# Core–Periphery Structure in the Overnight Money Market: Evidence from the e-MID Trading Platform

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- Accepted:

DOI: 10.1007/s10614-014-9427-x

- Cite this article as:
- Fricke, D. & Lux, T. Comput Econ (2015) 45: 359. doi:10.1007/s10614-014-9427-x

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## Abstract

We explore the network topology arising from a dataset of the overnight interbank transactions on the e-MID trading platform from January 1999 to December 2010. In order to shed light on the hierarchical structure of the banking system, we estimate different versions of a core–periphery model. Our main findings are: (1) the identified core is quite stable over time in its size as well as in many structural properties, (2) there is also high persistence over time of banks’ identified positions as members of the core or periphery, (3) allowing for asymmetric ‘coreness’ with respect to lending and borrowing considerably improves the fit and reveals a high level of asymmetry and relatively little correlation between banks’ ‘in-coreness’ and ‘out-coreness’, and (4) we show that the identified core–periphery structure could not have been obtained spuriously from random networks. During the financial crisis of 2008, the reduction of interbank lending was mainly due to core banks reducing their numbers of active outgoing links.

### Keywords

Interbank market Network models Systemic risk Financial crisis### JEL Classification

G21 G01 E42## 1 Introduction and Existing Literature

Interbank markets allow banks to exchange central bank money in order to share liquidity risks.^{1} At the macro level, however, a high number of bank connections could give rise to systemic risk.^{2} Since it is well known that the structure of a network is important for its resilience,^{3} policymakers need information on the actual topology of the interbank network.

The experiences of the last few years have made policymakers aware of the necessity of gathering information on the structure of the financial network in general and the interbank market in particular.^{4} One reason for the previous scarcity of research on the connections between financial institutions is certainly the limitation of available data, the other reason being the neglect of the internal structure of the financial system by the dominating paradigm in macroeconomics during the last quarter of a century, cf. Colander et al. (2009).

Recent research in the natural sciences has significantly advanced our understanding of the structure and functioning of complex networks. Network ideas have been applied to very diverse areas and datasets from the internet, epidemiology, ecosystems, scientific collaborations and financial markets, to name a few. Most previous studies on the topology of interbank markets have been conducted by physicists applying measures from the natural sciences to a network formed by interbank liabilities. Examples include Boss et al. (2004) for the Austrian interbank market, Inaoka et al. (2004) for the Japanese BOJ-Net, Soramäki et al. (2007) for the US Fedwire network, Bech and Atalay (2010) for the US Federal funds market, and De Masi et al. (2006) and Iori et al. (2008) for the Italian electronic market for interbank deposit (e-MID). Overall, the most important findings of this literature are: (1) interbank networks are sparse, i.e. their density is relatively low,^{5} (2) degree distributions appear to be scale-free (with coefficients between 2 and 3),^{6} (3) transaction volumes appear to follow scale-free distributions as well, (4) clustering coefficients are usually quite small, (5) interbank networks are close to ‘small world’ structures, and (6) the networks show disassortative mixing, i.e. high-degree nodes tend to trade with low-degree nodes, and vice versa.^{7} This indicates that small banks tend to trade with large banks, but rarely among themselves. Thus, we might expect the interbank network to display some sort of hierarchical community structure.

In passing, many authors have indeed mentioned the finding of certain community structures in the interbank network they analyzed. For example, Boss et al. (2004) note that the Austrian interbank network shows a hierarchical community structure that mirrors the regional and sectoral organization of the Austrian banking system. Soramäki et al. (2007) show that the network includes a tightly connected core of money-center banks to which all other banks connect. Thus there is some form of tiering in the interbank market. The empirical findings of Cocco et al. (2009) also show that relationships between banks are important factors to explain differences in interest rates.

Community detection is an important aspect in network analysis and in this paper we are concerned with the identification of the set of arguably systemically important (core) banks. In order to do so, we estimate various versions of core–periphery models in the spirit of Borgatti and Everett (2000).^{8} Similar to De Masi et al. (2006) and Iori et al. (2008) we use data from the e-MID trading platform, an electronic trading system for unsecured deposits based in Milan and mainly used by Italian banks for overnight interbank credit. Core–periphery models have been applied in a number of interesting fields before, for example to identify the spreaders of sexually transmitted diseases (see Christley et al. 2005), in protein interaction networks (see Luo et al. 2009), and to identify opinion leaders in economic survey data (see Stolzenburg and Lux 2011).

The literature on the structure and importance of financial networks is indeed expanding quickly (see among others Acemoglu et al. 2012; Langfield et al. 2012; Summer 2013). To our knowledge, Craig and von Peter (forthcoming) was the first contribution applying a core–periphery structure to an interbank market. Using this core–periphery framework to a dataset of credit relationships between German banks,^{9} their results speak in favor of a very stable set of core banks. Furthermore, they show that core membership can be predicted using bank-specific features such as balance sheet size.^{10}

In this paper we will apply the (unrestricted) discrete core–periphery model, the (restricted) tiering model due to Craig and von Peter (forthcoming) as well as symmetric and asymmetric versions of a continuous core–periphery model (hitherto not applied to interbank data) to a different set of interbank market data. Using a detailed dataset containing all overnight interbank transactions in the Italian interbank market from January 1999 to December 2010, we find that a core–periphery structure provides a concise characterization of this dataset with many characteristics implied by the core–periphery dichotomy displaying a high degree of persistence and relatively little variability over time. The identified core also shows a high degree of persistence over time, consisting of roughly 28 % of all banks before the global financial crisis and 23 % afterwards. We can classify the majority of core banks as intermediaries, i.e. as banks both borrowing and lending money in the market. Furthermore, allowing for asymmetric ‘coreness’ with respect to lending and borrowing activity considerably improves the fit, and reveals a high level of asymmetry and relatively little correlation between banks’ ‘in-coreness’ and ‘out-coreness’. In particular, overall coreness is mainly driven by the liquidity provision of core members to large parts of the banking system. In contrast, borrowing activity appears to play a less important part in explaining overall coreness. Comparing the empirical identification of the core–periphery structure with artificially generated random and scale-free networks with the same network density, we show that our results could not have been obtained spuriously from a completely random structure of links. Scale-free networks get closer to the empirical results in terms of the size of their (pseudo-) core and the similarity of the simulated networks to a CP structure, but at least simple generating mechanisms for scale-free networks would not be fully consistent with the complete set of our empirical findings. We also shed light on the development during the financial crisis of 2008, finding that the reduction of interbank lending was mainly due to core banks’ reducing their numbers of active outgoing links.

Our findings indicate that the core–periphery structure may well be a new ‘stylized fact’ of modern interbank networks. In fact, this finding is surprisingly robust given the substantial differences in the nature of the interbank data from Germany, as investigated by Craig and von Peter (forthcoming), and the Italian data employed here. In addition, we should stress that, at the time of writing, we are aware of similar findings using interbank data from other countries as well, including India, Mexico, the Netherlands, and the UK (see Markose et al. 2010; Martinez-Jaramillo et al. 2012; van Lelyveld and in’t Veld 2012; Langfield et al. 2012, respectively).

An open question is how individual bank behavior leads to the observed core–periphery structure. Note that the core–periphery structure implies that banks tend to restrict the set of potential trading partners, thus potentially restricting the amount of search and negotiation costs, cf. Wilhite (2001). Due to the anonymous nature of the dataset, we can only speculate that a key mechanism for the observed structure might be that core banks have a comparative advantage in gathering and distributing information about their counterparties.

The remainder of this paper is structured as follows: Sect. 2 gives a brief introduction to necessary terminology for the formalisation of (interbank) networks, Sect. 3 introduces the Italian e-MID interbank data and highlights some of its important properties. Section 4 introduces different variants of the core–periphery model. Section 5 presents the results and different robustness checks. Section 6 discusses the findings and Sect. 7 concludes. A set of Appendices provides more technical details as well as further robustness checks, which can be found in the online appendix to this article.

## 2 Networks

A network consists of a set of \(N\) nodes that are connected by \(M\) edges (links). Taking each bank as a node and the interbank positions between them as links, the interbank network can be represented as a square matrix of dimension \(N\times N\) (data matrix, denoted \(\mathbf D \)).^{11} An element \(d_{ij}\) of this matrix represents a gross interbank claim, the total value of credit extended by bank \(i\) to bank \(j\) within a certain period. The size of \(d_{ij}\) can thus be seen as a measure of link intensity. Row (column) \(i\) shows bank \(i\)’s interbank claims (liabilities) towards all other banks. The diagonal elements \(d_{ii}\) are zero, since a bank will not trade with itself. Off-diagonal elements are positive in the presence of a link and zero otherwise.

Interbank data usually give rise to directed, sparse and valued networks.^{12} However, much of the extant network research ignores the last aspect by focusing on binary adjacency matrices only. An adjacency matrix \(\mathbf A \) contains elements \(a_{ij}\) equal to 1, if there is a directed link from bank \(i\) to \(j\) and 0 otherwise. Since the network is directed, both \(\mathbf A \) and \(\mathbf D \) are asymmetric in general. In this paper, we also take into account valued information by using both the raw data matrix as well as a matrix containing the number of trades between banks, denoted as \(\mathbf T \). In some cases it is also useful to work with the undirected version of the adjacency matrices, \(\mathbf A ^u\), where \(a_{ij}^u=\max (a_{ij},a_{ji})\).

As usual, some data aggregation is necessary to represent the system as a network. In the following, we use quarterly networks. The next section summarizes the most important properties of our data, more detailed information can be found in Finger et al. (2013).

## 3 Dataset

The Italian e-MID is a screen-based platform for trading of unsecured money-market deposits in Euros, US-Dollars, Pound Sterling, and Zloty operating in Milan through e-MID SpA.^{13} The market is fully centralized and very liquid; in 2006 e-MID accounted for 17 % of total turnover in the unsecured money market in the Euro area. Average daily trading volumes were 24.2 bn Euro in 2006, 22.4 bn Euro in 2007 and only 14 bn Euro in 2008.

Available maturities range from overnight up to 1 year. Most of the transactions are overnight. While the fraction was roughly 80 % of all trades in 1999, this figure has been continuously increasing over time with a value of more than 90 % in 2010.^{14} As of August 2011, e-MID had 192 members from EU countries and the US. Members were 29 central banks acting as market observers, 1 ministry of finance, 101 domestic banks and 61 international banks. We will see below that the composition of the active market participants has been changing substantially over time. Trades are bilateral and are executed within the limits of the credit lines agreed upon directly between participants. Contracts are automatically settled through the TARGET2 system.

The trading mechanism follows a quote-driven market and is similar to a limit-order-book in a stock market, but without consolidation. The market is transparent in the sense that the quoting banks’ IDs are visible to all other banks. Quotes contain the market side (buy or sell money), the volume, the interest rate and the maturity. Trades are registered when a bank (aggressor) actively chooses a quoted order. The platform allows for credit line checking before a transaction will be carried out, so trades have to be confirmed by both counterparties. The market also allows direct bilateral trades between counterparties.

The minimum quote size is 1.5 million Euros, whereas the minimum trade size is only 50,000 Euros. Thus, aggressors do not have to trade the entire amount quoted.^{15} Additional participant requirements, for example a certain amount of total assets, may pose an upward bias on the size of the participating banks. In any case, e-MID covers essentially the entire domestic overnight deposit market in Italy.^{16} Researchers from the European Central Bank have repeatedly stated that the e-MID data is representative for the interbank overnight activity, cf. Beaupain and Durré (2012).

We have access to all registered trades in Euro in the period from January 1999 to December 2010. For each trade we know the two banks’ ID numbers (not the names), their relative position (aggressor and quoter), the maturity and the transaction type (buy or sell). As mentioned above, the majority of trades is conducted overnight and due to the global financial crisis (GFC) markets for longer maturities essentially dried up. We will focus on all overnight trades conducted on the platform, leaving a total number of 1,317,679 trades. The large sample size of 12 years allows us to analyze the network evolution over time. Here we focus on the quarterly aggregates, leaving us with 48 snapshots of the network.

The data show a trivial community structure in that foreign banks tend to trade with each other preferentially, and so do Italian banks. Due to the limited extent of trading between both components, and the smaller number of foreign banks, we will focus on Italian banks only in our subsequent analysis. This leaves a total number of 1,215,759 trades for the analysis.

The e-MID network has a relatively high density compared to other interbank networks investigated in the literature.

^{17}For the density of the network formed by Italian banks, a Chow-test and a CUSUM-test both indicate that there is a structural break after quarter 39 (i.e. at the onset of the financial crisis). Later on, we will see that the core–periphery structure was also influenced by the GFC.The aggregation period is important for economic applications as the network structure is less volatile with longer aggregation periods, cf. Finger et al. (2013). Since the network is sparse, short periods will only give an incomplete image of existing linkages, where many links between otherwise frequent trading partners may be dormant. In order to obtain a more comprehensive and less random picture of existing links, a larger aggregation period is required. We will, therefore, use quarterly data in the following (but our results are robust to somewhat shorter or larger aggregation periods).

There is very small (at times even negative) correlation between the banks’ in- and out-degrees. Hence, the directed version of the network might contain important additional information.

The underlying distributions of in- and out-degrees are unlikely to be scale-free at any aggregation level (including the daily level), cf. Fricke and Lux (2013). The same holds for the number of transactions and transaction volumes.

The network shows disassortative mixing patterns, so nodes with high overall degree (number of connections) tend to connect with low-degree nodes, cf. Fricke et al. (2013).

## 4 Models

Core–periphery network models have been proposed first by Borgatti and Everett (2000). The basic idea is that a network can be divided into subgroups of core and periphery members. The discrete model partitions banks such that core (periphery) banks are maximally (minimally) connected to each other. The concept of discrete group membership can be extended by considering the core and periphery as opposite ends of a continuum. The continuous model generalizes the binary structure of the discrete partitioning, by assigning a ‘coreness’ level to each bank. In the following we will first present the general discrete model, with the tiering model proposed by Craig and von Peter (forthcoming) as a special case, and then move on to the asymmetric continuous (AC) model for directed networks due to Boyd et al. (2010). Throughout the following we assume that a network cannot have more than one core.^{18}

### 4.1 The Discrete Model

#### 4.1.1 Formalisation

^{19}

The \(\mathbf {CC}\)-block contains the top-tier banks, while the \(\mathbf {PP}\)-block contains the periphery. Note that the off-diagonal blocks may be 1-blocks (each core member connected to all periphery-nodes), 0-blocks (no connection between core and periphery members) or something in between, depending on the problem. Borgatti and Everett (2000) argue that only the diagonal blocks are characteristic of CP structures and are thus the defining property. We will denote this version, without any restrictions on the off-diagonal blocks, as the (general) discrete model.

In some cases however, the underlying model explicitly dictates requirements on the \(\mathbf {CP}\) and \(\mathbf {PC}\) blocks. For instance, Craig and von Peter (forthcoming) propose a more strictly tiered interbank market than the benchmark discrete structure. In this model, a key characteristic of core banks (top tier) is that they intermediate between periphery banks. If at least a minimum level of intermediation activity is required of a ‘core’ bank, this means that \(\mathbf {CP}\) and \(\mathbf {PC}\) have to be row- and column-regular,^{20} respectively, i.e. at least one entry has to be non-zero in each row of \(\mathbf {CP}\) and in each column of \(\mathbf {PC}\).

#### 4.1.2 Optimization Problem

The discrete core–periphery framework amounts to assigning to each bank the property of membership in the core or the periphery. This classification can be summarized in a vector \(c\) of zeros and ones of length \(N\) (the total number of banks). The usual approach to find the optimal coreness vector, \(c\), referred to as the minimum residual (MINRES) approach, is to fit a pattern matrix \(\mathbf {P} = c c'\), which should be as close as possible to the observed network matrix \(\mathbf {A}\). This requires to identify the core banks, which are unknown a priori.

We start by defining a coreness vector, ordering the core banks first and writing the set of core members as \(\mathcal {C}=\{1,\ldots ,N_c\}\).^{21} Then we can measure the ‘fit’ of the corresponding core–periphery structure as the total number of inconsistencies between the observed network and the idealized pattern matrix \(\mathbf P _I\) of the same dimension. Depending on the problem, the distance involves certain restrictions on the off-diagonal blocks, \(\mathbf {CP}\) and \(\mathbf {PC}\). The optimal partition \(\mathcal {C}^{*}\) thus minimizes the residuals and gives the optimal set of core banks.

Residuals are obtained by simply counting the errors in each of the four blocks of Eq. (1) and aggregating over the blocks. The core-core block should be a complete \(\mathbf {1}\)-block of dimension \(N_c\), so any missing link represents an inconsistency (residual) with respect to the model.^{22} Likewise any link between two periphery banks constitutes an error relative to the benchmark. Obviously, we can introduce any constraints on the off-diagonal blocks, so the tiering model can be easily implemented here as well: errors in the off-diagonal blocks penalize zero rows and columns, because these are inconsistent with row- and column-regularity, respectively. For example, a zero column could be penalized by as many errors as there are banks in the periphery \((N-N_c)\).

^{23}Formally this can be written as

#### 4.1.3 Implementation

Fitting the discrete and the tiering model to a real-world network is a large scale problem in combinatorial optimization. Exhaustive search becomes impractical for large matrices, since the number of possible labeled bipartitions increases exponentially with the dimension of the matrix. More precisely, the number of nontrivial bipartitions (with both the core and the periphery having at least two members) is \(2^N-2N-2\). The term \(2^N\) corresponds to the number of all possible subsets, while the negative terms exclude partitions with only core or periphery banks. For example, with \(N=10\) banks there are \(1002\) nontrivial possible bipartitions. For a system with \(N=100\) banks there are already roughly \(10^{30}\) partitions.

A number of algorithms have been applied to tackle such problems. We will use a Genetic Algorithm (GA) to fit both the discrete and the tiering model.^{24} A GA uses operations similar to genetic processes of biological organisms to develop better solutions of an optimization problem from an existing population of (randomly initiated) candidate solutions. Typically the proposed solutions are encoded in strings (chromosomes) mostly using a binary alphabet, i.e. in our setting the strings have length \(N\) and consist of ones and zeros, depending on whether a bank is in the core or periphery. We use the rate of correct classifications (in terms of the error score) by a string \(l, f_l = 1-e(\mathcal {C}_l)\) as a fitness function that drives the evolutionary search. Details are explained in the online appendix to this article.

### 4.2 The Continuous Model

#### 4.2.1 Basic Structure

One limitation of the partition-based approach presented above is the excessive simplicity of defining just two homogeneous classes of nodes: core and periphery. Assuming that the network data consists of continuous values representing strengths or capacities of relationships (for banking data: credit volumes or number of transactions), it seems sensible to also consider a continuous model in which each node is assigned a measure of ‘coreness’. Since a continuous measure of coreness allows for more flexibility in capturing the importance of an institution, we apply this model to the valued matrix \(\mathbf {D}\) of interbank liabilities rather than the binary adjacency matrix \(\mathbf {A}\).

The usual approach in the symmetric continuous (SC) model is to find a coreness vector \(c\), where \(1 \ge c_i \ge 0 \text { }\forall i\), with pattern matrix \(\mathbf P =cc'\) that approximates the observed data matrix as closely as possible. Similar to the presentation of the discrete model, the optimal coreness vector in the SC model can be found using the MINRES approach.^{25} Again however, this method imposes a symmetric pattern matrix, i.e. \(p_{ij}=p_{ji} \text { , } \forall i,j\). Thus, it is assumed that the strength of the relation from \(i\) to \(j\) is the same as that from \(j\) to \(i\). To overcome this restriction, we also estimate an AC core–periphery model, as introduced by Boyd et al. (2010). This formulation involves two vectors, representing the degrees of outgoing and incoming coreness for each node. A certain disadvantage of the continuous models is that restrictions, such as the tiering model, cannot be implemented. In the following, we will briefly introduce both model versions. More details on the AC model can be found in the online appendix to this article.

#### 4.2.2 The Symmetric Continuous (SC) Model

#### 4.2.3 The Asymmetric Continuous (AC) Model

^{26}The PRE of the AC model can be defined similarly as in Eq. (8) as

^{27}Note that the split into in- and out-coreness is germane to a singular value decomposition of our matrix \(\mathbf {D}\) of interbank liabilities. This similarity is exploited in the empirical estimation of the vectors \(u\) and \(v\). Our numerical approach for estimating these two coreness vectors follows Boyd et al. (2010) and is detailed in the online appendix to this article.

## 5 Results

This section presents and discusses the results from the different versions of the core–periphery framework. In the following, as noted above, we focus on the quarterly networks formed by Italian banks only. Robustness checks, using different aggregation periods and sample banks can be found in the online appendix to this article. Recall that the discrete and tiering model use the (binary) adjacency matrices \(\mathbf A \), while the continuous model uses the (log-transformed) valued matrix of transaction volumes \(\mathbf D \), as defined in Sect. 2. The online appendix also discusses the results for the continuous model using the matrix containing the number of transactions \(\mathbf T \) and a number of additional robustness checks.

As a first step, we compare the coreness vectors between the different models. As it turns out, the discrete model with arbitrary structure of the CP and PC submatrices and the tiering model are almost identical throughout. Later on, we show that the AC model adds important information on the asymmetric nature of the network, since the in- and out-coreness vectors are far from being identical. Secondly, we investigate the properties of the core/periphery banks. We find that the core is large compared to the findings in Craig and von Peter (forthcoming), but also very persistent over time. Due to the high network density, we find that the error scores are also much higher compared to the German market. In particular, the model fit deteriorates over time due to the GFC. Formal tests suggest a significant worsening of the fit of the core–periphery model after the GFC, pointing towards the breakdown of part of the core–periphery structure. As a last step, we investigate whether our results could have been obtained from random and scale-free networks. To this end we apply a bootstrap algorithm generating synthetic random and scale-free networks calibrated to share key properties of the empirical ones. We find that the identified core–periphery structures strongly differ from those obtained under an Erdös–Renyi generating mechanism, while scale-free networks get somewhat closer to the results from the empirical data, at least for the discrete CP specification. Most importantly, the strong mismatch between the observed and random networks indicates that the identified core–periphery structure is not a spurious network property.

### 5.1 Model Similarity

^{28}The same is true for the discrete core and the out-coreness with a value of .73, whereas the correlation between the discrete core and the in-coreness is much smaller with a value of .26. Core banks from the discrete model are therefore more likely to be in the out-core of the continuous model as well, but not necessarily in the in-core. This result seems rather surprising at first, since, for example, Cocco et al. (2009) show that small (periphery) banks are net lenders, which offer their excess liquidity to a preferred set of large (core) banks. Our analysis shows that at least in the present dataset, the pattern of interbank linkages is more complex: again, periphery banks lend money to a relatively small set of selected core banks, but the core banks in turn tend to redistribute this liquidity not only among the other core banks, but also among a larger part of the periphery. Supporting the relevance of asymmetry in the coreness of lenders and borrowers, we find that the density in the CP-block is on average three times higher than the density in the PC-block (see Fig. 6 below), so for most core banks the out-degree clearly exceeds the in-degree. This asymmetry in the network is also captured by the negative correlation of \(-\).08 between the in- and out-coreness vectors, cf. Fig. 3.

In the following we present more detailed results for the discrete and tiering model, then moving on to the continuous model.

### 5.2 Discrete and Tiering Model

#### 5.2.1 The Size of the Core and Periphery

Note also the negative trend in the absolute size of the core over time. This is not surprising given that the number of active Italian banks has been decreasing over time. A Chow-test indicates the existence of a structural break in the (detrended) core sizes after quarter 10, but no evidence for a structural break around the time of the GFC. Given the overall trend in the number of active banks, it seems more interesting to consider the relative size of the core compared to the complete interbank network. Figure 5 shows that the relative size of the core is rather stable over time, fluctuating around 28 % before the GFC, and around 23 % afterwards. A Chow-test indicates that there is a structural break after quarter 39. However, under a CUSUM test this break is only marginally significant at the 5 % level for the discrete model, and insignificant for the tiering model. Thus, there is some evidence that the GFC has led to a structural break in the formerly relatively stable structure of intermediation in the interbank market. However, we also see a positive trend in the core sizes for the last 3 quarters of the sample period, so that the relative core size seemed to revert to its pre-GFC level. Not surprisingly, the size of the core is highly correlated with the density of the network (cf. Fig. 2). We should note that relative core sizes are very high compared to the value of 3 % found for the German interbank market by Craig and von Peter (forthcoming). This is driven by the very high overall network density of above 20 %, compared to only 0.61 % for the German market.^{29}

The left panel of Fig. 6 shows the densities of the complete network and the core–core and periphery–periphery subnetworks over time. Since results are virtually the same for both models, we only display those of the baseline discrete model^{30} with rather stable values for the pre-GFC period, but again with a structural break after quarter 39 for all time series in the Figure. The density in the CC-block is at least 2.5 times that of the entire network and at least 6 times that of the PP-block. The right panel of Fig. 6 shows the densities in the off-diagonal blocks. As already mentioned, the density in the CP-block is three times higher than the corresponding density in the PC-block. These values are very stable over time, and we do not find evidence for a structural break.

Transition matrix: discrete model. \(C, P\) and \(E\) stand for core, periphery and exit, respectively

\( C_{t}\) | \( P_{t}\) | \( E_{t}\) | |
---|---|---|---|

\( C_{t-1}\) | .8324 | .1565 | .0110 |

\( P_{t-1}\) | .0555 | .9055 | .0391 |

\( E_{t-1}\) | .0012 | .0104 | .9885 |

#### 5.2.2 Model Fit

Erdös–Renyi (ER) random graphs, where a link is formed with probability \(p\). The value of \(p\) will be set equal to the observed density of the empirical quarterly networks. These synthetic networks are completely random and we do not expect to find a convincing core–periphery structure in this case. The error scores should be relatively high, since identified cores would be completely spurious. Note that this is tantamount to a test of the null hypothesis of completely random link formation. If the error scores of the core–periphery model are below a certain percentile of the bootstrapped distribution of those obtained for the random networks, we could exclude with a significance level equal to the inverse of that probability, that our results are spuriously obtained from a completely random system of interbank liabilities. As it turns out, all error scores are always way below the minimum obtained for the random networks.

^{31}Scale-free (SF) random graphs, with scaling parameter of 2.3 for both in- and out-degrees.

^{32}Even though we found the degree distribution not to be scale-free, see Fricke and Lux (2013), many interbank market data have been reported to have a certain resemblance of their degree distributions to a scale-free distribution. Reported scaling parameters vary between 2 and 3, but are roughly similar for in- and out-degree. The most prominent mechanism to construct scale-free networks is that of preferential attachment, see Barabasi and Albert (1999), where high-degree nodes tend to attract more links than low-degree nodes over time. Here we generate directed SF networks using the algorithm of Goh et al. (2001), but the findings are similar when using other generating mechanisms.

The right panel of Fig. 8 shows the core sizes for the actual and random networks (again including one standard error for the simulated models). We see that the observed core is significantly larger than both the core of the ER and SF networks.^{33} We can safely conclude that the ER model is unable to account for the core–periphery distinction that we find in the data. While the CP structure is thus not consistent with link formation via an Erdös–Renyi algorithm, the relationship between CP and SF networks is less clear, as both frameworks focus on different aspects of the network topology that need not necessarily be mutually exclusive. The differences to the SF networks are less pronounced and more subtle. In part, these differences may be due to the subtle mixing patterns of core-periphery networks: it is intuitive that a CP network should be disassortative due to the preferred attachment of entities with few links to those with a large number of connections. However, this tendency is mainly represented by the entries entering the CP, PC, and PP blocks, while the CC block adds an element of positive assortativity (nodes with many links attaching themselves to each other). In Fricke et al. (2013) we show that SF networks are typically disassortative, but less so than the observed networks. This difference is largely driven by the smaller level of disassortativity in the off-diagonal blocks. At any rate, however, the low error scores of the SF networks indicate that their data-generating mechanism, while based on different principles, leads to network structures that are observationally close to core–periphery structures.

It is important to emphasize the different organizing principles underlying the CP and SF network models: the CP framework is based on a *mesoscopic* concept, basically providing conditions for the existence of links between individual nodes. In contrast, the scale-free models are based on a *macroscopic* concept, which is characterized by the overall degree distribution. In fact, one could imagine that, for instance, preferential attachment generates a structure similar to a CP pattern matrix: nodes with many links are typically also connected with each other (dense CC block), while being connected to many nodes with small degree (sparse CP and PC blocks). Finally, low degree nodes also have a particularly small probability to be linked to each other (close to an almost empty PP block). While SF models, therefore, have a tendency to generate network structures in which one might easily distinguish between core and periphery nodes, it is not clear that data exhibiting a CP structure should also display a power-law distribution of their degrees. Indeed the e-MID data appear to be far from a scale-free distribution, showing rather pronounced exponential decay of the degree distribution (Fricke and Lux 2013). From this perspective, the whole universe of SF models could be excluded as data-generating processes for this empirical record.

### 5.3 Continuous Model

^{34}Obviously, core banks have on average a higher in- and out-coreness. Indeed, we see a relatively sharp distinction between core and periphery banks. Core banks (red) of the discrete model are typically characterized by a sum of their continuous in- and out-coreness above .2, while this sum is lower for banks assigned to the periphery. For both categories, there might be a dominance of lending and borrowing or a more balanced composition of their transactions. The systemic importance of a bank, in terms of its in- and out-coreness, is therefore not identical in general.

An important question is by how much the fit of the model improves by using the AC model rather than the SC model. As a rule of thumb, Boyd et al. (2010) argue that the PRE of the SC model should be at least .5 in order to have a superior fit to an unstructured distribution of activity. Here we find small values around .2 for the SC model, but much higher values of around .58 for the AC model (cf. Fig. 11).^{35} Since the fit of the two-dimensional continuous approach (AC) is way better than that of the one-dimensional continuous approach (SC), we conclude that the directed version of the model contains important information about the structure of the interbank market. In line with our findings for the discrete and tiering model, the PRE of the AC model displays a structural break after quarter 39 (based on a Chow-test and a CUSUM test), but not in the SC model. Similar to the discrete and tiering models above, the fit of the model deteriorates somewhat with the GFC, with lower average values afterwards.

The poor fit of the SC model compared to the (symmetric) discrete model, is to a large extent due to the high level of asymmetry in the network. In the discrete model, only the CC- and PP-blocks enter the objective function, while the SC model also considers the CP- and PC-blocks by construction. However, these have already been shown above to be characterized by pronounced asymmetry which leads to the poorer fit of the SC model compared to the discrete specification.

In order to check the potential of other network models to resemble a CP-structure, we use the same approach as for the discrete and tiering model. Here we generate valued and directed synthetic networks of interbank liabilities.^{36} Figure 11 compares the PREs of the actual networks with the mean values from 100 realizations of random ER and SF networks (again with scaling parameter 2.3 for in- and out-degrees) minus and plus one standard deviation. As expected, the actual PREs of the SC and AC models again significantly exceed those from the ER networks by far, which are very low in general. In contrast, for the SF networks, the PREs of the SC model are close to the actual ones, while they are much lower than those obtained for the AC model.^{37} This finding underscores the observed asymmetries in the network, which are absent from baseline scale-free networks, where in- and out-degrees of individual banks are highly correlated by construction in the setup considered here, with in- and out-degrees (or volumes) following the same distribution. Hence, it seems to be more the asymmetry in the CP structure rather than the mere core–periphery distinction that limits the replicability of the structure of the present dataset by a baseline generating mechanism for SF networks. Altering the tail exponents of the in- and out-degree distributions might lead to SF networks which are closer to the observed ones.

### 5.4 What Defines a Core Bank?

In the following we will focus on the results from the discrete model.^{38} As a first step, we calculate the correlations between the coreness vectors and different observable variables (degree, size, and trading activity).^{39}

^{40}Here, in-size contains the total volume of borrowing transactions (per quarter and per bank), out-size the total volume of lending transactions and total size the sum of in-size and out-size. Similarly, the number of in-transactions (out-transactions) is the number of borrowing (lending) transactions per bank. The total number of transactions is the sum of the two. We find that the core banks are significantly larger and more active than periphery banks (unreported). However, the size measure appears to be a less reliable indicator than the simple number of transactions, since it is far more volatile. Both measures, however, confirm again the dominant aspect of the out-direction (lending activity) for the core membership of a bank.

We also constructed the same figures for the continuous model, see the online appendix to this article. As expected, the two coreness vectors can be better explained based on the directed version of the network. Most importantly, the correlation with the total degree is smaller compared to the correlation of in-coreness with in-degree and out-coreness with out-degree, respectively. Again, the correlations with the size measures are highly volatile.

We conclude that all measures point towards the lending activity as the more relevant aspect of core banks’ participation in the market. The much lower relevance of their borrowing activity, then, explains why in- and out-coreness vectors in the asymmetric model are virtually uncorrelated.

### 5.5 What Happened During the GFC?

In this section, we provide a more detailed analysis of the effects of a major shock to the interbank network, namely the collapse of Lehman Brothers in quarter 39. So far, our analysis shows that the GFC indeed had a substantial impact on the network along many dimensions, in particular in terms of the goodness-of-fit of the core–periphery models. To investigate the effects of the structural break in more detail, we split our sample into a short pre-crisis period (quarters 37 and 38) and a post-crisis period (40 and 41). Interestingly, despite the clear negative trend in the number of active banks during the complete sample period (cf. Fig. 1), the actual number during the analyzed subperiod is relatively stable with an average value of 98 banks. Thus the network sizes over this particular period are comparable, which allows to compare different network-related measures. As a first step, we will investigate network-related variables from a macro perspective. Then we take a closer look at the behavior of one particular exemplary core bank around the breakpoint.

As we have seen (cf. Fig. 6), the GFC affected the block-structures of the discrete and the tiering model: Core banks trade significantly less with each other (density in the CC-block smaller), and so do periphery banks (density in the PP-block smaller). In contrast, there is no evidence for a significant structural break in the densities of the off-diagonal blocks. Core banks also tend to lend less money to the periphery (density in the CP-block smaller), while there is no clear trend in the amount that peripheral banks lend to the core, thus the periphery tends to maintain their links to the core during and after the crisis. Given that the GFC, and the resulting tensions in money markets, can be seen as the result of a crisis of confidence, it comes as no surprise that core banks tend to reduce their risk exposure by cutting down the number of links going both to core and periphery banks.^{41} Concerning the market activity, we find that the total trading volumes (and also the total number of trades) in the CC- and the CP-blocks dropped substantially during the crisis, while it actually increased in the PP-block immediately after the GFC but then dropped substantially. In contrast, after a sharp drop of market activity in the PC-block right before the GFC, the total amount of credit flowing from the periphery to the core actually increased after the GFC.^{42} Thus it seems, that the crisis mainly affected the behavior of core banks, which began to hoard liquidity.^{43} In contrast, periphery banks tend to keep (at times even expand) the number of outgoing links with core banks, while reducing the exposure to the periphery. Overall, from the relatively stable Jaccard indices (see the online appendix to this article) it appears that no major disruption of the network pattern occurred, but that the aggregate volume of lending by core banks has declined substantially. Hence, most of the network structure remained intact, but continued its operations at a much lower level of activity. This finding speaks in factor of a positive effect of relationship lending that helped to prevent a complete collapse of the interbank market after the onset of the financial crisis (as suggested by Affinito 2012; Bräuning and Fecht 2011).

^{44}During this period, the particular bank had an average in-degree of 30, while its average out-degree was substantially higher with 64.

^{45}These mean values, however, hide the dynamic development, since there was a sharp drop in the banks’ out-degree during the GFC (the maximum level pre-GFC was 80, the minimum level at the end of the period is merely 32), while the in-degree actually increased during the crisis (the minimum level pre-GFC was 15, the maximum level at the end of the period is 45). In Fig. 15, we split up the bank’s links into outgoing links to core and periphery banks, respectively, and the same for the incoming links during the period under study. We see that the bank had reduced the number of outgoing links, both with core and periphery banks, but it had increased the number of incoming links, in line with the overall tendencies.

^{46}Interestingly, while the bank was a net-lender during most of the sample period, we see that the bank actually reversed its strategy during the GFC, since it became a net-borrower afterwards (see Fig. 16). Thus, the bank tried to attract liquidity, mainly from periphery banks, since core banks became reluctant to trade with other core banks.

Summing up, we conclude that the GFC both affected the behavior of core and periphery banks: Periphery banks seem to have increased their lending to the core, both in terms of the number of links and trading volumes. In contrast, core banks have reduced their lending, not only to other core banks, but also to the periphery. The decline in goodness-of-fit of the core–periphery structure is therefore mostly due to a loosening of the core. Core banks activated a smaller number of their previous outgoing links. Hence they started to hoard liquidity rather than distributing it in the system. Therefore, it seems that core banks tend to rely on the liquidity of periphery banks during times of distress, while in ‘normal’ times they would more freely redistribute liquidity in the complete system.

## 6 Discussion

The majority of studies on the structure of interbank networks has hitherto concentrated on the distribution of degrees. Many authors mention the finding of some form of community structure in the interbank market, suggesting a tightly connected core of money-center banks.^{47} The finding of a core–periphery structure in the Italian interbank market can be seen as a special case of community structure,^{48} where the core is a tightly connected part of the network, and the periphery is the loosely connected component.^{49} Taking into account similar recent findings for other countries it may well be that a core–periphery structure could be seen as a new ‘stylized fact’ of modern banking systems.

- (1)
In the sociological literature, models for the emergence of a CP network based on power structures have been developed (Persitz 2009). ‘Superior’ core members possess an intrinsic advantage over the ‘inferior’ periphery members, such that the core exerts power over the periphery. In order for a core–periphery structure to emerge, the advantage of the core members must be reflected in attributes affecting the linking behavior of all agents.

^{50}Then core agents would be able to translate their advantage into a positional advantage in the network.^{51}Transferring this idea to banking networks, one encounters several problems. First, it is not clear a priori which attributes might make core banks ‘superior’ to the periphery. We would also need to come up with an explanation why core banks share attributes that periphery banks do not have. Note also that this definition implies that it should be preferable for all banks to be part of the core, which is not very plausible. For example, a small bank (in terms of its balance sheet) would find it hard to intermediate between other core banks, simply because it does not command a sufficient amount of funds to do so. Therefore, this bank will always prefer being in the ‘inferior’ periphery, where it still might intermediate between other small banks. Moreover, the power-based explanation is not in line with the disassortative mixing patterns of interbank networks, as core banks should be reluctant to create links with periphery banks. - (2)
Core members have a comparative advantage in gathering (and spreading) information about other members of the network.

^{52}Thus, information costs are higher for periphery–periphery relationships compared to core–periphery relationships (in both directions). For the banking network, this would mean that periphery banks have an incentive in cutting down the number of links to other periphery banks, maintaining only a few links to core banks. Core banks on the other hand connect among themselves and to periphery banks. This explanation would not only be in line with the overall disassortative mixing patterns (including a more unassortative core), but also with the evidence in Cocco et al. (2009): small banks, with limited access to international capital markets and possibly limited investment/financing opportunities due to their more locally oriented business model, tend to rely on preferential relationships with (large) core banks. In this way, size would be one of the defining characteristics of core banks. These act as intermediaries between different parts of the periphery of the domestic banking system, resulting in indirect relationships between peripheral banks. Note that this explanation is also in line with the observed asymmetry between the densities in the CP- and the PC-blocks, since they imply that periphery banks cut down their credit risk by focusing on a few selected core banks, while they are prepared to borrow money from a larger set of core banks. - (3)
Brede and de Vries (2009) show that core–periphery structures might emerge from an evolutionary process as a compromise between network resilience (concentration makes the network more vulnerable) and network efficiency (concentration creates short average path lengths).

^{53}From a behavioral perspective, it is, however, unclear how the banking system would self-organize in a way to realize such design principles. From an economic perspective, the question would also be whether the self-organization of the interbank network into a core–periphery structure creates important externalities, such that policymakers might want to shift the balance towards higher resilience and somewhat lower efficiency.

^{54}This means, that given some overall statistics for the whole system, interbank credit is spread as evenly as possible across the system.

^{55}An idealized core–periphery structure amounts to pretty much the opposite in terms of concentration of interbank liabilities. If the data were closer to the latter type of structure, the entropy-based approach could give grossly misleading results for the expected aftereffects to shocks affecting single institutions. If, as we believe, the core–periphery structure turns out to be a stylized fact of the interbank market, stress-tests should take this particular topology into account. Unknown amounts of interbank liabilities could then be calibrated along the structural features of typical core–periphery models for available data (like those of the present paper and Craig and von Peter, forthcoming). As our results show, it might also be important to take into account asymmetries in the borrowing and lending attitudes of core banks. Even when comparing the effects of shocks between different network models with some tendency of concentration of links, important differences might exist.

In this regard, regulators should be interested in the network dynamics induced by the breakdown of individual nodes. This contagion effect is for example investigated by Caccioli et al. (2011) for different network structures. The authors analyze the extent of contagion in artificial banking systems after the random failure of individual institutions. Their main finding is that the likelihood of contagion, i.e. the breakdown of the entire system, is smaller for disassortative networks. Since in the latter, high-degree nodes tend to connect with low-degree nodes, the failure of a random node is unlikely to spread through the entire system. Conversely, the random breakdown of a high-degree node will severely affect other high-degree nodes in assortative networks. We should stress that these findings are not directly applicable to core–periphery interbank networks, which are disassortative in general, but display roughly unassortative mixing patterns within the core. As a consequence, a disassortative core–periphery framework might be more robust in ‘normal’ times, but more fragile under exceptional circumstances when key nodes are under stress or withdraw from the market. Hence, the ‘coreness’ translates to a certain extent into ‘systemic relevance’ of certain institutions.^{56}

The GFC seems to have been a major shock to the interbank network, as tests for structural breaks indicate. The observation that the fit of the core–periphery models significantly worsened with the GFC, might provide important information per se on the endogenous reaction of the system to stress which could be incorporated in stress-test scenarios. Furthermore the goodness-of-fit of the core–periphery framework might be seen as an indicator of tensions in the interbank market, so that various statistics based upon such a framework could be used as early warning signals of impending crises. Such information could supplement other stress indicators such as traded overnight money market interest rates, cf. Akram and Christophersen (2013), in monitoring the current state of the system.

## 7 Conclusions

The main findings of our paper are the following: we find a distinct core–periphery structure in the Italian interbank network for a sample period from January 1999 to December 2010. The identified core is very persistent over time, consisting of roughly 28 % of sample banks before the GFC and 23 % afterwards (discrete model). Given the substantial differences in the German and Italian interbank market data investigated by Craig and von Peter (forthcoming) and the present paper, e.g. in the underlying region and the maturity structure of the credit relationships, the finding of a structure close to a core–periphery network unlikely to be a coincidence. In line with other recent studies, we expect that other interbank markets display a similar hierarchical structure, which might be classified as a new ‘stylized fact’ of modern interbank networks and actually concretizes on a system level the role of money center banks. Going beyond the analysis of Craig and von Peter (forthcoming), we also investigate the continuous and asymmetric versions of core–periphery models and find evidence for strong asymmetries. In particular, overall coreness is mainly driven by the function of provision of liquidity to large parts of the banking system by the core members. Overall coreness is, therefore, largely identical to out-coreness, while its connection to in-coreness is very weak. Regulators should be aware of the fact that a bank which is part of the in-core but not of the out-core, may play a completely different role in the system than a bank with the reverse characteristics.

Formal tests favor the existence of a structural break in the last quarter of 2008, the time when Lehman Brothers collapsed. We investigated this time period in more detail and found that the deteriorating fit of the core–periphery structure in the post-GFC period is mainly due to the loosening of connections in the core, particularly on the lending side. Furthermore, it seems that during times of distress, core banks tend to rely on periphery banks as an important source of funding, since other core banks are reluctant to provide as much liquidity to other banks as in normal times.

Our findings provide some support for the view that the network structure is non-random due to the existence of preferential lending relationships. This is in line with the results of Cocco et al. (2009), Affinito (2012), and Braeuning (2011). Further evidence in this regard is provided by Finger and Lux (2014), who analyze the evolution of the banking network using the actor-oriented approach by Snijders (1996, 2001). The general conclusion is that preferential lending relationships at the micro-level lead to hierarchical structure at the macro-level. An open question is why the interbank network shows such a hierarchy. We argue that the comparative advantage of core banks in gathering and distributing information about their counterparties is likely to be a crucial factor. While core–periphery networks may be close in some respects to scale-free networks, this particular structure might not be entirely replicable by known data-generating mechanisms for SF networks. It would, thus, be important to develop alternative generating mechanisms for interbank networks. In this regard, it seems very promising to build up on existing work on the formation of trade networks, cf. Vriend (1995).

In the future we also plan to apply the models to other interbank data (going beyond interbank credit relationships), in order to evaluate whether the core–periphery structure is indeed a new stylized fact of banking systems. Furthermore, it would be interesting to relate the results to bank-specific variables, such as individual balance-sheet data. In any case, this approach can be seen as a contribution to identifying the systemically important banks in a quantitative way. We also believe that the methods presented here could be an important tool for regulators since they allow to reduce the complexity of large-scale network data, and to represent the salient structural features of the complicated web of dispersed activity in the interbank market in a compact way.

Systemic risk is closely related to financial contagion, see de Bandt and Hartmann (2000), and implies that an idiosyncratic shock causing the failure of one or few institutions may destabilize the entire system.

The density of a network is simply the fraction of existing links, relative to the maximum possible number of links. Ignoring the diagonal elements, the density can be calculated as \(M/(N^2 - N)\), with \(M\) being the number of observed links and \(N\) the number of active nodes (banks).

The in-degree is the number of incoming links, while the out-degree is the number of outgoing links per bank.

See Fricke et al. (2013) for an extensive discussion of the degree assortativity in scale-free networks.

We should mention another interesting approach in using network-based measures for financial regulation: Markose et al. (2010) use the eigenvector centrality in order to construct a ‘super-spreader’ tax.

The authors use comprehensive statistics from the so-called ‘Gross- und Millionenkreditstatistik’ (statistics on large loans and concentrated exposures) from the Deutsche Bundesbank. In Germany, financial institutions have to report (on a quarterly basis) their total exposure to each counterparty to whom they have extended credit of at least 1.5 million Euros or 10 % of their liable capital to the Bundesbank. These reports include outstanding claims of any maturity.

In the following, matrices will be written in bold, capital letters. Vectors and scalars will be written as lower-case letters.

Directed means that \(d_{i,j}\ne d_{j,i}\) in general. Sparse means that at any point in time the number of links is only a small fraction of the \(N(N-1)\) possible links. Valued means that interbank claims are reported in monetary values as opposed to 1 or 0 in the presence or absence of a claim, respectively.

This development is driven by the fact that the market is unsecured. The recent financial crisis made unsecured loans in general less attractive, with stronger impact for longer maturities. See below. It should be noted, that there is also a market for secured loans called e-MIDER.

The minimum quote size could impose an upward bias for participating banks. It would be interesting to check who are the quoting banks and who are the aggressors. Furthermore it would be interesting to look at quote data, as we only have access to actual trades.

Note that the density in the German interbank data analyzed by Craig and von Peter (forthcoming) is smaller for two reasons: first, the number of active banks is much larger, so it is more likely to observe missing links. Second, in our analysis we focus on overnight trades only, while Craig and von Peter (forthcoming) use aggregate credit volumes of all maturities (probably only with a small fraction of overnight trades). It seems plausible that the probability of observing a link between any two banks should be inversely related to the maturity of the loan.

The diagonal elements will be ignored in all that follows, since the network is not self-referential. Note that the arrangement of the adjacency matrix as in Eq. (1) is merely for the sake of illustration. The empirical implementation does not depend on such an arrangement.

Note that in order to have a core, \(N_c\) has to be \(\ge \) 2. Also note the difference between \(\mathcal {C}\) and \(c\): \(\mathcal {C}\) is the set of core banks and thus is a vector of dimension \(N_c\), while \(c\) is a vector of zeros and ones. Of course, both \(\mathcal {C}\) and \(c\) carry the same information.

The maximum number of possible inconsistencies in this block would be \(N_c (N_c-1)\) since the main diagonal is ignored. This upper bound is obviously never reached since otherwise there would be no core–periphery structure.

We cross-checked the results using the sequential algorithm applied in Craig and von Peter (forthcoming). Alternatives would be the algorithm of Kernighan and Lin (1970), see Boyd et al. (2006) for an application, and Branch-and-Bound Programming, see Brusco (2011).

An interesting alternative approach, based on the Kullback–Leibler distance, can be found in Muniz and Carvajal (2006) and Muniz et al. (2011).

This could be implemented by using standard algorithms for numerical optimization. Here we used a trust-region algorithm.

We also tried to fit the core–periphery models to the raw network matrices, however, the high level of skewness in the data results in a very poor fit in general. These results are hardly comparable to those presented below, see the online appendix to this article.

Correlations between individual coreness vectors of different models

Models | Correlation | |
---|---|---|

Discrete | Tiering | .9526 |

Discrete | Out-coreness | .7267 |

Discrete | In-coreness | .2567 |

Discrete | Sym. coreness | .7578 |

In-coreness | Out-coreness | \(-\).0809 |

Results from the tiering model are available upon request. We checked that the results from the tiering model are statistically not distinguishable from the results of the discrete model.

See the online appendix to this article for analytical results on the expected core size and error score in ER networks.

In actual interbank networks, the observed scaling parameters vary between 2 and 3. Here we take the value reported by De Masi et al. (2006) for the e-MID data. We emphasize, however, that these values are chosen only for the sake of defining a scale-free benchmark model. In a companion paper (Fricke and Lux 2013) we revisit the question of a power-law distribution of the degrees in this dataset and find no evidence in favor of anything similar to a scale-free distribution.

Interestingly, Craig and von Peter (forthcoming) found that the error scores for the German interbank market are significantly smaller than those obtained for synthetic SF networks. Thus, it seems remarkable that the SF error scores are typically smaller than the actual ones in the Italian case; however, when we impose the restriction that the SF cores be of the same size as the actual ones, the SF error scores will always exceed the actual ones.

In this approach, we first generated directed random ER and SF networks as explained above. Then, we randomly assigned observed transaction volumes from the actual networks (log-transformed) to the random ones. The results are essentially identical with and without replacement. Here we present the results without replacement. The last step consists of fitting the continuous CP models to the synthetic networks.

Note that the PREs of the AC model are always larger than those from the SC model, both for the actual and the random networks (even though for the random networks not always significantly). This is driven by the higher number of parameters (degrees of freedom) in the AC model.

It would be interesting to analyze the interest rates charged in the different blocks in more detail. This is, however, beyond the scope of this paper. Here we just note that the (volume-weighted) interest rates charged in the CC-block are typically quite large. Thus, it seems that core banks price in the systemic importance of other core banks. Interestingly, the interest rates in the CP-block are statistically indistinguishable from those in the CC-block, while the interest rates in the PC-blocks are significantly smaller. This indicates that core banks benefit from their role by borrowing relatively cheap from the periphery and lending at a higher rate to both core and periphery banks.

Interestingly, the number of reciprocal links, i.e. the fraction of links pointing in both directions, goes down due to the GFC. This is somewhat surprising, since we would expect that bilateral relationships become closer in crisis times.

The increase in the number of trades in the PC-block after the GFC is even more impressive, ending up above the pre-GFC level.

Interestingly, core banks lend more money than they borrow from the periphery, thus the core is a net lender to the periphery.

It would be interesting to see the quote data, rather than the transaction data. We suspect, that many quotes are simply never executed during the GFC.

Note that communities are usually defined as very dense subgraphs, with few connections between them. The periphery is thus more of an anti-community.

We also checked several standard community detecting algorithms for the Italian interbank network. The main finding is that, for the entire market, we find two separate communities consisting of foreign and Italian banks, respectively. Interestingly, it is impossible to split these communities further into smaller subcommunities.

For example, in a scientific network, the core agents are the highly productive agents being cited by many others. See Mullins et al. (1977).

Persitz (2009) provides a formal model for a power-based core–periphery network. The basic idea is that linking preferences are such that all agents prefer establishing links to ‘superior’ agents relative to ‘inferior’ agents.

For banks, the comparative advantage may stem from economies of scope and scale, but also from very frequent interactions on the market which small periphery banks usually do not have.

Note that the highest efficiency is realized in star-like configurations, while the highest resilience is related to the avoidance of short loops and degree homogeneity. See also Netotea and Pongor (2006).

See Sheldon and Maurer (1998), and Upper and Worms (2004). Mistrulli (2007) highlights some of the problems arising from this approach.

Note that this is equivalent to the benchmark against which the error reduction by the continuous core–periphery model is measured.

## Acknowledgments

The article is part of a research initiative launched by the Leibniz Community. We also acknowledge support by the European Union Seventh Framework Programme FP7/2007-2013 under CRISIS-ICT-2011 grant agreement no. 288501 and THEME-SSH-2013 grant agreement no. 612955. We are grateful to the participants of the SNA Workshop 2011 at the Vrije Universiteit Amsterdam, seminar participants at the Kiel Institute for the World Economy, Karl Finger, Sheri Markose, Uli Stolzenburg, and two anonymous referees for their helpful and detailed comments.