Introduction

Network models based on hyperbolic geometry have been successful in explaining the structural features of informational1, social2 and biological networks3. Such models provide a mathematical framework to resolve the conflicting paradigms of preferential attachment (attraction to popular nodes) and community effects (attraction to similar nodes) in networks4,5,6.

Just as the geometric structure of a social network determines the diffusion of news, rumors or infective diseases between individuals7, the geometric structure of a financial network influences the diffusion of financial stress between financial institutions, such as banks8,9,10,11. Indeed, the lack of understanding for risks originating from the systemic interaction of financial institutions has been identified as a major contributing factor to the global financial crisis of 200812. While many recent studies have analysed the mechanisms of financial contagion in theoretical or simulation-based settings, less attention has been payed to the structural characteristics and the geometric representation of real financial networks. Although evidence of hyperbolic structure has been uncovered for international trade networks13, no such analysis has been carried out for networks of financial institutions. Identifying a suitable geometric representation for such networks can help to monitor and quantify structural change, and—in the case of hyperbolic geometry—even distinguish structural change in terms of systemic importance from changes in the network’s peripheral structure. Moreover, a geometric representation can form the basis of analytic models of contagion processes and their optimal control in future research.

Here, we consider financial networks inferred from bank balance sheet data, as collected and made available by the European Banking Authority (EBA) within the European banking stress test and transparency exercises of 2014, 2016 and 201814,15. We show that these networks can be embedded into low-dimensional hyperbolic space with considerably smaller distortion than into Euclidean space of the same dimension, suggesting that the paradigm of latent hyperbolic geometry also applies to financial networks. In addition, we demonstrate that the hyperbolic geometric representation compares favorably to a degree-corrected stochastic block model16—a popular non-geometric network model. Furthermore—following Papadopoulos et al.4—we decompose the embedded hyperbolic coordinates into the dimensions of popularity and similarity and demonstrate that these dimensions align with systemic importance and membership in regional banking clusters respectively. Finally, the longitudinal structure of the data allows us to track changes in these dimensions over time, i.e., to track the stability of systemic importance and of the peripheral community structure over time.

Results

Inference of financial networks

Contagion in financial networks is a complex process, which can take place through several parallel (and potentially interacting) mechanisms and channels17. These mechanisms include direct bank-to-bank liabilities18, bank runs19, and market-mediated contagion through asset sales17,20,21,22 (‘fire-sale contagion’); see also (French et al.12, p. 21ff). Here, we focus on the channel of fire-sale contagion, which has been singled out—both in simulation21 and in empirical studies20—as a key mechanism of financial contagion. Moreover, the propensity of fire-sale contagion can be quantified from available balance sheet data, using liquidity-weighted portfolio overlap (LWPO)22,23 as an indicator (see “Methods” for details).

Our inference of financial networks follows a two-stage mechanism: First, we construct a weighted bipartite network in which banks \(B = (b_1, \ldots , b_n)\) are linked to a common pool of assets \(A = (a_1, \ldots , a_m)\), which consist of sovereign bonds classified by issuing country and by different levels of maturity. In the second step we perform a one-mode projection of this network on the node set B, using the LWPO of two banks \(b_i, b_j \in B\) to determine the weight \(w_{ij}\) of the link between the corresponding nodes. For any of the years \(y \in \{2014, 2016, 2018\}\), the result is an undirected, weighted network \(N_y\) of banks, in which two banks are connected if and only if they hold common assets. The link weight \(w_{ij}\), normalized to [0, 1], represents the susceptibility of two banks \(b_i, b_j\) to financial contagion, quantified by their LWPO.

Network features

The inferred networks are very dense (densities: \(\rho _{2014} = 0.86\), \(\rho _{2016} = 0.96\), \(\rho _{2018} = 1.00\)),   i.e., almost all pairs of banks hold some common assets. However, the distribution of weights is highly skewed (see Fig. 1A), with most of the connections exhibiting very small weights. In other words, the networks are dominated by a ‘sparse backbone’ of a few strong connections, which represent the dominant channels of potential contagion of financial distress. The same skew is present in the distribution of node strengths (see Fig. 1B) with a few strong nodes dominating over a majority of weaker nodes in all years.

Figure 1
figure 1

Densities of edge weights (A) and of node strengths (B) in the EBA financial networks of 2014, 2016 and 2018.

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Figure 2
figure 2

Degree distribution (I) and local clustering coefficients (II) in the 10%-backbone (a) and the 25%-backbone (b) of the EBA financial networks of 2014, 2016 and 2018.

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To extract more salient connectivity information from these highly connected networks, we also consider the ‘\(p\%\)-Backbone’ of each network, formed by the upper \(p\%\)-quantile of highest-weighted edges (We have also considered disparity filtering24 as an alternative method for backbone extraction; see ‘Comparison of methods and robustness checks’ below.). Figure 2 shows the degree distribution and the local clustering coefficient (in dependence on node degree) for the 10%- and the 25%-Backbone. While there is no evidence of a scale-free degree distribution, the clustering coefficient displays an interesting pattern: It is highest for medium-degree nodes and then decreases with increasing node degree. This indicates that high-degree nodes (i.e. highly connected banks) typically act as hubs between lower-degree nodes (i.e. ‘normal’ banks) without a direct link.

Latent network geometry

Network representation methods

Our next objective was to uncover the latent geometric network structure and to evaluate the suitability of a hyperbolic network model. (See “Methods” for background on hyperbolic geometry.) To this end, we applied four different network representation methods (one method embedding into two-dim. Euclidean space \({\mathbb{E}}_2\), two methods embedding into two-dim. hyperbolic space \({\mathbb{H}}_2\), and one non-geometric method) to the financial networks \(N_{2014}, N_{2016}\) and \(N_{2018}\) and their \(p\%\)-backbones for \(p = 10, 25, 50\). The first two methods, multidimensional scaling25 (MDS) and hydra+26,27, calculate stress-minimizing embeddings of the weighted network distances into Euclidean and hyperbolic geometry, respectively. The third method, Mercator28, is a connectivity-based method (i.e. ignoring network weights) and uses a mix of machine learning and maximum likelihood estimation to infer latent coordinates in a popularity-vs-similarity-model of hyperbolic geometry; see García-Pérez et al.28 for details. As non-geometric representation method, we used a degree-corrected stochastic block model (dSBM)16, as implemented in the R-package randnet29, which aims to represent network structure by inferring communities and their connection probabilities; see Karrer and Newman16 for details. Since Mercator and dSBM are connectivity-based, they can only be meaningfully applied to the network backbones. MDS and hydra+, on the other hand, can also be applied to the full weighted networks and are directly comparable, since they minimize exactly the same objective function, but only differ in their target geometries. Figure 3 shows a comparison of the embedding quality of the different methods. For the full networks, we use stress (i.e. the root mean square error between network distances and embedded distances) as an evaluation metric, while for the backbones we use the AUPR (area under the Precision-Recall-curve) from a network reconstruction task (see “Methods” for details).

Figure 3
figure 3

Panel (A): Stress (i.e. root mean square error) of network embeddings produced by hydra+ (hyperbolic target space) and multidimensional scaling/MDS (Euclidean target space). Lower stress values indicate better embedding quality. Panels (B)–(D): Area-under-Precision-Recall-curve (AUPR) for the task of reconstructing network backbones based on network representations produced by Mercator and hydra+ (hyperbolic target space), dSBM (non-geometric), and MDS (Euclidean target space). Higher AUPR values indicate better reconstruction performance.

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Comparison of methods and robustness checks

Our comparison shows that Mercator, based on hyperbolic geometry, outperforms all other methods in terms of network reconstruction performance, consistently over all three years of observation and independent of the threshold used for the extraction of the network backbone. The second hyperbolic embedding method, hydra+, yields results that are better or at least comparable to dSBM, while the Euclidean embedding method MDS performs worst. Also note that hydra+ and MDS are the only methods which can be directly applied to the full weighted network, in which case hydra+ clearly outperforms MDS in terms of embedding error.

To check the robustness of our results with respect to the method of backbone extraction, we have repeated the same analysis with backbones determined by disparity filtering24. The results are reported in supplementary Figure S1. While reconstruction quality deteriorates for all methods on the disparity filtered backbones, the advantage of the hyperbolic methods over the non-geometric and Euclidean methods becomes even more pronounced. Overall, we conclude that the latent geometry of the observed financial networks is—at least in low dimension—much better represented by negatively curved (hyperbolic) rather than flat (Euclidean) geometry. Moreover, the hyperbolic representations are superior even to the (non-geometric) degree-corrected stochastic block model in terms of network reconstruction quality.

Latent hyperbolic coordinates

As a result of the hyperbolic embeddings we obtain for each bank node \(b_i\) latent coordinates \((r_i, \theta _i)\) in the Poincaré disc model of hyperbolic space (see “Methods”). This allows us to connect the network embedding to the popularity-vs-similarity model of Papadopoulos et al.4 and the \({\mathbb{S}}^1\)-model of Ángeles Serrano et al.28,30. The hyperbolic embedding of the full banking network of 2018 produced by hydra+ is shown in Fig. 4A. The embedding of the 10%-Backbone of the same network produced by Mercator is shown in Fig. 4D. Note that the hydra+-embedding attempts to give a faithful representation of all distances in the weighted network, whereas Mercator only encodes connectivity information and is harder to interpret visually. This phenomenon is exacerbated by the laws of hyperbolic geometry, in which seemingly small differences in the radial coordinate can represent large differences in hyperbolic distance. With reference to Fig. 4A, the embedded network shows a clear core-periphery structure, in line with previous studies of financial networks31,32 and in agreement with the pattern exhibited by the local clustering coefficient in Fig. 2.

Figure 4
figure 4

Hyperbolic Embeddings of the EBA Financial Network of 2018. Nodes are labelled by country and bank ID and coloured according to region (see Table 1 for full names). Panel (A) shows the full network embedding produced by the \(\texttt{hydra+}\) method. Also shown is the top decile of strongest links, i.e., the connections with the largest liquidity-weighted portfolio overlap. Banks labelled as systemically important by the Financial Stability Board (G-SIBs) are indicated by asterisks. The black cross marks the capital-weighted hyperbolic center of the banking network. In panels (B) and (C) the Central/Eastern and the Nordic regional groups are highlighted to illustrate regional clustering. Panel (D) shows the hyperbolic embedding of the 10%-backbone of the same network, as produced by the Mercator method.

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Structural analysis

The popularity-vs-similarity model of Papadopoulos et al.4 and the \({\mathbb{S}}^1\)-model of Ángeles Serrano et al.28,30 used by Mercator offer a direct interpretation of the latent hyperbolic network coordinates in the Poincaré disc in terms of their popularity dimension (the radial coordinate r) and the similarity dimension (the angular coordinate \(\theta\)). In the context of financial networks, we hypothesized that the popularity dimension of a given bank aligns with its systemic importance, and that its similarity dimension is associated with sub-sectors of the banking system, e.g., along geographic and regional divisions. Also for the hydra+ embedding, a theoretical foundation for interpreting r as popularity dimension and \(\theta\) as similarity dimension has been given27. However, due to the asymmetric distribution of banks within the Poincaré disc (Fig. 4A) for the hydra+ embedding, we calculate its geodesic polar coordinates \((r_i,\theta _i)\) with respect to the network center-of-weight, rather than the center of the Poincaré disc; see “Methods” for details (The fact that both approaches—Mercator and re-centered hydra+—lead to qualitatively very similar results can be seen as a validation of this methodology.). For the Mercator method we directly use the coordinates \((r_i,\theta _i)\) from the embedding of the 25%-backbone and perform no additional centering.

To test the first hypothesis—the association between radial coordinate r and systemic importance—we labelled a bank as systemically important in a given year, whenever it was included in the contemporaneous list of global systemically important banks (G-SIBs) as published by the Financial Stability Board33,34,35; see also Table 1. Using a Wilcoxon–Mann–Whitney test, we find a significant association between radial rank and systemic importance in all years and for both methods (\(P_{2014} < .0001\), \(P_{2016} < .0001\) for both methods, \(P_{2018} = .0038\) for hydra and \(P_{2018} = 0.0001\) for Mercator). In Table 2 we report the five top-ranked banks (most central in terms of r) for each year.

Table 1 IDs and full names of banks in the 2018 EBA Network. Banks marked by asterisk (*) were G-SIBs in all years (2014, 2016, 2018); banks marked by dagger (\(\dagger\)) were G-SIBs in 2014 and 2016, but not in 2018.
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Table 2 For each year the five banks with the highest hyperbolic centrality (i.e., smallest r coordinate) are listed. The upper subtable corresponds to the hydra+ embedding of the full network and the lower subtable to the Mercator embedding of its 25%-backbone. Asterisks denote banks that are considered globally systemic relevant institutions (G-SIBs).
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To test the second hypothesis—the association between similarity dimension \(\theta\) and regional banking sub-sectors—we assigned banks to the following nine regional groups:

Spain (ES), Germany (DE), France (FR), Italy (IT), UK and Ireland (UK/IE), Nordic Region (EE/NO/SE/DK/FI/IS), Benelux Region (BE/NE/LU), Southern/Mediterranean (GR/CY/MT/PT), Central/Eastern Eur. (AT/BG/HU/LV/RO/SI).

These regions are reasonably balanced in terms of the number of banks included in the EBA panel. Using ANOVA for circular data (see “Methods”) we find a highly significant association between the angular coordinate \(\theta\) and the regional group in all three years considered (\(P < .0001\) in all years for both methods). This indicates that the peripheral community structure (away from the network core) of the EBA financial network is indeed strongly aligned with geographic and regional divisions in Europe. We have highlighted two different regional groups in Fig. 4B,C to illustrate the association between angular coordinate and regional structure.

Network structure over time

The longitudinal structure of the data set allows us to track changes in the network structure over the whole time span of observations from 2014 to 2018. Note, however, that the samples of banks included by the EBA vary substantially in size and—even when restricted to the smallest sample—are not completely overlapping; see Table 3. Nevertheless, the embedding quality of the hyperbolic methods (reported in Fig. 3) is surprisingly stable over all years. This suggests that the hyperbolic model does indeed capture intrinsic qualities of the network, rather than relying on transitory structural artefacts.

Table 3 Sample sizes of EBA data sets
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We proceed to analyze the temporal changes in the latent radial coordinate r and angular coordinate \(\theta\), corresponding to changes in systemic importance and community structure. Note that the small sample of banks included in the 2016 stress test restricts the number of banks that are included in this longitudinal analysis, cf. Table 3. The scatter plots in Fig. 5 and the corresponding Pearson’s correlations of .678 (hydra+, \(P < .0001\)) and .892 (Mercator, \(P < .0001\)) between 2014 and 2016, and .569 (hydra+, \(P = .0001\)) and .502 (Mercator, \(P =.0015\)) between 2016 and 2018 show a significant positive association between hyperbolic centrality in successive snap shots of the financial networks.

Figure 5
figure 5

Changes in radial coordinate r (low values indicate high centrality) between 2014 and 2016 (a) and 2016 and 2018 (b) for the hydra+-embedding (I) and the Mercator-embedding (II). Banks considered systemically relevant (G-SIBs) at the end of the time period are marked in red. Nordea bank is circled in the panel Ia and IIa; see text for background.

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In panel Ia of Fig. 5, Nordea bank can be identified as a clear outlier, moving from a very central position in 2014 to a peripheral position in 2016. Interestingly, Nordea was one of just two banks (together with Royal Bank of Scotland) which were removed from the list of G-SIBs in the subsequent update in 2018 due to decreasing systemic importance35. In the Mercator embedding (panel IIa) Nordea bank does not appear as an outlier, which is likely due to the fact that some structural information is lost when the full network is reduced to its backbone.

For the angular coordinate, we account for the circular nature of the variable and compute the circular correlation36 of the angular coordinates between successive years. Only moderate associations between successive years can be observed at absolute circular correlation values of 0.211 (hydra+, \(P = .1877\)) and 0.01 (Mercator, \(P = .9442\)) between 2014 and 2016 and 0.225 (hydra+, \(P = 0.1385\)) and 0.383 (Mercator, \(P = .0196\)) between 2016 and 2018.

Discussion

Based on data from the EBA stress tests of 2014, 2016 and the transparency exercise of 2018, we have presented strong evidence that the latent geometry of financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolic geometry. Calculating embeddings into the Poicaré disc model of hyperbolic geometry has allowed us to visualize this geometric structure and to connect it to the popularity-vs-similarity model of Papdopoulos et al.4 and the \({\mathbb{S}}^1\)-model of Ángeles Serrano et al.28,30. We find that the radial coordinate (‘popularity’) is strongly associated with systemic importance (as assessed by the Financial Stability Board) and the angular coordinate (‘similarity’) with geographic and regional subdivisions. A longitudinal analysis shows that—in the observation period from 2014 to 2018—systemic importance of banks within the European banking network has stayed rather stable and has been predominated by only gradual changes. The peripheral community structure has been more variable, but has remained strongly determined by geographical divisions in all years considered.

From a broader perspective, our results indicate that hyperbolic network representations could be important tools for regulators to monitor structural change in financial networks, as they are able to distinguish changes in the systemic importance (popularity) of financial institutions from ‘peripheral changes’ (similarity) which are less relevant from a regulator’s perspective. Furthermore, our research provides an empirical basis for using hyperbolic geometry as a model space for the modelling of contagion processes and their optimal control in financial (or other) networks. Instead of modelling such processes by simulation on individual networks, a geometric model space provides the opportunity of analytic models that provide deeper insights beyond a specific case.

Methods

Data preparation and inference of financial networks

The financial networks were extracted from three different publicly available data sets stemming from the stress tests (in 2014 and 2016) and the EU-wide transparency exercise (in 2018) of the European Banking Authority (EBA)14,15. The data sets contain detailed balance sheet information from all European banks (EU incl. UK + Norway) included in the stress test/transparency exercise of the EBA in the respective year. From these data sets we extracted the portfolio values of all sovereign bonds held by the banks, split by issuing country (38 countries) and three levels of maturity (short: 0M-3M, medium: 3M-2Y, long: 2Y-10Y+), resulting in \(m = 38 \times 3 = 114\) different asset classes.

For each year, this data was stored as the weighted adjacency matrix P (‘portfolio matrix’) of a bipartite network. The n rows of P correspond to the banks in the sample, the m columns to the different asset classes, and the element \(P_{ik}\) to the portfolio value (in EUR) of asset k in the balance sheet of bank i. To perform a one-mode projection of this bipartite network, we followed Cont and Wagalath23,37 as well as Cont and Schaanning22: We computed the liquidity-weighted portfolio overlap (LWPO) of bank i and bank j as

$$\begin{aligned} L_{ij} = \sum _{k=1}^{m} \frac{P_{ik} P_{jk}}{d_k}, \end{aligned}$$
(1)

where \(d_k\) is the market depth for asset k22. The LWPO measures the impact of a sudden liquidation of the portfolio of bank i on the portfolio value of bank j and vice versa. Hence, it quantifies the risk of fire-sale contagion between the banks in a financial stress scenario. The market depth of asset k was estimated from P as its total volume held by all banks in the sample, i.e., as \(d_k = \sum _{i=1}^{n} P_{ik}\). Writing D for the diagonal matrix of market depths, (1) can be succinctly written as matrix product \(L = P D^{-1} P^\top\). Finally, we set the link weight \(w_{ij}\) between bank \(b_i\) and \(b_j\) in the one-mode projection N of the banking network equal to the normalized LWPO between banks \(b_i\) and \(b_j\), i.e., \(w_{ij} := L_{ij} / \max _{i,j}L_{ij}\)

Background on hyperbolic geometry

The hyperboloid model

Hyperbolic geometry can be characterized as the geometry of a space of constant negative curvature, while the more familiar Euclidean geometry is the geometry of a flat space, i.e., a space of zero curvature. In the hyperboloid model of hyperbolic geometry38,39, d-dimensional hyperbolic space \({\mathbb{H}}_d\) is defined as the hyperboloid

$$\begin{aligned} {\mathbb{H}}_d = \left\{ x \in {\mathbb{R}}^{d+1}: x_0^2 - x_1^2 - \cdots - x_d^2 = 1, \; x_0 > 0\right\} \quad \text {equipped with distance} \quad {\mathrm {d}}_H(x,y) = {{\,\mathrm{arcosh}\,}}\Big (x_0 y_0 - x_1 y_1 - \cdots - x_d y_d\Big ). \end{aligned}$$

In fact, \({\mathbb{H}}_d\) endowed with the Riemannian metric tensor \(ds^2 = dx_0^2 - dx_1^2 - \cdots - dx_d^2\) is a Riemannian manifold and \({\mathrm {d}}_H(x,y)\) is the corresponding Riemannian distance38,39. The sectional curvature of this manifold is constant and equal to \(-1\). Thus, \({\mathbb{H}}_d\) is indeed a model of geometry of constant negative curvature.

The Poincaré disc model

While the hyperboloid model is convenient for computations, a more preferable (and popular) model for visualizations in dimension \(d=2\) is the Poincaré disc model38, which also forms the basis of the popularity-vs-similarity model of Papadopoulos et al.4. To obtain the Poincaré disc model, the hyperboloid \({\mathbb{H}}_2\) is mapped to the open unit disc (‘Poincaré disc’) \({\mathbb{D}} = \left\{ z \in {\mathbb{R}}^{2}:{z_{1}}^{2} + {z_{2}}^{2} < 1\right\}\), parameterized by hyperbolic polar coordinates as \(z_1 = \tanh (r/2) \cos \theta\), \(z_2 = \tanh (r/2) \sin \theta\), using the stereographic projection38

$$\begin{aligned} r = \log \left( x_0 + \sqrt{x_0^2 - 1}\right) , \qquad \theta = {{\,\mathrm{atan'}\,}}(x_2,x_1), \qquad x = (x_0,x_1,x_2) \in {\mathbb{H}}_2, \end{aligned}$$
(2)

where \({{\,\mathrm{atan'}\,}}\) is the quadrant-preserving arctangent (The quadrant-preserving arctangent \({{\,\mathrm{atan'}\,}}(x_2,x_1)\), well-defined unless \(x_1 = x_2 = 0\), returns the unique angle \(\theta \in [0,2\pi )\) which solves \(\tan \theta = x_2/x_1\) and points to the same quadrant as \((x_1,x_2)\). It is commonly implemented in scientific computing environments (e.g. in MATLAB or R) as atan2.). In the Poincaré disc model, the hyperbolic distance becomes

$$\begin{aligned} {\mathrm {d}}_B((r_1,\theta _1),(r_2,\theta _2)) = {{\,\mathrm{arcosh}\,}}\left( \cosh (r_1) \cosh (r_2) - \sinh (r_1)\sinh (r_2) \cos (\theta _1 - \theta _2)\right) \end{aligned}$$

and geodesic lines are represented by arcs of (Euclidean) circles intersected with \({\mathbb{D}}\).

Stress-minimizing embeddings and hyperbolic centering

Stress-minimizing embeddings

Stress-minimizing embedding methods aim to find—for each network node \(b_i\)—latent coordinates \(x^{i}\) in a geometric model space G, such that the geodesic distance between \(x^{i}\) and \(x^{j}\) in G matches—as closely as possible—a given dissimilarity measure \({\mathrm {d}}_{ij}\) (such as the weighted network distance) between nodes \(b_i\) and \(b_j\). This is achieved by minimizing the stress functional

$$\begin{aligned} {\text {Stress}}(x^{1}, \ldots , x^{n}) = {\sqrt{\frac{1}{n(n-1)}\sum _{i,j} \left( {{\mathrm {d}}_{ij}}^{\mathrm{network}} - {{\mathrm {d}}_{G}^{\mathrm{geom}}}(x^{i}, x^{j})\right) ^{2}}}, \end{aligned}$$
(3)

which measures the root mean square error between given network distances and the corresponding distances in the model space. For Euclidean geometry, this method is well-known as multidimensional scaling25,40, or—using a weighted stress functional—as Sammon mapping41. For hyperbolic space, i.e., when \({\mathrm {d}}_G^\text {geom} = {\mathrm {d}}_H\), several optimization methods for (3) have been proposed26,27,42. We use the hydra+ method implemented in the package hydra for the statistical computing environment R43.

Hyperbolic centering

For a point cloud \(x^1, \dots , x^n\) in \({\mathbb{H}}_d\) and non-negative weights \(w_1, \ldots , w_n\) summing to one, the hyperbolic mean36 or hyperbolic center of weight44 can be determined as follows: Calculate the weighted Euclidean mean \({\bar{x}} = \sum w_i x^i\), and its ‘resultant length’ \(\rho = \sqrt{(\bar{x}_0)^2 - ({\bar{x}}_1)^2 - \cdots - ({\bar{x}}_d)^2}\), which is a measure of dispersion for the point cloud. The hyperbolic center c is then determined as \(c = {\bar{x}} / \rho\) and is again an element of \({\mathbb{H}}_d\). The point cloud can be centered at c by transforming each point as \({\tilde{x}}^i = T_{-c} x^i\), where \(T_{c}\) is the hyperbolic translation matrix (‘Lorentz boost’)

$$\begin{aligned} T_{c} = \begin{pmatrix} c_0 &{} {\bar{c}}^\top \\ {\bar{c}} &{} \sqrt{I_d + {\bar{c}} {\bar{c}}^\top }\end{pmatrix} \qquad \text {with} \qquad c=(c_0,{\bar{c}}) = (c_0, c_1, \ldots , c_d). \end{aligned}$$

In dimension \(d = 2\), the stereographic projection (2) can then be applied to convert the centered coordinates \({\tilde{x}}^i\) to centered polar coordinates \((r_i, \theta _i)\) in the Poincaré disc.

Application to financial networks

For the hydra+ embedding, the described methods were applied to the financial networks inferred from the EBA data as follows: We converted the similarity weights \(w_{ij}\) (normalized LWPO) to dissimilarities \({\mathrm {d}}_{ij} = 1 - w_{ij}\). We embedded these dissimilarities by minimizing the stress functional (3), using the R-package hydra. For the resulting network embeddings, we calculated the capital-weighted network center c as the weighted hyperbolic mean with weights \(w_i\) proportional to the total capital \(\sum _{k=1}^m P_{ik}\) of bank i invested in all assets \(a_1, \dots a_m\). After centering at the hyperbolic center c, we calculated the coordinates \((r_i, \theta _i)\) by the stereographic projection (2).

For the Mercator embedding and the dSBM, we first extracted network backbones, both by simple thresholding and by disparity filtering24. The resulting backbones were used as input for the methods provided at https://github.com/networkgeometry/mercator and the implementation of dSBM in the R-package randnet. Mercator outputs latent coordinates \((r_i, \theta _i)\) in the Poincaré disc, and the output of the dSBM method is a matrix of connection probabilities \({\hat{p}}_{ij}\) for each node pair.

For multi-dimensional scaling (MDS) the same methodology as for hydra+ was used, except that Euclidean distance (instead of hyperbolic distance) was used as \({\mathrm {d}}^\text {geom}_G\) in the objective function (3).

Analysis of embedding results

AUPR and network reconstruction

To evaluate the embedding results of the network backbones, we used the following network reconstruction task: To each pair of nodes \((b_i, b_j)\), assign the score \(d^\text {geom}_G(x_i,x_j)^{-1}\), where \(d^\text {geom}_G\) is the geodesic distance in the geometric model space G, or—in case of the degree-corrected stochastic block model—the score \({\hat{p}}_{ij}\), where \({\hat{p}}_{ij}\) is the estimated connection probability between nodes i and j. Based on these scores we predict whether an edge is present between nodes \((b_i, b_j)\) or not, and construct the Precision-Recall(PR)-curve45 of this classifier. The area under the PR-curve (AUPR) measures the quality of this predictor, with an AUPR of 1.0 representing perfect prediction.

Wilcoxon–Mann–Whitney test

The Wilcoxon–Mann–Whitney46 test is a non-parametric test to decide whether the distributions of two populations are identical without assuming them to follow the normal distribution. Let X be a sample of size m from the first population and Y be a sample of size n from the second population. Consider the combined sample of size \(m+n\) ordered from least to greatest and denote the ranks of \(Y_{i}\) in this joint ordering by \(S_{i}\), \(i = 1, \ldots , n\). Then the test statistic \(W = \sum _{i=1}^{n} S_{i}\) is the sum of the ranks assigned to the values of Y.

ANOVA for circular data

With the Analysis of Variance for circular data36, we test for the equality of p mean directions from independent circular (i.e. taking values on the unit circle) populations with von-Mises (M) distribution and the same (unknown) concentration parameter \(\kappa\). We test the null hypothesis \(H_{0} :\mu _{1} = \ldots = \mu _{p}\), where \(\mu _{i}\) are the mean directions for the p populations following a \(\text {M}(\mu _{i}, \kappa )\) distribution. For any circular observation \(\theta\), denote \(s = \sin (\theta )\), \(c = \cos (\theta )\) and let \({\bar{s}}_i, {\bar{c}}_i\) be the averages within the i-th population. Let \(n_{i}\) be the sample size, \(R_{i} = \sqrt{\bar{s}_i^2 + {\bar{c}}_i^2}\) the mean resultant length of the i-th population, and let \(n = \sum _{i=1}^{p} n_{i}\) be the size of the combined sample and R the overall mean resultant length based on all n observations. The identity

$$\begin{aligned} n-R = \left( n - \sum _{i=1}^{p} R_{i} \right) + \left( \sum _{i=1}^{p} R_{i} - R \right) = \left( \sum _{i=1}^{p} (n_{i} - R_{i}) \right) + \left( \sum _{i=1}^{p} R_{i} - R \right) \end{aligned}$$

has the approximate \(\chi ^{2}\) decomposition \(\chi ^{2}_{n-1} = \chi ^{2}_{n-p} + \chi ^{2}_{p-1}\)36 and therefore, the test statistic \(F = \frac{(\sum _{i=1}^{p} R_{i} - R) / (p-1)}{\sum _{i=1}^{p} (n_{i} - R_{i}) / (n-p)}\) can be derived. The null hypothesis is rejected for a given confidence level \(\alpha\), when \(F > F_{p-1, n-p; \alpha }\), where \(F_{p-1, n-p; \alpha }\) is the \(\alpha\)-quantile of the F-distribution with \(p-1\) and \(n-p\) degrees of freedom36.