## 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.

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## Notes

- 1.
- 2.
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.

- 3.
See also Allen and Gale (2000).

- 4.
- 5.
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).

- 6.
The in-degree is the number of incoming links, while the out-degree is the number of outgoing links per bank.

- 7.
See Fricke et al. (2013) for an extensive discussion of the degree assortativity in scale-free networks.

- 8.
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.

- 9.
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.

- 10.
We cannot carry out such an analysis since we do not observe bank IDs, see below.

- 11.
In the following, matrices will be written in bold, capital letters. Vectors and scalars will be written as lower-case letters.

- 12.
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.

- 13.
The vast majority of trades (roughly 95 %) is conducted in Euro.

- 14.
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.

- 15.
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.

- 16.
More details can be found on the e-MID website, see http://www.e-mid.it/.

- 17.
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.

- 18.
Everett and Borgatti (2000) include the possibility of multiple cores.

- 19.
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.

- 20.
See Doreian et al. (2005).

- 21.
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.

- 22.
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.

- 23.
Note that \(M\) is the maximum error possible in a network consisting only of a periphery.

- 24.
- 25.
- 26.
This could be implemented by using standard algorithms for numerical optimization. Here we used a trust-region algorithm.

- 27.
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.

- 28.
Estimating the continuous model with binary rather than discrete network matrices yields very similar results, see the online appendix to this article. The correlation of around 0.75 between both specifications can, thus, mainly be seen as the imprint of the rounding of continuous coreness values towards 0 and 1 by the discrete model.

- 29.
Recall that the number of banks in the German market is roughly 1800, so the network is at least 10 times larger than the Italian network. Thus it is not surprising, that the density is much higher in the Italian case. Since the e-MID sample presumably contains mainly large banks, our core might be the core of the overall banking network. The online appendix to this article contains a network illustration for one particular quarter.

- 30.
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.

- 31.
See the online appendix to this article for analytical results on the expected core size and error score in ER networks.

- 32.
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.

- 33.
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.

- 34.
Recall that the coreness values from the continuous model are standardized values.

- 35.
Obviously the fit has to be better in the AC model, since we have twice as many parameters. Interestingly, the fit is mostly more than twice as good as the fit of the SC model.

- 36.
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.

- 37.
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.

- 38.
Again the results for the tiering model are very similar and available upon request.

- 39.
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.

- 40.
See De Masi et al. (2006).

- 41.
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.

- 42.
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.

- 43.
Interestingly, core banks lend more money than they borrow from the periphery, thus the core is a net lender to the periphery.

- 44.
In fact, this bank (ID number ‘IT0278’) was in the core during the complete sample period.

- 45.
These numbers just underline the observed asymmetry between the CP- and PC-blocks.

- 46.
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.

- 47.
- 48.
Note that communities are usually defined as very dense subgraphs, with few connections between them. The periphery is thus more of an anti-community.

- 49.
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.

- 50.
For example, in a scientific network, the core agents are the highly productive agents being cited by many others. See Mullins et al. (1977).

- 51.
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.

- 52.
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.

- 53.
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).

- 54.
- 55.
Note that this is equivalent to the benchmark against which the error reduction by the continuous core–periphery model is measured.

- 56.
See also Markose et al. (2010).

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## 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.

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Fricke, D., Lux, T. Core–Periphery Structure in the Overnight Money Market: Evidence from the e-MID Trading Platform.
*Comput Econ* **45, **359–395 (2015). https://doi.org/10.1007/s10614-014-9427-x

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### Keywords

- Interbank market
- Network models
- Systemic risk
- Financial crisis

### JEL Classification

- G21
- G01
- E42