Network structure analysis of the Brazilian interbank market

EMEMAR-00437; No of Pages 23 Emerging Markets Review xxx (2016) xxx–xxx

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Network structure analysis of the Brazilian interbank market☆ Thiago Christiano Silva a,b,c,⁎,1, Sergio Rubens Stancato de Souza a, Benjamin Miranda Tabak c,1 a b c

Research Department, Central Bank of Brazil, Brasília, Brazil Department of Computing and Mathematics, Faculty of Philosophy, Science, and Literature at Ribeirão Preto, São Paulo, Brazil Universidade Católica de Brasília, Brasília, Brazil

a r t i c l e

i n f o

Article history: Received 1 July 2015 Received in revised form 13 November 2015 Accepted 20 December 2015 Available online xxxx Keywords: Network analysis Core-periphery Complex network Interbank market Systemic risk Financial stability

a b s t r a c t In this paper, we provide a detailed analysis of the roles financial institutions play within the Brazilian interbank market using a network-based approach. We present a novel methodology to assess how compliant networks are to being perfect core-periphery structures. The approach is flexible, allowing for the identification of multiple cores in networks. We verify that the interbank network presents a high disassortative mixing pattern, suggesting preferential attachment of highly connected financial institutions to others with few connections. We use the clustering coefficient to assess the substitutability of financial institutions. We find that large banking institutions are counterparties that are easily substitutable in normal times. We uncover that the rich-club effect is strongly present in the community comprising the large banking institutions, as they normally form near-clique structures. Since they play the role of liquidity providers in the interbank market, this interconnectedness effectively endows the network with robustness, as participants that are with liquidity issues can easily substitute counterparties that are liquidity suppliers. This substitutability will likely vanish during periods of stress, increasing systemic risk and the likelihood of cascade failures. © 2016 Elsevier B.V. All rights reserved.

☆ The views expressed in this work are those of the authors and do not necessarily reflect those of the Banco Central do Brasil nor of its members. We wish to thank Editor Jonathan A. Batten, Associate Editor Wai-Sum Chan, Eduardo J. A. Lima, Aquiles R. de Farias, João B. R. B. Barroso, and the anonymous referee for the constructive comments, which have helped in improving the paper. ⁎ Corresponding author. E-mail addresses: [email protected] (T.C. Silva), [email protected] (S.R.S. de Souza), [email protected] (B.M. Tabak). 1 Thiago C. Silva and Benjamin M. Tabak gratefully acknowledge financial support from the CNPq foundation.

http://dx.doi.org/10.1016/j.ememar.2015.12.004 1566-0141/© 2016 Elsevier B.V. All rights reserved.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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1. Introduction The subprime crisis highlighted the important role that money markets play in providing liquidity for the banking system. The lack of liquidity can hamper a financial institution's (FI's) maturity transformation performance. If it funds long-term assets with short-term liabilities, and for some reason, cannot rollover these liabilities, it may be unable to repay its creditors. If those creditors are facing the same liquidity shortage, this scenario may trigger a cascade of repayment failures. Consequently, those FIs under stress may be forced to sell less liquid assets for cash, in order to settle their repayments, suffering losses and worsening their situation. If this process spreads along the financial system, it can trigger a confidence crisis and reduce the ability of the banking system to perform its financial intermediation role, affecting the real sector. Network topology is one of the factors that can favor the propagation described above and, thus, the consequences that follow. In this work, we intend to characterize the Brazilian interbank network from 2008 to 2014 in terms of complex network measurements, providing economic meaning for these measurements whenever possible. In addition, we present a novel methodology to assess how compliant networks are to being perfect coreperiphery structures. To better motivate its introduction, we contrast our methodology to that of Craig and Von Peter (2014) in terms of the relative advantages and shortcomings. Complex networks are a research area that lies at the intersection between graph theory and statistical mechanics, which endows it with a truly multidisciplinary nature (da F. Costa et al., 2005; Silva and Zhao, 2016). As highlighted by Newman (2010), one prominent advantage of employing network-based theory is that it is able to capture topological and structural characteristics of the data relationships. In this work, we take advantage of the characteristics provided by the networked representation of the interbank market to describe it in a structural and systematic way. Using classical network analysis tools, we find that the degree measures of the interbank market participants show that the most connected FIs are large banks and that non-large banks are less connected than the others, but still participate in the market both as borrowers and lenders. In contrast, non-banks of all sizes are mostly borrowers that have, on average, zero lending relationships. We use the clustering coefficient network measurement to assess the substitutability of FIs from the viewpoint of the observed network structure. We find that large banks usually possess larger clustering coefficients than non-banks, indicating that they present a relevant non-sparse network structure in their surroundings. In other words, their neighborhood may invest in several other counterparties. Consequently, these large banks are more substitutable than large non-banks. In normal times, banks can fund themselves using roughly any one of the core banks, as they are easily substitutable. However, there is an inherent instability in this substitutability as with the emergence of a crisis or enhancement of asymmetric information. In these cases, substitutability will likely vanish and these core regions within the network will likely suffer from contagion and cascade defaults. These are two sides of the same coin. In normal times, large banks – liquidity providers – can be easily replaced as the network around them is dense. However, if a default is triggered, there is a potential for large increases in systemic risk due to the network topology. The analysis of the assortativity shows that the network present strong disassortative mixing patterns. This fact indicates a financial system in which highly connected FIs are frequently connected to others with very few connections. The joint analysis of assortativity and degree measures suggests that links between non-banks and banks are far more typical than links among non-banks. In addition, the strong disassortative trace displayed by the Brazilian interbank market suggests the presence of a network with a core-periphery structure. In such kind of network, the peripheral vertices only connect to the core, and the core is allowed to connect to the remainder of the network, acting as intermediators in the interbank borrowing and lending processes. In fact, Lux (2015) shows that the underlying network generating process for interbank markets is better approximated by a core-periphery network generation model than by other classical network formation approaches. Even though the disassortative pattern shown by the Brazilian interbank market suggests the existence of a core-periphery structure, that network measurement only takes into account the degree correlations between market participants. That is, it does not confirm the existence of a well-defined network core, in which its members are strongly interconnected. We can extract this type of information using another kind of network measure called “rich-club” coefficient. This index verifies the existence of the “rich-club” phenomenon, which refers to the tendency of FIs with several interbank operations to be tightly connected to each Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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other. In fact, we find a strong “rich-club” effect for the community of FIs with large degrees. This core community is mainly composed of large banking institutions in the network. In this way, our novel method to assess for the existence of core-periphery structures in networks combines the assortativity and the rich-club coefficient. We show that our method is more flexible and computationally efficient than that of Craig and Von Peter (2014). Unlike our approach, Craig and Von Peter (2014) use a rigid hypothesis of the existence of a single core in the network. Our method, in contrast, naturally works with multiple cores because it uses the notion of multiple “rich clubs” inside the network. In addition, Craig and Von Peter's (2014) methodology relies on a stochastic optimization process to perform the coreperiphery fit into the network. As such, to confer robustness to their results, we must run the optimization process multiple times. In this sense, our methodology is more efficient because it is deterministic and can be quickly computed. Therefore, it is more suitable for large scale networks. As large banking institutions play the role of liquidity providers in the interbank market and are members of a network core, the large connectivity between themselves and also to the remainder of the network effectively endows the interbank network with robustness, as participants that are with liquidity issues can easily substitute one liquidity provider to another in the absence of a crisis or increase in asymmetric information. This finding is in line with our conclusion that large banking institutions are easily substitutable because they present large clustering coefficient values. In stressful times, nonetheless, substitutability can be largely reduced, which in turn can lead to cascade failures. From a policy-making viewpoint, it is desirable to have indicators that could be employed to identify the proper time to take macroprudential actions. However, to the extent of our knowledge, there are no works in the literature defining critical values for network measures that could be used to trigger macroprudential policy actions. This absence can be partly explained by the inherent complexity of the issue. One aspect of this issue is that there are other contagion channels and circumstances that interfere with the direct exposures channel we study here. For instance, market and funding liquidity issues, procyclicality issues leading to spirals of constraints or losses, market confidence issues or access to credit facilities that may help FIs to overcome the situation. The simultaneity of these processes requires complex models in which the direct contagion channel is just one of the many risk factors. In this context, it seems that vulnerability-based network measures, i.e., those that relate exposures to loss-absorbing buffers, would be useful in a decisionmaking process related to macroprudential actions. At the present stage, that decision would need to consider other risk factors and would also require judgment. An example of macroprudential measures decision making under these complexities is (BCBS, 2010), who recommends the following to national authorities operating the countercyclical buffer: “Authorities are expected to apply judgment in the setting of the buffer in their jurisdiction, often using the best information available to gauge the build-up of system-wide risk.” This recommendation can be applied to some of the other macroprudential tools, and conveys that the available information about systemic risk, be it derived from network measures or not, cannot be (yet) the exclusive criterium to define actions to be taken, regarding the operation of macroprudential tools. 1.1. Literature and related works In a pioneer paper, Allen and Gale (2000) show that network topology is a key feature in understanding financial contagion. In line with them, Chinazzi et al. (2013) suggest that networks that have high interconnectedness may allow adverse shocks to dissipate quicker. Understanding financial contagion is crucial for policymaking and for assessing systemic risk. Developing stress tests to evaluate the likelihood of financial distress due to shocks and their propagation has been on the top of the research agenda since the financial crisis of 2008. Mastromatteo et al. (2012) show that understanding network topology is crucial to designing proper stress tests and to developing methods to accurately estimate risk. There is a recent discussion on how and to what extent network topology matters. Bargigli and Gallegati (2012) argue that network risk depends on the interbank structure. Langfield et al. (2014) and Lee (2013) show how network topology is one of the determinants of systemic risk. Krause and Giansante (2012) find that the spread of contagion and its impact depend on aspects of the network structure such as interconnectedness and tiering. Roukny et al. (2013) show that network topology matters when markets are illiquid. However, they present some evidence that network topology does not determine the stability of the system alone. They also show evidence that no network topology is superior to others in terms of systemic risk. Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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Overall, most papers find that there is a role for the network topology as it has a direct impact on how distress can be propagated through the network. Following these papers, Elliott et al. (2014) study cascades of failures in a network of interdependent financial organizations, analyzing how discontinuous changes in asset values, due to defaults and shutdowns, trigger further failures, and how this depends on network structure. In turn, Acemoglu et al. (2015) investigate the relationship between the financial network structure and the likelihood of systemic failures due to contagion of counterparty risk, and show that contagion within the interbank market exhibits a form of phase transition. If shocks that affect FIs are small, denser financial networks enhance financial stability. However, if the magnitude of these shocks increases beyond a certain point, denser networks then improve the shock propagation, leading to more fragile financial systems. Hence, the study of the interbank market structure and its evolution stands as an important issue in the field. Nier et al. (2007) investigate how the structure of the financial system relates to systemic risk by varying the key parameters that define the structure of the financial system, such as level of capitalization, the degree to which banks are connected, the size of interbank exposures and the degree of concentration of the system. They show that the effect of the degree of connectivity is non-monotonic, that is, initially a small increase in connectivity increases the contagion effect; but after a certain threshold value, connectivity improves the ability of a banking system to absorb shocks. The relationship between network structure and robustness is also studied by Li and He (2012); Lee (2013); Georg (2013). Battiston et al. (2012) investigate endogenous factors of increasing systemic risk through the contagion channel formed by credit exposure interconnections. A comprehensive review of this topic is performed by Upper (2011). This study is here conducted using complex network measurements to characterize the Brazilian interbank network. Cajueiro and Tabak (2008) present an early application of such techniques to study the Brazilian interbank market, where they show that the Brazilian interbank market possesses weak evidence of community structure, high heterogeneity and that this market is characterized by money centers having exposures to many banks. In contrast, Boss et al. (2004) provide an empirical analysis of the network structure of the Austrian interbank market, where they show that the interbank network presents community structure that exactly mirrors the regional and sectoral organization of the actual Austrian banking system. Moreover, they show that the Austrian interbank network has low clustering coefficient and a relatively short average shortest path length. Several other works investigate how different network measures can contribute to evaluating the systemic risk. Tabak et al. (2014) show that the directed clustering coefficient may be used as a measure of systemic risk in complex networks. Papadimitriou et al. (2013) identify core banks as those entities with the largest degrees that are present in a minimum spanning tree constructed from the correlation of the assets return variable. With this respect, the authors make a clear distinction between big and core banks in the sense that the latter are central in the network as they are shown to be crucial for network supervision.

1.2. Contributions The contributions of this work are threefold. First, we provide a novel, efficient, and flexible methodology to assess for the existence of core-periphery structures. Second, we systematically report how the Brazilian exposures network evolves with respect to several network measurements, such as: in- and out-degrees, degree distribution, clustering coefficient, assortativity, rich-club coefficient, among many others. Third, we provide economic interpretation for these network measurements, in a way that we can understand the features of a financial network using only these complex network descriptors. In contrast to graphs that cannot be easily compared with each other,2 these measurements can be easily comparable throughout time, permitting us to make assertions regarding the evolution of the financial network. Moreover, only the Brazilian payments system has been investigated in the literature (cf. Castro Miranda et al. (2014)). In contrast, in this paper, we investigate the network structure of the Brazilian interbank network.

2 The problem of verifying whether two graphs are similar is classified as a graph isomorphism problem in graph theory. Köbler et al. (1993) indicates this task as an NP-problem and, hence, intractable for large-scale graphs, such as complex networks.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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1.3. Notation In the process of analyzing the network, we extract some network measures from the graph G ¼ hV ; Ei that is constructed from the borrowing and lending relationships of the interbank market members. To build up such network, let V denote the set of vertices (FIs) and E, the set of edges (financial operations). The cardinality of V , N ¼ jV j, represents the number of vertices or FIs in the network. The matrix L represents the liabilities matrix (weighted adjacency matrix), in which the (i, j)-th entry represents the liabilities of the FI (vertex) i towards j. The set of edges E is given by the following filter over L: E¼ fLij N0 : ði; jÞ∈V 2 g. In our analysis, there is no netting between i and j.3 As such, if an arbitrary pair of FIs owe to each other, then two directed independent edges linking each other in opposed directions will emerge. An interesting property of maintaining the gross exposures in the network is that, if an FI defaults, its debtors remain liable for their debts. We also define the matrix of exposures or assets between the FIs as A =LT, where T is the transpose operator. In this paper, when we do not mention the type of network that we are using, it is assumed to be the liabilities matrix L. 1.4. Organization The paper proceeds as follows. In Section 2, we elucidate the methodology employed in our experiments. In Section 3, we provide meta-information about the exposures interbank network data set. In Section 4, we discuss the main results. Finally, in Section 5, we draw some conclusions about our findings. 2. Methodology In this section, we discuss the network measurements that we employ to extract topological information from the interbank market. In addition, we present a novel method for determining how similar financial networks are from being core-periphery structures. 2.1. Network measurements In this part, we present the network measurements we employ to extract topological information of the interbank network. In this paper, we analyze networks using classical network measurements borrowed from the complex network theory. In addition, we classify all of these network measurements with respect to the type of information they depend on, which can be: • Strictly local measures: they are related to the inherent characteristics of the vertex itself. In this way, measures qualified as strictly local do not take into account the neighboring nor global features. They are vertexlevel measures. • Quasi-local measures: they take into account the neighborhood's structural or topological characteristics to render information. Considering a reference vertex, the neighborhood may be its direct neighbors or indirect neighbors. They are vertex-level measures. • Global measures: they make use of all of the relationships contained in the network to derive information. They are network-level measures. For every network measure, we also provide subsidies to help in understanding its economic meaning in the context of interbank markets.

2.1.1. Degree (strictly local measure) The degree or valency of a vertex i∈V , indicated by ki, is related to its connectivity, or number of links, to the remainder of the network. In directed graphs, this notion can be further decomposed into the in-degree, (in) (out) (in) (out) ki , and out-degree, ki , such that the identity ki = ki + ki holds. The domain of ki correspond to the 3 Pairwise liabilities are not netted out so as to maintain consistency with the Brazilian law, because financial compensation is not always legally enforceable.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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discrete-valued interval {0, … , 2(N− 1)}. When ki = 0, we say that vertex i is a singleton vertex. Conversely, when ki is sufficiently large, we say that vertex i is a hub. ðinÞ The in-degree of the vertex i is defined as ki ¼ ∑ j∈V 1fLji N 0g, where 1{A} represents the indicator function that yields 1 if the logical expression A evaluates to true, and 0, otherwise. In the liabilities network L, the indegree represents the number of FIs in which participant i has invested (is exposed to). Hence, it can be regarded as a measure of investment diversification. With a similar reasoning, the out-degree of the vertex ðoutÞ i is defined as ki ¼ ∑ j∈V 1fLij N 0g. The out-degree symbolizes, in a network of liabilities, the number of creditor FIs participant i has (they are exposed to i). As such, it can be an indicator of funding diversification. 2.1.2. Strength (strictly local measure) The strength of a vertex i∈V , indicated by si, represents the total sum of weighted connections of i towards its neighbors. Likewise the degree, the notion of strength can be further decomposed into the in(in) (out) (in) (out) strength, si , and out-strength, si , such that the identity si = si + si holds. The domain of si corresponds to the continuous interval [0, ∞). ðinÞ The in-strength of the vertex i is defined as si ¼ ∑ j∈V Lji. In a network of liabilities, the in-strength represents the amount of money that an FI has invested in that market, providing an indicator of total exposure of that entity to a specific market segment. Thus, the in-strength can be seem as a measure of activeness from the investment perspective or lending dependency on the interbank market segment. ðoutÞ In contrast, the out-strength of the vertex i is defined as si ¼ ∑ j∈V Lij. Hence, it is a signal of activeness from the funding perspective of FIs or borrowing dependency on the interbank market segment. In a network of liabilities, the out-strength symbolizes the amount of money an FI has received from players of that market segment. Finally, it is worth noting that the strength is a generalization of the degree measure for weighted networks. When we are dealing with non-weighted networks, the strength reduces to the degree concept. 2.1.3. Link distribution (strictly local measure) The link distribution can be defined both in terms of the borrowing and lending perspectives. The link distribution may further be decomposed in degree- or strength-based distributions. The former is defined as the histogram of the vertices’ out-degrees (borrowing or funding) and in-degrees (lending or investment). The strength distribution follows the same reasoning for the degree distribution, but here we use the vertex strength indicator. In the interbank network, the link distribution is helpful for evaluating the concentration of relationships of the market represented by the network. A possible application of this analysis is to suggest the existence of money centers. Changes in the distribution itself also carry information on changes in the market players’ characteristics. 2.1.4. Weighted clustering coefficient of vertices (quasi-local measure) The clustering coefficient is a measure of the extended degree to which vertices in a graph tend to cluster together, which quantifies the number of loops of order three (transitivity). Originally designed for nonweighted networks, it has been extended and adapted to weighted directed networks, as is the case of the interbank market network. The weighted clustering coefficient of a vertex i∈V is given by (Barthélemy et al., 2005): CC i ¼

X Wij þ Wik 1 Aij Aik Ajk ; si ðki −1Þ 2 2

ð1Þ

ð j;kÞ∈V

in which CCi ∈ [0, 1]; Wij is the edge weight from i to j; Aij = 1{Wij N 0}. When CCi → 1, vertex i presents dense topological structures in the vicinities in the sense of triangular modules. In contrast, when CCi → 0, it only contains sparse structures, possibly with long linear chains of vertices. In the context of interbank markets, the clustering coefficient can be conceptualized both for the borrowing and lending perspectives. From the borrowing perspective, the edge weights in Eq. (1) are set to Wij =Lij, (in) and the strengths si are set to s(out) . From the lending perspective, we fix Wij = (LT)ij and si = si . A large CCi i means that bank i is easily substitutable because the neighborhood of bank i has other nearby options to invest in (lending perspective) or borrow from (borrowing perspective). This is because neighbors of i communicate with each other as well. Conversely, if CCi is small, few options are available for the neighbors of i, implying Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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that bank i is important in the neighborhood because its removal would drastically reduce the investment or funding alternatives in the nearby network surroundings. Consequently, CCi can be seen as a measure of the diversification of the counterparties of i. It is worth mentioning that these effects should hold in normal times, whereas during stress periods the propagation of failures is likely to increase. 2.1.5. Weighted clustering coefficient of the network (global measure) Eq. (1) quantifies the clustering coefficient of a single vertex. The idea can be extended to the entire network by averaging the weighted clustering coefficient over all of the network vertices, that is (Newman, 2003b): CC ¼

1X CC i ; N i∈V

ð2Þ

in which CC assumes the same domain of CCi, i.e., CC ∈ [0,1], because Eq. (2) is a convex combination of terms in the region [0, 1]. A financial system network with small CC has several entities with singular properties, i.e., they are not easily substitutable. Conversely, if CC is high, almost all of the participants have alternative options to invest in or borrow from and, therefore, the majority of FIs are substitutable. 2.1.6. Criticality (quasi-local measure) Criticality is a measure of the impact of an FI on its neighbors if it defaults. It is the sum of the vulnerabilities of its creditors regarding their exposures to the FI and is given by: Ci ¼

X

Vji ¼

j∈V

X Lij j∈V

Ej

ð3Þ

where Ej indicates the available resources or capital buffer of bank j∈V and Vji is the vulnerability of FI j to i. The vulnerability Vji measures the impact on j if i defaults. The local importance of an FI in terms of the criticality index is not directly related to its size; rather, it represents its creditors' vulnerabilities, measured by their exposures to capital buffer ratios. 2.1.7. Assortativity (global measure) Assortativity is a network-level measure that, in a structural sense, quantifies the tendency of vertices to link with similar vertices in a network. The assortativity coefficient r is computed as the Pearson's correlation of degrees of vertices in each connected pair. We compute the correlation between out- and in-degrees of links between debtor and creditor FIs. Positive values of r indicate that the network's pairs of vertices have vertices in the endpoints with similar degrees, while negative values indicate endpoints with different degrees (Newman, 2003a). In general r ∈ [−1, 1]. When r = 1, the network has perfect assortative mixing patterns, while, it is completely disassortative in the case r = − 1. Negative assortativity is often correlated with the existence of money centers in financial networks. Considering that iu and ku are the degrees of the two vertices at the endpoints of the u-th edge of a non-empty graph G, and that l = | E | is the number of edges of G, the network assortativity r is evaluated as follows (Newman, 2002): h ‐1 X i2 l i k − ð i þ k Þ u u u u 2 u∈E u∈E r¼ " #2 :  l‐1 X  2 l‐1 X 2 ð Þ − i þ k i þ k u u u∈E u u∈E u 2 2 l‐1

X

ð4Þ

According to Silva and Zhao (2012), understanding the assortative mixing patterns in complex networks is important for interpreting vertex functionality and for analyzing the global properties of the networks' components. Concerning the interbank market network, assortativity is important for the following reasons: • When the network indicates high disassortative mixing, i.e., r → − 1, the majority of FIs connect with other entities with dissimilar degrees. In practical terms, only a few FIs have several connections (money centers). In this network topology, the onset of a default in these money centers directly affects a large portion of the other FIs. In this configuration, neighbors that are vulnerable to these money centers may in turn default, leading to the beginning of a contagion process throughout the network. Since there are many highly Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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connected money centers, the network diameter tends to be small, favoring the spread of the contagion process. Now if an FI with small degree defaults, it will probably affect money centers in a direct manner. Since their capital buffers are high, they are likely to absorb all loses coming from that defaulted entity. As such, the chaining effect of subsequent defaults is retained by money centers. • When the network indicates high assortative mixing, i.e., r → 1, we expect that two types of clusters or communities will emerge in the interbank network: communities with large and small degrees. In each community, there will be more edges connecting each pair of FIs of the same group than edges cross-connecting different communities. We expect that the network will not present money centers, in the sense that they are a few highly connected FIs. The community with large-degree vertices tend to be an almost complete (clique) graph, because the number of large institutions is small and they almost connect to all of their similar pairs. With regard to the community with small-degree vertices, it is expected that the network structure will present long linear chains of FIs (large network diameter), because there is only a small quantity of edges per each vertex. In this way, each FI in that group will exhibit very low investment and funding diversification. If a member of the community with large-degree vertices defaults, it will affect all of the connected neighbors in that community and also a few vertices of the other community with small-degree vertices, which are more likely to be vulnerable given their lower diversification. Additionally, because the community with small-degree vertices has long linear paths of less diversified FIs, if they are vulnerable (have low capital buffers and larger exposures), there is a higher probability that the process will cascade through the neighbors in a domino-like effect, onsetting a process of contagion with long paths. In contrast, if a default occurs in a member of the community with small-degree vertices, then it is very unlikely that it will propagate to the money center cluster. However, depending on the FI's vulnerability, the same dominolike effect can occur in the community with small-degree vertices. In any case, we do not see reports in the literature stating financial networks with positive assortativity. • When the network does not indicate edge attachment preferences in a global sense, we expect the graph will have a slight mixture of assortative and disassortative trends in its various different regions. During the calculation of the assortativity, these terms are canceled out, resulting in r ≈ 0. In this configuration, we do not expect to encounter communities in the strong sense, as in the previous case. This effect can happen, for instance, when the number of money centers is roughly equal to the number of peripheral members. In practice, however, money centers are the minority.

2.1.8. Rich-club coefficient (global measure) The rich-club coefficient measures the structural property of complex networks called “rich-club” phenomenon. This property refers to the tendency of vertices with large degrees (hubs) to be tightly connected to each other, forming clique or near-clique structures. This phenomenon has been discussed in several instances in both social and computer sciences. Essentially, vertices with a large number of links – usually known as rich vertices – are much more likely to form dense interconnected subgraphs (clubs) than vertices with small degree. Considering that ENk is the number of edges among the NNk vertices that have degree higher than a given threshold k, the rich-club coefficient is expressed as Zhou and Mondragon (2004): ϕðkÞ ¼

2E N k ; N N k ðN N k −1Þ

ð5Þ

where the factor NNk(NNk − 1)/2 represents the maximum feasible number of edges that can exist among NNk vertices. If the subgraph formed by only those vertices with degree larger than k is disconnected, then we evaluate the rich-club coefficient for each of the graph components. In this situation, we may have several closed rich clubs forming in the network. We note that, while the network assortativity captures how connected similar vertices are in terms of degree connectivity, the rich-club coefficient can be viewed as a more specific notation of associativity, where we are only concerned with the connectivity of vertices beyond a certain richness metric. For example, if a network consisted of a collection of hubs and spokes, where the hubs were well connected, such a network would be considered disassortative. However, due to the strong connectedness of the hubs in the network, the network would demonstrate a strong rich-club effect. Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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For financial networks, the rich-club coefficient of a network is useful as a heuristic indicator of the robustness of a network. A large rich-club coefficient implies that money centers are well connected, so that liquidity can easily flow between market participants through the intermediation of these money centers. 2.2. Assessment of core-periphery structures in networks In this section, we provide a novel way to assess for the existence of core-periphery structures in networks. Core-periphery structures are a very common network topology in financial networks. Several empirical works show that real domestic interbank markets fit quite well in a core-periphery model. Among these researches, we can highlight that of the Netherlands (in ‘t Veld and van Lelyveld, 2014), Germany (Craig and Von Peter, 2014), Italy (Fricke and Lux, 2015), among many others. Networks with a core-periphery have two mesoscale structures: the core and the periphery. While core members are highly interconnected with each other and intermediate financial operations among peripheral members, members of the periphery are only allowed to have connections with core members and not among other peripheral members. Ideally, core members form a complete graph (clique) and peripheral members either borrow or lend to core members but not both. We now discuss how to check for the existence of core-periphery structures in networks. In our methodology, we use a combination of two global network measures, namely the assortativity and the rich-club coefficient, to verify if the network topology resembles a core-periphery structure. The procedure consists of two steps: 1. Verify if there exists a “rich club” inside the network for members with large degree. In a core-periphery network, the core members must be strongly interconnected, preferentially forming a near-clique or a clique structure. We choose a large degree because core members should intermediate several financial operations. Moreover, we must check for the absence of a “rich club” for members with small degree. It is imperative to check this condition because peripheral members cannot interconnect to each other. Hence, they should not form complex nor clique structures among themselves. 2. Verify if the network presents a disassortative mixing pattern behavior. If the previous step holds and the network is disassortative, it means that peripheral members tend to connect to core members and vice versa. This condition hence verifies for the existence of the two mesoscale structures: the core and the periphery. Note that the network cannot be perfectly disassortative, otherwise we would not have a clique structure between core members.4 For networks with small cores, we can confirm in favor of core-periphery structures when the assortativity is inside the interval [− 0.9, − 0.3]. The larger in magnitude the assortativity is, the smaller is the number of connections between two members of the periphery. Our methodology is qualitative and can be used to quickly check whether the network has a coreperiphery structure. In addition, the procedure is flexible as it encompasses the case of multiple cores in networks. For instance, consider a network with two cores that are highly interconnected. Since we independently compute the rich-club coefficient for each of the subgraphs, the rich-club coefficient for each of these cores is expected to be large. Hence, our method still works. We contrast our approach to that of Craig and Von Peter (2014). They provide a quantitative approach that relies on a sequential optimization algorithm to fit the network into a core-periphery model. However, the solution is not optimal because the global optimal can only be guaranteed in a combinatorial optimization process (NP-hard) that is roughly infeasible for networks with more than 30 vertices. Unlike our approach, the optimization process consists of a rigid hypothesis of the existence of a single core in the network. Our approach, in contrast, is flexible in this point by allowing multiple cores in the network. Essentially, Craig and Von Peter's (2014) optimization process consists of minimizing the error score of the core-periphery fit, which is defined as the number of connections

4 A perfectly disassortative network means that vertices with large degree strictly connect to other vertices with small degree, and vice versa. In this configuration, core members that have similar degrees would never connect to each other.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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between two members in the periphery. The optimization process uses gradient descent for exploring the search space and also simulated annealing to prevent local minima. In this way, the output is stochastic and also depends on the initial condition of the system. In our simulations, we find that the initial condition is sensitive and the core definition usually changes from one run to another. Hence, we need to run the procedure multiple times to get robust results. In this way, this procedure becomes infeasible for large networks. Our approach in turn does not rely on any initial conditions, is deterministic, and can be quickly computed. Thus, the procedure can be applied to large scale networks. It is deterministic because, given a network, it can only output an answer for the core members. Craig and Von Peter's (2014) methodology is stochastic because it can supply multiple results for a given network as it relies in a optimization process that randomly evolves due to the embedded simulated annealing process. Our method can be quickly computed because the assortativity and the rich-club coefficients only require a single pass over the edges of the network. In this way, the method runs in linear time with respect to the number of edges. To confer robustness to the results, Craig and Von Peter's (2014) methodology, in contrast, may need to run several times. Thus it cannot be quickly computed. Looking at the shortcomings, our procedure is qualitative and Craig and Von Peter's (2014) is quantitative. However, we show in Section 4.2 that, given the existence of a rich club in the network, then the assortativity magnitude can be a good proxy to tell us how good the core-periphery fit is in the network. In this respect, we show that the correlation of the Craig and Von Peter's (2014) error score and the assortativity remains greater than 0.9, showing that they are practically identical. Hence, the more disassortative the network becomes, the better the core-periphery fit is. 3. Data In this paper, we use a unique Brazilian database with supervisory and accounting data.5 From this database, we take quarterly information on Brazilian domestic interbank market exposures, supervisory variables and balance sheet statements. We use accounting information to evaluate the FIs' capital buffers for the period from March 2011 through December 2014, in addition to September 2008. We convert all of the monetary values in the financial operations into present values. Although exposures among FIs may be related to operations in the credit, capital and foreign exchange markets, here we solely focus on operations in the money market. The money market comprises operations with public and private securities. Both types of operations are registered and controlled by different institutions. We have information on operations with private securities provided by Cetip 6: interfinancial deposits, debentures and repos collateralized with securities issued by the borrower FI. We consider the repos from our sample as unsecured because the collateral is issued by the own borrower. These operations are unsecured and their trajectories in the money market are given in Fig. 1a. We can see the amount of interfinancial deposits is prevalent against debentures and repos in the period. The total amount invested in this market by its participants varies from US$19.33 billion to US$33.77 billion in the analyzed period (without considering September 2008), corresponding to 1.5% of the FIs' total assets and 14% of their aggregated Tier 1 Capital. In Fig. 1b we compare the total interbank market assets with the net cross-border flows. There is cross-border inflow in the period, i.e., the foreign-exchange reserves are changing each month. We note that the correlation between total interbank market assets and cross-border inflow in the period is 0.38 and that the total amount flowing in the interbank market is much higher than the payments over the cross-border balance. We use exposures among financial conglomerates and individual FIs that do not belong to a conglomerate. Intra-conglomerate exposures are not considered. They can be either banking or non-banking financial institutions. Banking FIs can be commercial banks, investment banks, savings banks and development banks. The 5

The collection and manipulation of the data were conducted exclusively by the staff of the Central Bank of Brazil. Cetip is a depositary of mainly private fixed income, state and city public securities and other securities representing National Treasury debts. As a central securities depositary, Cetip processes the issue, redemption and custody of securities, as well as, when applicable, the payment of interest and other events related to them. The institutions eligible to participate in Cetip include commercial banks, multiple banks, savings banks, investment banks, development banks, brokerage companies, securities distribution companies, goods and future contracts brokerage companies, leasing companies, institutional investors, non-financial companies (including investment funds and private pension companies) and foreign investors. 6

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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Fig. 1. Properties of the interbank network. (a) Total amount registered in the interbank market per financial instrument. (b) Interbank market assets compared to the overall cross-border payments balance. The overall cross-border balance information is extracted from the Time Series Management System, whose series code is 8266, maintained by the Central Bank of Brazil. 250

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Fig. 2. Evolution of several features extracted from the Brazilian interbank network. We discriminate the trajectories by size (large or nonlarge) and type of entity (banking or non-banking). (a) Total number of FIs that are active in the interbank market. (b) Total market share. (c) Average capital buffer. (d) Average leverage. All monetary values are expressed in US dollars.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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non-banks are credit cooperatives, credit unions and brokers/dealers. Banks and non-banks are classified by size according to the Central Bank of Brazil's Financial Stability Report published in the second semester of 2012 (see BCB, 2012), as follows7: 1) we group together the micro, small, and medium banks into the “non-large” category, and 2) the official large category is maintained as is in our simplified version. Therefore, instead of four segments representing the banks' sizes, we only employ two. Fig. 2a displays the evolution of the number of interbank participants in the analyzed period. Non-large banks are the majority in the entire period. Large non-banks and non-large non-banks are present in similar quantities in the sample. The number of large banks is the minority and remains roughly constant throughout the period. Although the proportions of banks and non-banks and size groups as defined above remain roughly the same, the total number of FIs varies. In special, the number of FIs consistently grows until March 2013, date in which it suffers a considerable drop due to a reduction in the number of large non-banks. Afterwards, we can verify again the upward trend in the number of FIs. Fig. 2b shows the share of interbank assets for each segment. We note that large banks have a higher average market share. The seven largest banks hold more than 60% of the assets, which suggest they have a role as funds providers in the market. Fig. 2c presents the mean capital buffer for the same categories of FIs. Large banks also have much higher capital buffers, which also reflect their size differences. Fig. 2d portrays the FIs' average leverage by category. For the interbank market, we define leverage as the ratio of the sum of the FI's interbank liabilities to its capital buffer. It may be seen as a measure of dependence of FIs on this market for funding purposes. In spite of being funds providers, large banks are not leveraged in this market because their liabilities are significantly lower than their capital buffers. In contrast, observe that non-large banking institutions are the most leveraged entities in this market. As such, they are active funders in the interbank network. Therefore, they are important from the systemic risk viewpoint, as they are exposed to that market. In turn, non-banking institutions present low leverage in the interbank market. This is because they operate in the interbank market mainly as investors. Thus, if they default, the interbank market will not suffer relevant losses. In any case, when an FI defaults, the network topology plays a crucial role in how losses are absorbed or propagated. For this matter, the reader is referred to the work in Souza et al. (2015), in which this propagation process is analyzed in the Brazilian interbank market. 4. Results In this section, we use the network measures previously presented to gather information on the interbank market network. In addition, we use our novel methodology to check for the existence of a core-periphery structure in the Brazilian interbank market and contrast our procedure to that of Craig and Von Peter (2014). 4.1. Network measurements Here we discuss the topological characteristics of the Brazilian interbank market using network measurements borrowed from the complex network theory. 4.1.1. Degree We first use degree measures to obtain information on the roles played by the market participants. Fig. 3a and b presents the average in-degree of large and non-large participants in the interbank market, while Fig. 3c and d displays the out-degree of large and non-large participants. Since we are evaluating these measures using the liabilities matrix L, the in-degree of FI i represents the number of investors it has in the market. In contrast, the out-degree of FI i denotes the number of counterparties that are funding i. These measures, hence, give a sense of investment and funding diversifications. According to Fig. 3, large banks are the most connected FIs, taking both creditor and debtor positions in the market with about the same number of network links. The most connected large banks are linked to approximately 25% of market participants, clearly characterizing their money center role. Banks are more active on 7 The Financial Stability Report ranks FIs according to their positions in a descending list ordered by FIs' total assets. The Report builds a cumulative distribution function (CDF) on the FIs' total assets and classifies them depending on the region that they fall in the CDF. It considers as large FIs that fall in the 0% to 75% region. Similarly, medium-sized FIs fall in the 75% to 90% category, small-sized, 90% to 99% mark, and those above are micro-sized.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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Fig. 3. Average in- and out-degree of large and non-large FIs as a function of time in the interbank market. We classify both measures as strictly local measurements. (a) Average in-degree of large FIs. (b) In-degree of non-large FIs. (c) Out-degree of large FIs. (d) Out-degree of non-large FIs.

the market than non-banks, in that their in-degrees and out-degrees are higher than those of non-banks. We also note that non-banks are usually investors in this market and do not diversify, for their in-degrees are almost always less than two in the analyzed period. In addition, their out-degree is zero, meaning that they do not receive funding in this market. Considering also that they are among the most vulnerable FIs, we can conclude that if they suffer asset losses from the interbank market and default as a consequence, the resulting losses will necessarily propagate to other markets, and not to the interbank market because no players are exposed to them. 4.1.2. Link distribution We now consider the distribution of pairwise investment values so as to verify the concentration of borrowing relationships in the financial system and also to check for the possible occurrences of significant changes of these distributions along time. To obtain this information, we use the cumulative distribution function (CDF) of the in-strength (total interbank investment) for the interbank market network in five distinct periods, where September 2008 (crisis period) is included. Fig. 4 shows this analysis, in such a manner that we can compare the investments distribution of the network from the crisis period against other periods. Note that the pairwise exposures in September 2008, on average, are smaller than those in the other four periods. In the other extreme, the network presents the largest pairwise exposures in December 2012. As time progresses from September 2008 onwards, we see that the CDF moves downwards, showing that the Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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1 0.9 0.8 0.7 September 2008 December 2011 December 2012 December 2013 December 2014

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Fig. 5. Trajectory of the network assortativity as a function of time in the interbank network. We see that the network presents strong disassortative mixing pattern throughout the entire studied period. In addition, this feature becomes even stronger during 2012.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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exposure values between FIs increase, on average. This phenomenon is maintained until December 2012, time in which the CDF starts to now move upwards, revealing that the average pairwise exposures begin again to decrease in magnitude. 4.1.3. Assortativity Fig. 5 depicts the trajectory of the network assortativity from 2008 to 2014. This chart shows that the interbank market network is disassortative because r b 0. This indicates a financial system in which highly connected FIs are frequently connected to others with very few connections. The joint analysis of assortativity and degree measures suggests that links between non-banks and banks are far more usual than links among nonbanks. This is consistent with the presence of money centers in the network. We can see two distinctive behaviors in the network assortativity in Fig. 5: 1) a tendency to become more disassortative from March 2011 to September 2012, and 2) a tendency to become more assortative from December 2012 to December 2012. We also note that in September 2008 and in the second semester of 2013 the network presents a less disassortative mixture than the other periods. This means that the money centers are slightly more interconnected to each other and less connected to non-large FIs. These, in turn, are also more interconnected to each other (similar pairing). 4.1.4. Rich-club coefficient Fig. 6 portrays the trajectory of the rich-club coefficient in the Brazilian interbank market evaluated for the degree thresholds k ∈ {5,10,20,30} from 2008 to 2014. We note that, on average, as the degree threshold gets larger, we see a more consistent “rich-club” effect in the network. For instance, when k = 5, the rich-club coefficient assumes small values because almost all of the banking institutions are taken into consideration. Because non-large entities tend to connect mostly to large entities due to the high disassortative mixing pattern in the network, the rich-club effect is low. However, as we filter institutions by raising the degree threshold k, only more connected entities are considered. We note that, when k ≥ 20, only entities with several operations in the interbank market are eligible in the evaluation of the rich-club coefficient in Eq. (5). These, in turn, are

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Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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composed of the large banking institutions, as we can see from the in- and out-degree shown in Fig. 3. Because of the large values of the rich-club coefficient, these large entities present the “rich-club” effect and therefore tend to form near-clique structures (complete graphs). Since they often play the role of liquidity providers in the interbank market, this phenomenon effectively gives robustness to the interbank network, as participants that are with liquidity issues can easily substitute one liquidity provider to another. Interestingly, we see that, during the crisis in September 2008, the “rich-club” formed by all of the institutions with more than 30 operations is a complete graph (clique). We also note that for k = 30, the rich-club coefficient has roughly the same trajectory of the network assortativity. Thus, it seems that, specially about the half of 2012, part of the assortativity decrease observed was due to the decrease in the number of connections between large FIs.

4.1.5. Weighted clustering coefficient of FIs Another point in financial stability analysis is the resilience of the financial system to the removal of an FI, which is also an overall measure of the diversification of its counterparties. To obtain this information, we analyze the network weighted clustering coefficient. We note that to compute the clustering coefficient of an FI, we can consider clusters formed from the perspectives of its borrowing or lending relationships, i.e., the computation is related to the FI's out- or in-strength, respectively.

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Fig. 7. Trajectory of the weighted clustering coefficient (CC) of FIs both in the lender and borrower perspectives as a function of time in the interbank network. We classify both measures as quasi-local measurements. (a) Borrower CC of large FIs. (b) Borrower CC of non-large FIs. (c) Lender CC of large FIs. (d) Lender CC of non-large FIs.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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Fig. 7a and b displays the average clustering coefficient (CC) of large and non-large financial institutions from the borrower perspective. According to Fig. 7a, large banks usually possess higher clustering coefficients than non-banks, revealing that they present a relevant non-sparse network structure in their surroundings, in the sense that their lenders communicate with each other. Moreover, large banks are more substitutable than large non-banks because of their large CC. That means that their neighbors can choose other counterparties to fund themselves. In contrast, according to Fig. 3, large non-banks play roughly the role of investors. Most of the time, their average out-degree is zero, meaning that they are connected to the network by inward links (investment relationships). These non-banking FIs usually invest (in-degree) in two large banks, which, in turn, have large in- and out-degrees. Therefore, they contribute to increasing the borrower CC of large banks. This only holds true during normal times. In contrast, if a shock hits the banking sector, the likelihood of a freeze in the interbank markets may lead to propagation of stress within the network, reducing to a large extent the substitutability property of large banks. Therefore, the current topology works well during normal times but carries an intrinsic instability. In Fig. 7b, which shows the CC for non-large institutions, we see the same picture: banks have higher clustering coefficients than non-banks due to the their higher average out-degrees and their tendency to relate to large banks, which in turn are the most diversified FIs. Looking at the interbank market, we can see that each of the non-large banks typically borrows from two or more large banks and also takes investments from nonbanking institutions. Fig. 7c and d displays the average clustering coefficient of large and non-large financial institutions from the lender perspective. Large non-banks have higher clustering coefficient than large banks from this perspective. The reason behind that is as follows. First, they have very few, but not zero, investments.8 Moreover, considering that they often invest in large banks and that the graph component comprising only large banks is almost a clique (complete graph), there will be some kind of relationship between the two invested large banks with high probability. Therefore, large non-banks will form much of the potential triangles between the (often two) counterparties9 and, as a result, will present a large clustering coefficient.

4.1.6. Weighted clustering coefficient of the network Fig. 8a and b presents the weighted clustering coefficient of the network from the borrowing and lending perspectives, respectively. We note a downward trend in both perspectives, meaning that members in the network are becoming harder to be substituted. Given an FI i, this happens because neighbors of i are less likely to be interconnected if the clustering coefficient of i is small. This is true because, in this scenario, on average, there are less triangular structural patterns in the network. The decrease in the network weighted clustering coefficient can also be related to the diminish of the number of active operations in the money market, as we can see from the downwards trends of the in- and out-degree of FIs in Fig. 3.

4.1.7. Criticality We now analyze the criticality (cf. Eq. (3)), which is a network measure that can be used as a quasi-local importance indicator of participants in the interbank network. For a given FI, this equation takes into account only the vulnerabilities of that FI's direct neighbors. We present mean criticalities by FI size category in Fig. 9a and b. On average, large banks have a higher number of creditors (cf. Fig. 3); thus, even if they are not especially vulnerable individually, the sum of the vulnerabilities tends to be higher. Non-large banks present lower criticality than large banks, because they have a smaller number of creditors on average. Non-banking institutions show zero average criticality as their average number of creditors in the interbank market is zero, because they do not take funds in the money market. According to this criterion, an FI may have only small vulnerable FIs as counterparties and be critical; however, the FI's impact on the entire financial system will not be important.

8 In reality, from Fig. 9 (investment diversification) and 11 (funding diversification), we see that non-banking institutions do not take funds from the money market. Instead, they act solely as investors in this market segment. 9 In this case, only two triangles are possible. If the communication between the invested counterparties is bidirectional, then the large non-bank will have maximum clustering coefficient.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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Fig. 8. Trajectory of the weighted clustering coefficient (CC) of the network as a function of time in the interbank network. We classify both measures as global measurements. (a) Borrower CC of the network. (b) Lender CC of the network.

4.2. Does the Brazilian interbank market have a core-periphery structure? In this section, we check whether the Brazilian interbank network has a core-periphery structure using two methodologies: 1) our method discussed in Section 2.2 and 2) Craig and Von Peter's (2014) sequential optimization procedure that fits the network into a core-periphery network. Recall that Craig and Von Peter's (2014) methodology assumes that the network has a single core, otherwise the optimization process fails in attempting to identify core members. To check that, we simply verify the number of subgraphs that result when applying the rich-club coefficient with a large degree. Performing this task, we confirm the existence of a single core, in such a way that Craig and Von Peter's (2014) methodology can be applied. Note that we do not need such test in our methodology, because it allows for multiple cores. Using the methodology in Section 2.2, we first verify the existence of the rich-club community for FIs with large degree. Looking at Fig. 6, it is clear that we get near-clique and sometimes clique subgraphs composed of FIs with large degree. In this way, there exists a core with members that intermediate several financial operations in the interbank network. Following the procedure, we now inspect for the absence of the rich-club effect in the network composed of FIs with small degree. Looking again at Fig. 6, we see that the rich-club coefficient turns out to be small when the degree is small. Lastly, we verify if the connections between the 7

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Fig. 9. Trajectory of the criticality as a function of time in the interbank network. We classify both measures as quasi-local measurements. (a) Criticality of large FIs. (b) Criticality of non-large FIs.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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peripheral and core members are well-behaved. For that, we check the assortativity graph portrayed in Fig. 5. We can see that the network shows strong signals of disassortativeness, revealing that FIs with small degree (periphery) prefer to connect to FIs with large degree (core). Putting these two facts together, we conclude that the Brazilian interbank market has a core-periphery structure. We now conduct the same exercise but using Craig and Von Peter's (2014) methodology that uses a greedy optimization process. The fitness function of this methodology is the error score that is defined as the fraction of connections between two members in the periphery. Fig. 10 exhibits the average value and standard deviation of the error score when we independently run the stochastic optimization model 200 times. We see that the core-periphery better adjusts to the interbank network in 2012 due to the smaller error scores. Nonetheless, the error scores stay below the 38% mark, meaning that overall members of the periphery connect to core members, though with some violations from the ideal core-periphery model (connections between two members of the periphery). We now compare the core members that we identify using both methodologies. In order to compare the data partitions into core and periphery from both methodologies, we use the Rand index, whose technical details we supply in Appendix A. Essentially, the Rand index R measures the similarity between two data partitions. It assumes values between the interval [0, 1], in which R = 0 means total disagreement between the partitions and R = 1 indicates a perfect match. In our case, there are two partitions composed of the core and peripheral members. Fig. 11 displays the Rand index of the partitions that our method and Craig and Von Peter (2014)’s procedure. Since the latter is a stochastic method, we opt to use the partition with the smallest error score to perform the comparison. Overall, we see that the Rand index assumes large values, revealing that both approaches provide similar results. In addition, we can check that the core members identified by both methodologies are identical throughout 2012. Recall that, according to Fig. 10, the Brazilian interbank network seems to adjust better to a core-periphery model than the other years. We can observe another evidence of that by looking at the assortativity curve in Fig. 5. In 2012, we see that the Brazilian interbank network shows stronger disassortative mixing pattern than in the other periods, suggesting that the boundaries of periphery and the core were more well-defined.

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Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004

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In order to try to link both methodologies, Fig. 12 shows the correlation between the assortativity and the error score of Craig and Von Peter's (2014) methodology. We get a very high correlation index of 0.9,10 suggesting that the assortativity can give us a sense of how well-defined the core and the periphery of the network are. In turn, this information gives us a hint of how large the error score in our methodology will be without going into optimization processes. Considering that Craig and Von Peter's (2014) methodology is stochastic and hence we need to run it multiple times, our approach stands as a reasonable candidate to quickly assess the core-periphery structure of a network. It is deterministic and runs in linear time. Moreover, it can naturally accommodate several cores in the network, while Craig and Von Peter's (2014) methodology fails in identifying core-periphery structures with multiple cores. In summary, given that the network presents a rich club for members with large degree, then the assortativity magnitude can tell us how good the core-periphery fit is in our model. In addition, we can see that it seems that both methodologies give identical results when the core-periphery fit is good. As we add noise or violations in the ideal core-periphery model, both approaches seem to provide different results.

5. Conclusion In this work, we have investigated the roles FIs play within the Brazilian interbank market using an approach based on complex networks. One prominent advantage of employing network-based theory is that it is able to capture topological and structural characteristics of the players' relationships from the data representation itself. Using a classical network analysis approach on the Brazilian interbank market from 2008 to 2014, we have seen that the network presents strong disassortative mixing along the entire studied period. This characteristic suggests that highly connected FIs are frequently connected to others with very few connections. This observation is consistent to the fact that interbank networks frequently have money centers, which act as hubs to non-large institutions. Using this fact together with the link distribution, we have seen that connections between non-banking and banking institutions are far more typical than links among pairs of non-banking institutions. Furthermore, we have employed the weighted clustering coefficient as an FI substitutability indicator, showing that large banks are more substitutable than large non-banks in the borrowing perspective. Employing the classical criticality measure, we have seen that large banking institutions are more critical to the remainder of the FIs. In special, we have seen that, during the crisis, the criticality of large banking institutions achieves its maximum value in the studied period. It is important to highlight that the substitutability property is likely to occur only during normal times. During stress periods in which a bank may default, substitutability should be reduced or may disappear, which can lead to cascade failures and a freeze in the interbank markets. Therefore, the network topology is inherently unstable and policy actions may be needed to provide liquidity to the market under such abnormal conditions. We have also studied the “rich-club” effect on the Brazilian interbank network. We have found a strong presence of the “rich-club” property in the community formed by the large banking institutions, as they tend to form near-clique structures (complete graphs). Since they often play the role of liquidity providers in the interbank market, this phenomenon effectively gives robustness to the interbank network, as participants that are with liquidity issues can easily substitute their counterparties that are liquidity providers in normal times. Moreover, we have seen that the rich-club coefficient assumed its maximum theoretical value in September 2008, showing that the core entities in the “rich-club” formed a complete graph. We have also proposed a novel method to assess for the existence of core-periphery structures in networks combines the assortativity and the rich-club coefficient. We have showed that our method is more flexible and efficient than that of Craig and Von Peter (2014). Unlike our approach, Craig and Von Peter (2014) use a rigid hypothesis of the existence of a single core in the network. Our method, in contrast, naturally works with multiple cores because it uses the notion of multiple “rich clubs” inside the network. In this paper, we have focused on topological measures that take into account the linkages, either weighted or not, between FIs, with only a glimpse on vulnerability measures, such as criticality. A natural development from this point is to quantify, given an FI's vulnerabilities, how vulnerable and harmful that FI is to the financial system, specially if these features have only local reach or if they also reach indirect neighbors. This is left 10

Running with other partitions of Craig and Von Peter's (2014) procedure, we also get similar correlation indices.

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for future work. Furthermore, this paper deals with one contagion channel, the direct contagion channel. The study of interactions between this channel and others will also enrich our comprehension on contagion risk in financial systems, improving the information that can be used by the decision making process related to the operation of macroprudential tools. Appendix A. Rand index The Rand index is a measure of the similarity between two data partitions (Rand (1971)). Consider a set (core) (periphery) of V elements V ={V 1 , . . . ,V V } and two partitions of V that we desire to compare: P 1 ={P1 , P1 } (core) (periphery) and P 2 ={P2 , P2 }. Define the following quantities: • • • •

a: the number of pairs of elements in V that are in the same set in P 1 and in the same set in P 2 ; b: the number of pairs of elements in V that are in distinct sets in P 1 and in different sets in P 2 ; c: the number of pairs of elements in V that are in the same set in P 1 and in different sets in P 2 ; d: the number of pairs of elements in V that are in distinct sets in P 1 and in the same set in P 2 ; Then, we compute the Rand index R as follows:

R¼

aþb aþb ¼  aþbþcþd V 2

ð6Þ

The numerator a + b registers the number of agreements between the two partitions P 1 and P 2 and c + d is the number of disagreements between P 1 and P 2 . The Rand index assumes values inside the interval [0, 1]. While R= 0 indicates that the two partitions do not agree on any pairs of items, R= 1 states that the two partitions are identical (perfectly match). References Acemoglu, D., Ozdaglar, A., Tahbaz-Salehi, A., 2015. Systemic risk and stability in financial networks. Am. Econ. Rev. 105, 564–608. Allen, F., Gale, D., 2000. Financial contagion. J. Polit. Econ. 108, 1–33. Bargigli, L., Gallegati, M., 2012. Finding communities in credit networks. Economics Discussion Paper 2012–41. Kiel Institute for the World Economy. Barthélemy, M., Barrat, A., Pastor-Satorras, R., Vespignani, A., 2005. Characterization and modeling of weighted networks. Physica A 346, 34–43. Battiston, S., Gatti, D.D., Gallegati, M., Greenwald, B., Stiglitz, J.E., 2012. Liaisons dangereuses: increasing connectivity, risk sharing, and systemic risk. J. Econ. Dyn. Control. 36, 1121–1141. BCB, 2012. Financial stability report. vol. 11 pp. 1–63. BCBS, 2010. Guidance for national authorities operating the countercyclical capital buffer. BCBS Publications n. 187 (Dec 2010). Bank for International Settlements. Boss, M., Elsinger, H., Summer, M., Thurner, S., 2004. Network topology of the interbank market. Quantitative Finance 4, 677–684. Cajueiro, D.O., Tabak, B.M., 2008. The role of banks in the Brazilian interbank market: does bank type matter? Physica A 387, 6825–6836. Castro Miranda, R.C., Stancato de Souza, S.R., Silva, T.C., Tabak, B.M., 2014. Connectivity and systemic risk in the Brazilian national payments system. J. Complex Networks 2, 585–613. Chinazzi, M., Fagiolo, G., Reyes, J.A., Schiavo, S., 2013. Post-mortem examination of the international financial network. J. Econ. Dyn. Control. 37, 1692–1713. Craig, B., Von Peter, G., 2014. Interbank tiering and money center banks. J. Financ. Intermed. 23, 322–347. da F. Costa, L., Rodrigues, F.A., Travieso, G., Boas, P.R.V., 2005. Characterization of complex networks: a survey of measurements. Adv. Phys. 56, 167–242. Elliott, M., Golub, B., Jackson, M.O., 2014. Financial networks and contagion. Am. Econ. Rev. 104, 3115–3153. Fricke, D., Lux, T., 2015. Core-periphery structure in the overnight money market: evidence from the e-MID trading platform. Comput. Econ. 45, 359–395. Georg, C.-P., 2013. The effect of the interbank network structure on contagion and common shocks. J. Bank. Financ. 37, 2216–2228. in ‘t Veld, D., van Lelyveld, I., 2014. Finding the core: network structure in interbank markets. J. Bank. Financ. 49, 27–40. Köbler, J., Schöning, U., Torán, J., 1993. The Graph Isomorphism Problem: Its Structural Complexity. Birkhauser Verlag, Basel, Switzerland, Switzerland. Krause, A., Giansante, S., 2012. Interbank lending and the spread of bank failures: a network model of systemic risk. J. Econ. Behav. Organ. 83, 583–608. Langfield, S., Liu, Z., Ota, T., 2014. Mapping the UK interbank system. J. Bank. Financ. 45, 288–303. Lee, S.H., 2013. Systemic liquidity shortages and interbank network structures. J. Financ. Stab. 9, 1–12. Li, S., He, J., 2012. The impact of bank activities on contagion risk in interbank networks. Adv. Complex Syst. 15, 1250086.

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Lux, T., 2015. Emergence of a core-periphery structure in a simple dynamic model of the interbank market. J. Econ. Dyn. Control. 52, A11–A23. Mastromatteo, I., Zarinelli, E., Marsili, M., 2012. Reconstruction of financial networks for robust estimation of systemic risk. J. Stat. Mech: Theory Exp. 2012, P03011. Newman, M.E.J., 2002. Assortative mixing in networks. Phys. Rev. Lett. 89, 208701. Newman, M.E.J., 2003a. Mixing patterns in networks. Phys. Rev. E 67, 026126. Newman, M.E.J., 2003b. The structure and function of complex networks. SIAM Rev. 45, 167–256. Newman, M.E.J., 2010. Networks: An Introduction. Oxford University Press, Inc., New York, NY, USA. Nier, E., Yang, J., Yorulmazer, T., Alentorn, A., 2007. Network models and financial stability. J. Econ. Dyn. Control. 31, 2033–2060. Papadimitriou, T., Gogas, P., Tabak, B.M., 2013. Complex networks and banking systems supervision. Physica A 392, 4429–4434. Rand, W.M., 1971. Objective criteria for the evaluation of clustering methods. J. Am. Stat. Assoc. 66, 846–850. Roukny, T., Bersini, H., Pirotte, H., Caldarelli, G., Battiston, S., 2013. Default cascades in complex networks: topology and systemic risk. Sci. Rep. 3, 2759. Silva, T.C., Zhao, L., 2012. Network-based high level data classification. IEEE Trans. Neural Netw. Learn. Syst. 23, 954–970. Silva, T.C., Zhao, L., 2016. Machine Learning in Complex Networks. Springer International Publishing. Souza, S.R., Tabak, B.M., Silva, T.C., Guerra, S.M., 2015. Insolvency and contagion in the Brazilian interbank market. Physica A 431, 140–151. Tabak, B.M., Takami, M., Rocha, J.M., Cajueiro, D.O., Souza, S.R., 2014. Directed clustering coefficient as a measure of systemic risk in complex banking networks. Physica A 394, 211–216. Upper, C., 2011. Simulation methods to assess the danger of contagion in interbank markets. J. Financ. Stab. 7, 111–125. Zhou, S., Mondragon, R.J., 2004. The rich-club phenomenon in the internet topology. IEEE Commun. Lett. 8, 180–182.

Please cite this article as: Silva, T.C., et al., Network structure analysis of the Brazilian interbank market, Emerg. Mark. Rev. (2016), http://dx.doi.org/10.1016/j.ememar.2015.12.004