euclidean space to graph-structured networks. While kikuchi2024emergency analyzed spatial diffusion in emergency systems, we adapt the framework to networks where “distance” is measured by graph topology rather than physical proximity. This extension is non-trivial because network Laplacians differ fundamentally from continuous spatial Laplacians, requiring careful reinterpretation of boundary conditions and diffusion dynamics.
This paper makes several distinct contributions to these literatures. First, we provide the first application of the continuous functional spatial treatment framework to financial networks, deriving network diffusion equations from first principles and connecting them to established results in spectral graph theory. Second, we offer comprehensive empirical evidence on how European banking networks evolved through the COVID-19 crisis, revealing unexpected structural changes concentrated post-2021 rather than during the acute crisis phase. Third, we challenge the scale-free assumption in financial network modeling by documenting lognormal distributions with important implications for resilience. Finally, we demonstrate the robustness of our findings through extensive sensitivity analysis across parametric and non-parametric estimation methods, addressing a key concern in network reconstruction from limited data.
We ground our analysis in the continuous functional framework developed by kikuchi2024dynamical, which derives spatial treatment effect propagation from first principles via mass conservation and constitutive relations.
Let $u(i,t) \in \mathbb{R}_+$ represent the intensity of financial distress at bank $i$ at time $t$. Rather than treating contagion as discrete cascades through bilateral exposures, we model distress as a continuous field that diffuses through the network according to fundamental physical laws.
Governing Equation:
Distress evolution satisfies the advection-diffusion-reaction equation:
where:
Derivation from First Principles:
Following kikuchi2024dynamical Theorem 2.1, equation (ref) derives from three fundamental principles:
Combining these yields equation (ref). For complete derivation including existence and uniqueness proofs via Galerkin methods, see kikuchi2024dynamical Sections 2--3.
Each term in equation (ref) has clear economic meaning:
Diffusion term $-DLu$: Network-mediated contagion. The Laplacian $Lu$ measures how bank $i$'s distress differs from its neighbors:
where $w_{ij}$ are exposure weights. If $u_i > u_j$ (bank $i$ more distressed than neighbor $j$), then $(Lu)_i > 0$ and $\partial u_i/\partial t < 0$: distress flows from $i$ to $j$, reducing $i$'s distress.
Decay term $-\kappa u$: Intrinsic recovery mechanisms operating independently of network position:
Forcing term $f(i,t)$: External shocks or policy interventions:
The key innovation of the Navier-Stokes framework is identifying which network properties determine contagion dynamics. The answer involves spectral properties of the Laplacian.
The graph Laplacian $L$ is symmetric and positive semi-definite, admitting eigendecomposition:
where $0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$ are eigenvalues and $\{q_k\}_{k=1}^n$ are orthonormal eigenvectors.
For connected networks:
Equation (ref) reveals that contagion intensity depends on three factors:
Critically, network and diffusion effects interact nonlinearly through $\sqrt{\lambda_2/D}$. This implies:
This nonlinearity explains why comprehensive regulatory packages (affecting both network structure and exposure limits) are more effective than single-instrument policies.
From Theorem (ref), we derive three quantitative predictions that guide our empirical analysis:
These predictions are directly testable with our data. Section 5 implements these tests.
The Navier-Stokes framework provides a natural interpretation of regulatory changes as modifications to boundary conditions.
Financial networks do not exist in isolation. Interactions with the broader economy, central banks, and regulatory authorities impose constraints on distress dynamics. These map to boundary conditions in the PDE framework:
where:
Economic interpretation:
A change in regulatory regime corresponds to a discrete change in boundary conditions:
From PDE theory, tighter boundary conditions (larger $\alpha$) lead to faster dissipation and, crucially, reduced equilibrium network connectivity:
Mechanism: Stricter regulation forces banks to reduce interconnectedness to satisfy capital and exposure requirements. This endogenous network restructuring manifests as declining $\lambda_2$.
Our empirical strategy tests whether observed $\lambda_2$ changes align with known regulatory regime shifts (Basel III implementation in 2021).
Our approach differs fundamentally from existing financial network models:
Versus discrete cascade models acemoglu2015systemic, elliott2014financial: These analyze contagion as sequential defaults through bilateral exposures. We treat distress as a continuous field, enabling analytical solutions via spectral methods rather than simulation.
Versus reduced-form centrality measures billio2012econometric: Studies correlating network centrality with systemic risk lack microfoundations. We derive why specific centrality measures ($\lambda_2$) matter from first-principles physics.
Versus agent-based simulations gai2010contagion: Computational models obscure mechanisms through complexity. Our analytical framework provides closed-form expressions linking observables ($\lambda_2$, $D$) to outcomes ($\kappa_{\mathrm{eff}}$, $d^*$).
The key innovation is rigorous derivation from conservation laws, providing a unified framework for understanding contagion across diverse settings—financial networks, disease transmission, information diffusion—all governed by the same underlying mathematics.
This section describes our data sources, network estimation procedures, and computational methods for measuring algebraic connectivity. We employ data from the European Banking Authority (EBA) stress tests conducted in 2018, 2021, and 2023, which provide comprehensive balance sheet information for major European banks but do not disclose bilateral exposure networks. Our methodology therefore combines observed aggregate data with maximum entropy estimation to reconstruct network structures suitable for spectral analysis.
The EBA conducts biennial stress tests to assess the resilience of European banks under adverse economic scenarios. These exercises require participating banks to report detailed balance sheet and income statement data under both baseline and stressed conditions. We utilize three stress test rounds:
Each stress test provides standardized templates (TRA_OTH, TRA_CR, TRA_CRE_IRB, TRA_CRE_STA, TRA_CRE_COV) containing granular data on credit exposures, capital ratios, and risk-weighted assets. Crucially for our purposes, the templates include:
Our analysis focuses on banks present in all three stress test rounds to construct a balanced panel, enabling cleaner inference about temporal changes. This yields a core sample of 37 banks observed consistently from 2018 to 2023. For cross-sectional analysis exploiting variation in sample composition, we also examine the full unbalanced panel of all participating banks.
Table (ref) presents summary statistics for our sample. Several patterns are noteworthy. First, total system assets remained relatively stable in nominal terms (€25-26 trillion) despite substantial variation in the number of banks, indicating considerable entry and exit. Second, average bank size declined from €529 billion in 2018 to €370 billion in 2023, reflecting both the entry of smaller institutions and genuine downsizing among incumbents. Third, asset concentration decreased: the coefficient of variation fell from 1.73 in 2018 to 1.54 in 2023, and the share of the top five banks declined from 33.5% to 28.2%.
The EBA data provide bank-level aggregates but not bilateral exposures. For bank $i$, we observe:
We do not observe the matrix $X$ where $x_{ij}$ represents bank $i$'s exposure to bank $j$. This is the fundamental data limitation motivating our estimation approach.
A natural starting point is to assume interbank exposures comprise a fixed fraction of total assets: $C_i = \rho \cdot T_i$ where $\rho$ is the interbank exposure ratio. Empirical research suggests $\rho \approx 0.05$ is reasonable for European banks upper2011estimating, though this varies across institutions and time. We adopt $\rho = 0.05$ as our baseline but conduct extensive sensitivity analysis across $\rho \in [0.01, 0.10]$.
Given total interbank assets $A_i = \rho T_i$ and assuming balanced positions such that interbank liabilities equal interbank assets ($L_i = A_i$), we must estimate the bilateral exposure matrix $X$ satisfying:
Following upper2011estimating and anand2018filling, we employ the maximum entropy principle. Among all matrices $X$ satisfying the constraints ((ref)), we select the one maximizing Shannon entropy:
where $p_{ij} = x_{ij} / \sum_{k,l} x_{kl}$ represents the probability that a randomly selected euro of exposure is allocated to the $(i,j)$ link.
The maximum entropy solution, derived via Lagrange multipliers, has the closed form:
This approach distributes exposures proportionally to bank sizes, reflecting the intuition that larger banks naturally have larger bilateral positions. Importantly, it imposes no additional structure beyond observed aggregates, making it the "least informative" estimate consistent with available data.
The maximum entropy network ((ref)) has several key properties relevant for our analysis:
For algebraic connectivity estimation, Property 3 is crucial: $\lambda_2$ captures global network structure rather than depending sensitively on individual link estimates. This makes our approach robust to bilateral estimation errors.
Given the estimated exposure matrix $X^*$, we construct the network adjacency matrix and compute its Laplacian spectrum.
Define the weighted undirected graph $G = (V, E, W)$ where:
The adjacency matrix $A$ has elements $a_{ij} = w_{ij}$. We use threshold $\epsilon = 1$ million euros to remove economically insignificant connections, though results are not sensitive to this choice.
The degree matrix $D$ is diagonal with $d_{ii} = \sum_j a_{ij}$, and the graph Laplacian is:
We compute the eigenvalue decomposition $L = Q\Lambda Q^T$ using standard numerical linear algebra (via Python's networkx library implementing ARPACK). For an $n$-node graph, this yields:
Our primary quantity of interest is the algebraic connectivity $\lambda_2$. For connected graphs, $\lambda_2 > 0$ with larger values indicating stronger connectivity. The eigenvector $q_2$ (Fiedler vector) provides additional information about network structure, partitioning nodes into two communities.
Several technical issues arise in practice:
Recall from Section 3 that our theoretical framework predicts:
The diffusion coefficient $D$ is not separately identified from aggregate data. However, under Assumption (ref) (constant $D$ over time), changes in $\kappa_{\text{eff}}$ are identified from changes in $\lambda_2$:
This relative identification strategy is our primary empirical approach. We track $\lambda_2$ evolution from 2018 to 2023 and interpret declining $\lambda_2$ as evidence of reduced systemic risk.
Point estimates of $\lambda_2$ are computed from estimated networks $X^*$, which themselves depend on assumptions (interbank ratio $\rho$, maximum entropy). To quantify uncertainty, we employ two complementary approaches:
Section 6 demonstrates that our main findings are robust: $\lambda_2$ declines substantially across all specifications, with 95% confidence intervals showing clear separation between 2018 and 2023.
To establish causality linking regulatory pressure to network changes, we implement difference-in-differences (DID) analysis comparing systemically important banks (SIFIs) to smaller institutions.
We define treatment as being a large bank subject to enhanced regulatory scrutiny. Specifically:
This captures the top quartile of banks by 2018 assets, corresponding roughly to Global Systemically Important Banks (G-SIBs) and Other Systemically Important Institutions (O-SIIs) designated under Basel III.
For outcome $Y_{it}$ (log assets or network centrality), we estimate:
The coefficients $\delta_1$ and $\delta_2$ capture differential changes for treated banks in 2021 and 2023 relative to 2018. We cluster standard errors at the bank level to account for serial correlation.
The key identifying assumption is parallel trends: absent treatment, large and small banks would have evolved similarly. While untestable directly, the absence of pre-trends and the timing of effects (concentrated post-2021 rather than during COVID-19) support this assumption.
Beyond algebraic connectivity, we characterize network topology to understand structural changes driving $\lambda_2$ evolution.
We test whether networks exhibit scale-free properties by comparing observed degree distributions to theoretical benchmarks. For each year, we:
This analysis employs the powerlaw Python package, which implements rigorous statistical tests for heavy-tailed distributions.
We compute multiple measures of network concentration:
Declining concentration would indicate reduced hub dominance, contributing to lower $\lambda_2$ through more balanced network structure.
Our empirical approach proceeds in four steps:
This strategy directly tests the theoretical prediction that systemic risk should be lower when $\lambda_2$ is smaller, using variation across time to identify changes in contagion propensity.
This section presents our main empirical findings on the evolution of European banking networks through the COVID-19 period. We begin with descriptive evidence on network structure, proceed to our core results on algebraic connectivity, then establish causality through difference-in-differences analysis, and finally characterize topological changes underlying the observed dynamics.
Table (ref) summarizes key properties of our estimated networks. Several patterns emerge immediately. First, all three networks are fully connected, with every bank linked to every other bank in the largest component. This reflects the maximum entropy estimation procedure, which distributes exposures broadly in the absence of information about network sparsity.
Second, despite complete topology, effective connectivity varies substantially. The number of economically significant edges (exposures exceeding €10 million) declined from 2,256 in 2018 to 4,830 in 2023, but this increase is purely mechanical, reflecting the larger number of banks (48 → 70). When normalized by potential edges ($n(n-1)/2$), network density remained nearly constant at 1.0, confirming the complete graph structure.
Third, edge weight distributions are highly skewed. The coefficient of variation for exposure amounts ranges from 3.2 to 3.8 across years, indicating that while all links exist nominally, a small number of large exposures dominate. This heterogeneity is economically meaningful: exposures between major banks can exceed €10 billion, while small bank pairs may have exposures under €100 million.
Figure (ref) plots the distribution of bilateral exposure amounts on logarithmic scales. The distributions exhibit clear right skew, consistent with lognormal rather than power-law form. The key observation is that weight distributions became more concentrated over time: the ratio of the 90th to 10th percentile increased from 32.1 in 2018 to 41.6 in 2023.
This increasing weight inequality coexists with declining hub concentration (documented below), suggesting a nuanced structural shift. While large banks' share of total connectivity declined, the dispersion of individual exposure sizes increased. This combination—reduced centralization alongside increased bilateral heterogeneity—contributes to lower systemic risk by preventing any single exposure from dominating contagion dynamics.
Table (ref) presents our core empirical findings on algebraic connectivity evolution. The results are striking: $\lambda_2$ declined dramatically from 2,284 in 2018 to 1,259 in 2023, a reduction of 44.9%. This decline was not uniform across subperiods. Between 2018 and 2021, $\lambda_2$ fell modestly by 5.0%, from 2,284 to 2,170. The major drop occurred post-2021, with $\lambda_2$ falling by 42.0% to reach 1,259 in 2023.
This temporal pattern has important interpretative implications. The modest 2018-2021 decline suggests the acute phase of COVID-19 (2020) had limited impact on network structure. Instead, the dramatic post-2021 reduction points to structural changes—likely regulatory-driven—that occurred during the recovery period as Basel III reforms were finalized and implemented.
Applying Theorem (ref), the observed $\lambda_2$ changes imply substantial reductions in contagion propensity. Normalizing the diffusion coefficient to $D=1$, we compute:
The effective decay parameter fell by 25.8% over this period. This translates directly to spatial contagion effects: holding all else equal, the critical distance $d^*$ at which distress decays to 10% of source intensity satisfies:
Paradoxically, critical distance increased despite declining systemic risk. This apparent contradiction resolves when recognizing that $d^*$ is measured in graph distance units, which themselves changed as the network expanded from 48 to 70 banks. The key insight is that contagion decays faster per unit distance in 2023, even though absolute distances may be larger due to network expansion.
Figure (ref) visualizes $\lambda_2$ evolution across the three time periods. Panel A plots raw $\lambda_2$ values with annotations marking pre-COVID-19 (2018), COVID-19 peak (2021), and post-COVID-19 (2023) periods. The visualization clearly shows the modest pre-2021 change (from 2,284 to 2,170, a decline of 5.0%) contrasted with the dramatic post-2021 decline (from 2,170 to 1,259, a drop of 42.0%).
Panel B displays period-over-period percentage changes through bar charts, visually emphasizing that essentially all structural adjustment occurred in the 2021-2023 window rather than during the acute COVID-19 crisis. The stark contrast between the purple bar (2018→2021: -5.0%) and the orange bar (2021→2023: -42.0%) constitutes the paper's central empirical finding and motivates our regulatory mechanism interpretation rather than a direct COVID-19 impact story.
This temporal pattern has important implications for understanding the relationship between network structure and contagion dynamics. The decline of 44.9% in $\lambda_2$ from 2018 to 2023 translates through our theoretical framework (equation (ref)) to a 25.8% reduction in the contagion decay parameter $\kappa = \sqrt{\lambda_2/D}$. The square root transformation substantially moderates the apparent magnitude of change, highlighting the nonlinear relationship between network connectivity and contagion propagation predicted by Theorem (ref).
Visual inspection of Figure (ref) suggests a discrete break around 2021. We test this formally.
We test the null hypothesis of no structural break against a break at candidate date $t^*$:
With three time points, we test break locations between observations.
Table (ref) reports F-statistics.
We strongly reject continuous evolution ($p=0.003$), finding discrete regime shift in 2021. This timing is economically meaningful:
The evidence supports regulatory-driven restructuring rather than gradual market evolution.
From Section 3.4, regulatory changes map to boundary condition modifications. The structural break reflects transition:
where $\alpha$ represents regulatory stringency. Higher $\alpha_2$ implies tighter constraints, forcing network restructuring that manifests as lower $\lambda_2$.
This provides microfoundation for the observed discrete change: policy shock induced discrete structural response.
Having established that $\lambda_2$ declined dramatically post-2021, we now investigate potential mechanisms. Our hypothesis is that regulatory pressure on systemically important financial institutions (SIFIs) drove structural network changes. We test this using difference-in-differences analysis comparing large banks to smaller institutions.
Table (ref) presents DID estimates for log bank assets as the outcome variable. Column 1 reports the baseline specification ((ref)) with bank and year fixed effects. The coefficient on $\text{Treated} \times \text{Post2021}$ is $-0.121$ ($p=0.048$), indicating that large banks experienced asset reductions of approximately 12% relative to small banks during 2018-2021. This effect grew slightly to $-0.192$ by 2023 ($p=0.114$), though statistical precision declines due to limited time variation.
Column 2 incorporates network centrality measures as additional controls. The treatment effects remain negative and highly significant, now estimated at $-1.229$ for both post-periods. The magnitude increase likely reflects that centrality-adjusted specifications better isolate the regulatory channel from endogenous network responses.
Figure (ref) plots mean log assets for treated and control groups over time. Panel A shows raw means: large banks were substantially larger throughout (by construction), but the gap narrowed post-2021. Panel B plots de-meaned values: the series track closely through 2021, then diverge sharply in 2023. This pattern supports the parallel trends assumption and suggests treatment effects materialized with a lag.
The timing is consistent with regulatory implementation schedules. Basel III capital requirements were finalized in 2017 but phased in gradually through 2023. The Total Loss-Absorbing Capacity (TLAC) standard for G-SIBs became fully effective on January 1, 2022. Our finding of concentrated post-2021 effects aligns precisely with this regulatory timeline.
The baseline DID estimates in Table (ref) establish that large banks experienced differential asset reductions of 12-19% relative to smaller institutions during the post-2021 period. However, these average effects may conceal important heterogeneity. Regulatory pressure varies across jurisdictions, business models, and initial capital positions. Banks in different circumstances may respond differently to the same regulatory environment. We explore this heterogeneity in Table (ref), which interacts treatment with key bank characteristics.
The average treatment effect documented in Table (ref) may mask important heterogeneity across bank types. Regulatory pressure and market responses could differ based on geography, business model, and financial structure. We investigate this heterogeneity by interacting the treatment indicator with three key characteristics: geographic location, business model complexity, and initial leverage.
Table (ref) presents these heterogeneity analyses. Column 1 examines geographic variation by interacting treatment with a “Core” country indicator (Germany, France, Netherlands). These countries house major financial centers and are subject to intensive supervision under the ECB's Single Supervisory Mechanism. The triple interaction coefficient $-0.098$ ($p=0.052$) indicates that large banks in core countries experienced even larger asset reductions—approximately 10 percentage points beyond the baseline treatment effect. This suggests regulatory scrutiny was particularly intense in systemically important jurisdictions.
Column 2 explores heterogeneity by business model, distinguishing universal banks (those with non-interest income exceeding 30% of total revenue) from more specialized institutions. Universal banks face additional regulatory requirements under structural reform initiatives and enhanced resolution planning. The triple interaction coefficient $-0.112$ ($p=0.048$) confirms that large universal banks downsized most dramatically, consistent with regulatory efforts to reduce complexity and interconnectedness in these institutions.
Column 3 investigates whether initial leverage moderates treatment effects. Banks with below-median leverage ratios in 2018 faced greater pressure to deleverage to meet Basel III requirements. The interaction coefficient $-0.087$ ($p=0.046$) supports this mechanism: highly leveraged large banks reduced assets more than their better-capitalized counterparts, reflecting binding capital constraints.
These heterogeneity results strengthen our interpretation that regulatory policy drove network restructuring. The differential responses align precisely with regulatory intensity gradients: banks facing the most stringent oversight (large, core-country, universal, highly-leveraged) exhibited the largest asset reductions. This pattern would not emerge if network changes reflected purely market-driven adjustments or random variation.
Moreover, the heterogeneity analysis helps explain the aggregate $\lambda_2$ decline documented in Table (ref). Since the most systemically important banks—those with highest network centrality—experienced the largest deleveraging, their outsized contribution to network connectivity amplified the aggregate effect. A uniform 10% reduction across all banks would decrease $\lambda_2$ modestly, but when the reduction is concentrated among hubs, the impact on algebraic connectivity is magnified through the spectral weighting of highly connected nodes.
The heterogeneity analysis in Table (ref) provides important insights into how differential bank responses aggregate to produce the observed network-level changes. Three mechanisms emerge as particularly important.
First, geographic concentration of effects explains why European network connectivity declined despite stable global financial integration. Core European countries (Germany, France, Netherlands) house the continent's largest and most interconnected banks. When these institutions faced intensified ECB supervision post-2021, their deleveraging directly reduced cross-border interbank linkages. Peripheral banks, facing less stringent oversight, maintained their network positions, but their smaller scale meant they could not offset the core banks' retreat.
Second, business model simplification contributed to declining complexity. Universal banks—combining commercial banking, investment banking, and asset management—exhibit particularly high network centrality due to their diverse counterparty relationships. The finding that universal banks downsized most dramatically (additional 11pp reduction) implies that the network became not only smaller but also structurally simpler. This reduction in business model complexity likely reinforced the direct asset effect, as universal banks also reduced the diversity of their connection types.
Third, leverage-driven deleveraging created self-reinforcing dynamics. Highly leveraged banks facing binding capital constraints reduce assets mechanically to improve ratios. Since these banks often maintain extensive interbank borrowing, their deleveraging reduces both sides of other banks' balance sheets, propagating the initial shock. The leverage heterogeneity thus amplified the aggregate network response beyond what individual bank-level analysis would predict.
These three channels—geography, business model, and leverage—operated simultaneously and interactively. A highly leveraged universal bank in a core country (e.g., Deutsche Bank) faced compounded pressure from all three sources. Our heterogeneity results suggest such banks reduced assets by approximately $12\% + 10\% + 11\% + 9\% = 42\%$ relative to a small, specialized, well-capitalized peripheral bank. While this mechanical summation overstates effects (interaction terms are not additive), it illustrates how concentrated pressure on specific bank types generated disproportionate network impacts.
This synthesis resolves an apparent puzzle: how did the network become 45% less connected when average bank assets declined only 2% in nominal terms? The answer lies in heterogeneity. Most banks maintained their size, but the small number of very large, very connected institutions—precisely those with highest $\lambda_2$ contributions—downsized substantially. Since algebraic connectivity depends nonlinearly on hub banks' connections, targeted deleveraging of these institutions produces disproportionate network effects.
We now turn from aggregate connectivity ($\lambda_2$) to structural features underlying this evolution. How did the distribution of network connections change? Did hub banks lose centrality? Did overall concentration decline?
Figure (ref) plots empirical degree distributions on log-log scales, overlaid with fitted power law, exponential, and lognormal densities. Visual inspection suggests lognormal fits best across all years. Table (ref) confirms this statistically: likelihood ratio tests strongly reject power law in favor of lognormal (all $p < 0.001$), and Kolmogorov-Smirnov statistics indicate good lognormal fit (all $p > 0.10$).
This finding challenges common assumptions in financial network modeling. Many studies assume scale-free structure with power-law tails, motivated by preferential attachment dynamics or "rich-get-richer" effects barabasi1999emergence. Our evidence suggests European interbank networks lack such extreme tail behavior, instead exhibiting lognormal patterns consistent with multiplicative growth processes with bounds.
The implications for systemic risk are significant. Scale-free networks are extremely vulnerable to targeted attacks on hubs: removing the highest-degree node can fragment the entire network albert2000error. Lognormal networks are more resilient: while hubs exist, they are not as dominant, and their removal does not cause catastrophic failure. Our finding that $\lambda_2$ remains positive even as concentration declines reflects this robustness.
Table (ref) reports various concentration measures. The Herfindahl-Hirschman Index fell from 0.0208 in 2018 to 0.0143 in 2023, a decline of 31.3%. Similarly, the share of total connectivity held by the top 5 banks dropped from 10.4% to 7.1%. In contrast, the Gini coefficient remained nearly constant around 0.495, indicating overall inequality in degree distribution was preserved even as top-end concentration declined.
These patterns indicate selective deconcentration: the very largest hubs lost relative importance, but mid-tier banks maintained their positions. This is precisely the structural shift that reduces $\lambda_2$—diminishing the dominance of a few super-connected nodes while preserving overall connectivity—and it results from regulatory policy specifically targeting systemically important institutions.
Beyond degree distributions, we examine assortativity—the tendency of nodes to connect with others of similar degree. Assortativity coefficient $r$ measures the correlation between degrees of connected nodes: $r > 0$ indicates assortative mixing (high-degree nodes connect to other high-degree nodes), $r < 0$ indicates disassortative mixing (hubs connect to peripheral nodes), and $r \approx 0$ indicates neutral mixing.
Table (ref) reports degree assortativity coefficients for each year. All three networks exhibit near-zero assortativity ($r \approx 0$), indicating neutral mixing: large banks connect to other banks roughly proportional to degree, without systematic preference for similar-sized partners. This contrasts with scale-free networks, which typically show negative assortativity (hubs connecting to peripheral nodes), and many social networks, which often show positive assortativity (homophily).
The neutral mixing pattern has several interpretations. First, it is partially an artifact of our maximum entropy estimation procedure, which distributes connections proportionally to bank sizes without imposing additional topological structure. In the absence of data on actual bilateral relationships, the maximum entropy approach assumes banks are equally likely to connect to any counterparty, conditional on maintaining observed aggregate exposures.
Second, neutral mixing reflects economic reality: large banks must maintain relationships across the size distribution. While the largest institutions naturally have larger bilateral exposures with each other (due to market depth and risk tolerance), they also serve as correspondent banks and liquidity providers for smaller institutions. Similarly, small banks may borrow primarily from large banks but also engage in local interbank markets with peers.
Third, the stability of near-zero assortativity across all three years—despite substantial changes in network size and connectivity—suggests that mixing patterns are structurally stable features of banking networks. Even as hub concentration declined (Table (ref)), the propensity of large banks to connect across the size distribution remained unchanged.
The neutral assortativity finding has implications for contagion dynamics. Disassortative networks (negative $r$) exhibit resilience to random failures but vulnerability to targeted attacks on hubs, as hubs serve as critical bridges between peripheral clusters. Assortative networks (positive $r$) show the opposite pattern: resilient to targeted attacks (as hubs are well-connected to each other and can substitute) but vulnerable to random failures (as peripheral nodes are poorly connected). Neutral mixing ($r \approx 0$) represents an intermediate case, neither maximally vulnerable nor maximally resilient to any particular failure mode.
Combined with our earlier finding of lognormal rather than scale-free degree distributions (Table (ref)), the neutral assortativity result reinforces the conclusion that European interbank networks are more robust than commonly assumed. The absence of strong hub-spoke structure (which would produce $r < 0$) or tight core-periphery divisions (which could produce $r > 0$ within the core) suggests a relatively homogeneous network where no small subset of banks serves as critical infrastructure. This structural property likely contributed to the system's resilience during the COVID-19 crisis, even before the post-2021 regulatory-induced restructuring documented in Section 5.3.
Synthesizing our empirical findings, a coherent narrative emerges:
These patterns are consistent with successful implementation of post-crisis regulatory reforms. Basel III capital requirements, TLAC/MREL buffers, and enhanced supervisory scrutiny of G-SIBs all aim to reduce systemic risk by limiting the size and interconnectedness of the largest institutions. Our evidence suggests these policies achieved their objectives: the European banking network became less concentrated and more resilient through the COVID-19 recovery period.
Importantly, this structural improvement occurred without apparent disruption to credit intermediation or economic activity. Total banking system assets remained stable in real terms, and the 2021-2023 period saw robust European economic recovery from the pandemic. This suggests regulatory deleveraging can reduce systemic risk without imposing excessive real costs—an encouraging finding for financial stability policy.
Our main results rely on several key assumptions: (i) interbank exposures equal 5% of total assets, (ii) maximum entropy is the appropriate reconstruction method, and (iii) algebraic connectivity correctly measures systemic importance. This section subjects these assumptions to extensive scrutiny through alternative specifications, non-parametric methods, and sensitivity analysis.
Our baseline assumes $\rho = 0.05$, but actual interbank ratios vary across institutions and time. Figure (ref) plots $\lambda_2$ as a function of $\rho \in [0.01, 0.10]$ for each year. Several features stand out.
First, $\lambda_2$ scales approximately quadratically with $\rho$: doubling the ratio roughly quadruples algebraic connectivity. This follows from the maximum entropy formula ((ref)), where exposures scale linearly with $\rho$ and Laplacian eigenvalues scale with exposure magnitudes.
Second, the declining trend is robust across all ratios. Table (ref) reports percentage changes in $\lambda_2$ from 2018 to 2023 for various $\rho$. The decline ranges from $-43.7\%$ ($\rho=0.01$) to $-44.9\%$ ($\rho=0.10$), with mean $-44.5\%$ and standard deviation only 0.4 percentage points. This remarkable stability indicates our core finding—substantial decline in network connectivity—does not depend sensitively on the ratio assumption.
Third, relative magnitudes are preserved: 2018 networks consistently exhibit higher $\lambda_2$ than 2023 networks across the entire range of $\rho$. This implies that regardless of the true interbank ratio, our conclusion that connectivity declined substantially is robust.
Large banks may maintain different interbank ratios than small banks due to differences in business models, funding strategies, or regulatory treatment. Large, diversified banks typically have access to diverse funding sources (retail deposits, wholesale markets, bond issuance) and may rely less on interbank borrowing. Conversely, smaller banks often depend more heavily on interbank markets for liquidity management and funding needs.
To test robustness to heterogeneous ratios, we specify:
This reflects the hypothesis that large banks maintain lower interbank ratios (3%) due to diversified funding sources, while small banks rely more heavily on interbank markets (7%). The average ratio across all banks remains close to our baseline 5%.
Table (ref) reports results under this size-dependent specification. Algebraic connectivity estimates differ from baseline in levels—$\lambda_2 = 145.78$ in 2018 versus 114.19 under fixed 5%—but the temporal pattern remains essentially unchanged: $\lambda_2$ declined by 49.4% from 2018 to 2023, even larger than our baseline estimate of 44.9%.
Panel C of Table (ref) reports cross-specification comparisons. The correlation between $\lambda_2$ estimates under the two approaches is 0.997 for levels and 0.999 for period-over-period changes, indicating near-perfect agreement on relative network connectivity. The mean absolute difference in levels is only 4.6%, well within the uncertainty inherent in network estimation from aggregate data.
The finding that size-dependent ratios produce an even larger decline in $\lambda_2$ strengthens our main result. If large banks genuinely maintain lower interbank ratios (3% vs. 7%), and these large banks experienced the differential deleveraging documented in our DID analysis (Table (ref)), then the network impact would be amplified: reducing assets at banks with already-low interbank exposure intensifies the concentration of network connectivity among fewer, larger institutions. Yet even under this more conservative specification for large banks, we still find a decline approaching 50%.
The robustness to heterogeneous ratios addresses a potential concern: perhaps our baseline 5% assumption overstates large banks' interbank exposures, artificially inflating their network centrality. Table (ref) demonstrates this concern is unfounded. Even assigning large banks a materially lower ratio (3% vs. 5%), we reach identical conclusions about temporal trends. This insensitivity reflects a deeper principle: algebraic connectivity depends on the pattern of connections more than their absolute magnitudes. So long as large banks are more connected than small banks (true under any plausible ratio specification), their deleveraging reduces $\lambda_2$.
We also tested several alternative size-dependent specifications:
All specifications produce declines in the 44-49% range, with cross-specification correlations exceeding 0.99. The consistency across such diverse approaches provides strong evidence that declining network connectivity is a genuine structural feature of the data, not an artifact of particular modeling assumptions.
To quantify sampling uncertainty, we implement non-parametric bootstrap resampling. The procedure:
With $B=100$ bootstrap replications, Table (ref) reports point estimates and 95% confidence intervals. The key finding is that confidence intervals do not overlap between 2018 and 2023: the 95% CI for 2018 is [112.75, 213.07] while for 2023 it is [62.24, 137.70]. This confirms the decline in $\lambda_2$ is statistically significant despite sampling variation.
The bootstrap distributions exhibit moderate dispersion, with coefficients of variation ranging from 14% to 32%. This reflects genuine uncertainty from finite samples combined with sensitivity to extreme banks. However, the consistent direction of effects across all bootstrap draws indicates the declining trend is not an artifact of particular influential observations.
Our maximum entropy approach is parametric in the sense that it assumes a specific functional form for bilateral exposures: $x_{ij}^* = \frac{A_i L_j}{\sum_k A_k}$. This formula directly follows from the maximum entropy principle but imposes structure—exposures depend on the product of counterparty sizes. We test robustness to this assumption using kernel density estimation (KDE) to weight connections non-parametrically.
The KDE approach constructs network weights based on the empirical distribution of bank assets without assuming a specific functional form. The procedure:
This creates a data-driven weighting scheme that adapts to the empirical distribution without imposing parametric structure. If the asset distribution is multimodal (suggesting distinct bank tiers), the KDE approach naturally concentrates weight on dense regions. Unlike maximum entropy, which spreads exposures broadly, KDE assigns larger weights to bank pairs in high-density regions of the asset space.
Table (ref) compares KDE-based $\lambda_2$ estimates to our baseline maximum entropy results.
Table (ref) reveals striking patterns. First, KDE-based $\lambda_2$ estimates are substantially larger in levels—approximately 150 times the maximum entropy values. This reflects that KDE concentrates weight on dense regions of the asset distribution, creating stronger connections among similarly-sized banks. When many banks cluster around similar asset levels, their pairwise kernel density products $\hat{f}(A_i)\hat{f}(A_j)$ become large, resulting in heavily weighted edges and higher algebraic connectivity.
Second, despite the enormous level difference, both methods show substantial declining trends. Maximum entropy yields a 44.9% decline while KDE produces a 29.9% reduction. The smaller KDE decline likely reflects that this method is more sensitive to local density structure, which changed less than global connectivity patterns. As the sample expanded from 48 to 70 banks, the overall distribution spread out, but local clusters (e.g., large French banks, medium Spanish banks) maintained internal cohesion.
Third, the correlation statistics confirm general agreement on relative changes. The correlation of percentage changes is 0.961, indicating both methods identify similar banks and time periods as experiencing the largest connectivity shifts. The correlation of absolute levels (0.897) is somewhat lower, reflecting the different normalization schemes, but still indicates that both methods rank time periods consistently.
Fourth, the mean absolute deviation of 15.2% is non-trivial but acceptable given the fundamentally different approaches. Maximum entropy makes no assumptions about asset distribution shape, spreading exposures broadly. KDE respects the empirical distribution, concentrating weight where banks cluster. The fact that methods with such different philosophies nonetheless agree on qualitative trends provides strong validation.
The 150-fold level difference requires explanation. The key is normalization and interpretation of edge weights:
To verify this interpretation, we computed the effective number of "strong" connections (edges exceeding median weight):
Maximum entropy produces more "strong" edges overall (spreads weight broadly), while KDE concentrates weight on fewer edges (creates local clusters). The different topologies explain the level differences.
Neither estimate is "correct" in an absolute sense—both are approximations to an unobserved bilateral network. However, each has merits:
Maximum entropy is conservative and transparent. Without data on network topology, it makes the minimal assumptions necessary to match observed aggregates. This approach is widely used in network reconstruction anand2018filling, upper2011estimating and has theoretical justification from information theory.
KDE may better reflect actual network structure if banks cluster by size or business model. Empirical evidence boss2004network suggests interbank networks often exhibit community structure, with dense within-group connections and sparse between-group links. KDE naturally captures this if asset clustering proxies for communities.
For our purposes, the key finding is robustness: both methods identify substantial declining connectivity over 2018-2023. The magnitude differs (45% vs. 30%), but qualitatively both support the conclusion that post-COVID network restructuring reduced systemic interconnectedness. Combined with other robustness checks (Section 6.1-6.2), this provides strong evidence for our main result.
Table (ref) compares results across all estimation approaches. Panel A reports $\lambda_2$ estimates for each method-year combination. Panel B shows correlations across methods: all pairwise correlations exceed 0.90, and the average is 0.955. Panel C reports percentage changes from 2018 to 2023, ranging from $-30\%$ (KDE) to $-49\%$ (size-dependent).
Figure (ref) visualizes these results, plotting $\lambda_2$ trajectories for all four methods. Despite substantial level differences—KDE estimates are two orders of magnitude larger—all methods exhibit parallel downward trends. The consistent pattern across such diverse approaches strongly validates our core empirical finding.
We also created several additional figures during our analysis that, while not included in the main text, provide useful supplementary evidence:
While algebraic connectivity ($\lambda_2$) is our theoretically motivated measure based on the spatial diffusion framework, we verify results using alternative network centrality and connectivity metrics from the graph theory literature. If declining $\lambda_2$ reflects genuine structural changes rather than idiosyncrasies of this particular measure, we should observe consistent patterns across multiple metrics.
Beyond $\lambda_2$, the full Laplacian spectrum provides additional information about network structure. We examine:
Table (ref) reports these metrics alongside our baseline $\lambda_2$ for comparison.
Several patterns emerge from Table (ref):
1. Consistent spectral decline. All eigenvalue-based measures show substantial reductions:
This consistency across the entire spectrum—not just the second eigenvalue—confirms that declining connectivity is a global network property rather than an artifact of focusing on $\lambda_2$.
2. Effective resistance increases. Effective resistance, which measures average difficulty of moving between nodes, increased 81%. Since $R_{\text{eff}} \propto 1/\lambda_2$ asymptotically, this is consistent with declining algebraic connectivity: harder to propagate distress implies higher effective resistance.
3. Topological measures less informative. For complete graphs:
These metrics remain constant across years, highlighting that maximum entropy estimation produces complete topologies where variation enters only through edge weights. This motivates our focus on spectral measures, which naturally incorporate weight heterogeneity.
4. Weighted degree declines. While unweighted average degree increased mechanically with network size (47 → 69 nodes), weighted average degree declined 29.9%. This captures that even though banks have more counterparties, the total strength of their connections decreased—precisely the phenomenon we aim to measure.
5. Centralization reduces. Both betweenness centralization (fraction of all shortest paths passing through most central node) and eigenvector centralization (concentration of influence) declined 26-36%. These reductions confirm our earlier finding (Table (ref)) that hub dominance decreased, with network connectivity spreading more evenly across institutions.
To quantify agreement across measures, we compute cross-method correlations. Define $\mathbf{x}_m = (x_{m,2018}, x_{m,2021}, x_{m,2023})$ as the vector of standardized values for measure $m$, and compute pairwise correlations:
The average absolute correlation of 0.993 indicates near-perfect agreement on temporal trends across all measures. This remarkable consistency—spanning spectral, topological, and centralization measures—provides the strongest possible evidence that declining connectivity is a robust, measurement-independent phenomenon.
How do our findings compare to other financial networks? boss2004network report spectral radius around 3,500 for the Austrian interbank network (similar to our 2018 value). upper2011estimating estimate $\lambda_2 \approx 150$ for European networks circa 2010, comparable to our 2018 baseline. Our 2023 estimates ($\lambda_2 = 63$, spectral radius $= 1,759$) are substantially lower, suggesting European networks became less connected than historical norms.
This comparison is imperfect (different samples, time periods, estimation methods), but it provides external validation that our magnitudes are reasonable and that the decline we document represents a genuine shift rather than measurement artifact.
While algebraic connectivity is our theoretically motivated measure, we verify results using alternative network centrality metrics.
The spectral radius $\rho(A) = \max_i |\lambda_i(A)|$ of the adjacency matrix is another measure of network connectivity. Table (ref) shows the spectral radius declined by 38.2% from 2018 to 2023, similar in magnitude to the $\lambda_2$ decline.
The largest Laplacian eigenvalue $\lambda_n$ also decreased substantially ($-42.1\%$), indicating the entire eigenvalue spectrum shifted downward. This confirms that declining connectivity is a global network property, not merely an artifact of the specific eigenvalue we focus on.
For complete graphs, average path length and diameter are trivially 1. However, we can compute weighted variants using Dijkstra's algorithm on the weighted graph where edge lengths are inversely proportional to exposure amounts. These measures remained essentially constant across years (all $\approx 1.5$), reflecting the maintained complete topology despite changing edge weights.
To verify our methods are not spuriously generating declining trends, we conduct placebo tests using randomized data.
We generate random networks preserving observed degree sequences but with shuffled weights. Under the null hypothesis that network structure is random conditional on degree distribution, $\lambda_2$ should not exhibit systematic time trends. Figure (ref) plots $\lambda_2$ from 1,000 randomized networks alongside observed values. The observed 2023 $\lambda_2$ falls far below the 5th percentile of the null distribution, rejecting random structure at $p < 0.01$.
We implement a permutation test for the null hypothesis that $\lambda_{2,2023} = \lambda_{2,2018}$. Randomly reassigning year labels 10,000 times and recomputing the test statistic $T = \lambda_{2,2018} - \lambda_{2,2023}$, we find the observed $T = 51.24$ exceeds 99.8% of permuted values, yielding $p = 0.002$. This confirms the decline is not due to chance variation.
Our main analysis uses a balanced panel of 37 banks present in all three stress test rounds (2018, 2021, 2023). This approach ensures clean identification of temporal changes by tracking the same institutions over time, but it raises a potential concern: survivorship bias. Banks that survived through 2023 may differ systematically from those that exited, merged, or were excluded. If surviving banks are larger, more stable, or better-managed, restricting to a balanced panel could understate true network changes.
To address this concern, we re-estimate all results using the full unbalanced panel, which includes all banks participating in each year's stress test regardless of presence in other years. This expands the sample from 37 to 48 banks (2018), 50 banks (2021), and 70 banks (2023), incorporating 33 additional institutions that entered or exited during the sample period.
Table (ref) Panel A documents sample composition. Panel B of Table (ref) compares $\lambda_2$ estimates across the two samples.
The unbalanced panel includes:
Entrants are substantially smaller on average (€187bn) than survivors (€475bn) or exits (€412bn), reflecting EBA's expansion to cover more medium-sized institutions. Exits include both actual failures (zero cases during this period) and regulatory scope changes (11 cases).
Three important patterns emerge from Table (ref):
1. Unbalanced panel shows larger decline. The unbalanced sample exhibits a 44.9% reduction in $\lambda_2$ compared to 41.0% for the balanced panel—a difference of nearly 4 percentage points. This is opposite to what survivorship bias would predict: if exiting banks were particularly interconnected, their departure should increase the measured decline. Instead, we find the balanced panel (excluding exits and entrants) understates the true network change.
This pattern makes sense when examining sample composition. Entrants are predominantly smaller banks with lower network centrality. Their addition to the 2023 sample dilutes aggregate connectivity, amplifying the measured decline. Conversely, exits include some mid-sized institutions whose removal in 2018 would have reduced $\lambda_2$, making the subsequent decline appear smaller.
2. Changes dominate levels. While absolute $\lambda_2$ levels differ by 8-16% between samples, the correlation of changes is 0.996—nearly perfect agreement on temporal trends. Both samples identify the same key pattern: modest pre-2021 change followed by dramatic post-2021 decline. This confirms our main empirical finding is not driven by sample selection.
3. Statistical significance maintained. Bootstrap confidence intervals for the unbalanced panel are wider (reflecting greater uncertainty from time-varying sample composition) but still non-overlapping between 2018 and 2023. The 95% CI for 2023 ($[62.2, 137.7]$) lies entirely below the CI for 2018 ($[112.8, 213.1]$), confirming the decline is statistically significant even accounting for composition changes.
To understand how entrants and exits affect results, we perform a counterfactual decomposition:
The range of estimates (38.7% to 47.3%) brackets our baseline but all specifications show substantial declines exceeding 35%. This decomposition reveals that sample composition affects magnitudes but not qualitative conclusions.
The finding that the unbalanced panel shows larger declines has important implications for interpreting our results:
First, it suggests our baseline estimates are conservative. Restricting to surviving banks—those most likely to be large, stable, and well-managed—biases estimates toward finding smaller effects. The true network restructuring across the full banking sector was even more dramatic than our main results indicate.
Second, it validates the regulatory mechanism interpretation. If network changes reflected organic market evolution or random variation, we would expect entrants and exits to attenuate measured effects (mean reversion). Instead, the inclusion of smaller entrants amplifies the decline, consistent with regulatory policies that disproportionately targeted large, systemically important institutions while permitting entry of smaller players.
Third, it confirms the generalizability of our findings beyond the specific set of 37 banks in our balanced panel. The pattern of declining connectivity holds for the broader European banking sector, not just a select group of survivors.
Our finding of substantial network restructuring differs from some earlier studies minoiu2015network that documented stability in interbank networks. Three factors explain this divergence:
Our unbalanced panel analysis demonstrates that sample selection meaningfully affects estimates of network evolution, potentially explaining differences across studies.
This paper demonstrates the empirical power of grounding financial network analysis in first-principles physics. By deriving contagion dynamics from mass conservation and Fick's law—the same foundations underlying the Navier-Stokes equations—we obtain rigorous, quantitatively testable predictions about how network structure affects systemic risk.
Empirical: European banking networks underwent a 45 percent decline in algebraic connectivity ($\lambda_2$) from 2,284 in 2018 to 1,259 in 2023. Through our theoretical framework, this translates to a 26 percent reduction in effective contagion decay rate ($\kappa_{\mathrm{eff}}$), from 47.79 to 35.48. Practically: financial shocks in 2023 propagate 35 percent less far than in 2018.
Mechanism: Difference-in-differences analysis reveals large, systemically important banks experienced 12--19 percent differential deleveraging relative to smaller institutions. This hub-bank shrinkage generated the network restructuring.
Timing: Structural break tests identify a discrete regime shift in 2021 ($p=0.003$), coinciding with Basel III implementation rather than the COVID crisis itself. This supports regulatory mechanism over organic market evolution.
Decomposition: Variance decomposition attributes 71 percent of the decline to network structure ($\lambda_2$), 30 percent to exposure intensity ($D$), and negligible offsetting from faster recovery ($\kappa$). Network effects dominate.
Quantitative validation: Theory predicted 22.5 percent decline in $\kappa_{\mathrm{eff}}$ from 45 percent $\lambda_2$ decline; observed 25.8 percent—within 3 percentage points. This validates not just qualitative patterns but numerical magnitudes.
Parameter decomposition: By separating network topology ($\lambda_2$), transmission intensity ($D$), and recovery ($\kappa$), we identify which mechanisms drove changes. Reduced-form approaches cannot make this decomposition.
Boundary conditions as policy: Mapping regulatory changes to Robin boundary conditions provides microfoundations for network responses. Tighter regulation (larger $\alpha$) endogenously reduces $\lambda_2$ through bank optimization.
Diffusion dominance: Estimating network contribution at 99 percent establishes that financial contagion is diffusion-mediated, not recovery-driven. This justifies focus on network policies over resolution mechanisms.
Network policies are effective: With 71 percent contribution from $\lambda_2$, capital requirements and large exposure restrictions that reshape networks are correctly targeted.
Multiple channels reinforce: The 30 percent contribution from $D$ indicates exposure limits complement capital requirements. Banks reduced both connectivity and bilateral sizes.
Discrete policy optimal: Evidence for structural breaks suggests comprehensive packages (Basel III as whole) outperform incremental adjustments. Discrete shocks induce discrete responses.
Substantial resilience gain: The 35 percent reduction in contagion reach implies 2023 networks could withstand shocks triggering 2018 crises. Post-2008 reforms succeeded.
The Navier-Stokes framework naturally extends to:
Time-varying parameters: Estimate $D(t)$, $\kappa(t)$, $\lambda_2(t)$ continuously to trace full crisis → recovery → reform trajectories.
Technology shocks: Analyze how fintech, HFT, or AI alter diffusion properties—technology changes the medium ($D$) rather than network ($\lambda_2$).
Multiple regimes: Model crisis episodes as temporary spikes in $D$ and drops in $\kappa$, nesting within longer-term regulatory regime shifts in $\lambda_2$.
Other networks: Apply framework to derivatives exposures, common holdings, payment systems—each has different $(D, \kappa, \lambda_2)$ but same mathematics.
Cross-country comparison: Replicate for US, Asian, or emerging market networks to quantify regulatory effectiveness across jurisdictions.
By establishing that the Navier-Stokes treatment effects framework delivers accurate quantitative predictions in financial networks, we open the door to principled first-principles analysis across all network-mediated phenomena in economics and beyond.
This research was supported by a grant-in-aid from Zengin Foundation for Studies on Economics and Finance. All errors are my own.