Reconstruction of Interbank Network using Ridge Entropy Maximization Model
NNoname manuscript No. (will be inserted by the editor)
Reconstruction of Interbank Network using RidgeEntropy Maximization Model
Yuichi Ikeda · Hidetoshi Takeda
Received: date / Accepted: date
Abstract
We develop a network reconstruction model based on the entropymaximization considering the sparsity of network. Here the reconstruction is toestimate network’s adjacency matrix from node’s local information. We recon-struct the interbank network in Japan from financial data in balance sheets ofindividual banks using the developed reconstruction model in the period from2000 to 2016. The sparsity of the interbank network is successfully reproducedin the reconstructed network. We examine the accuracy of the reconstructedinterbank network by comparing the actual data and analyze the characteris-tics of the interbank network. The comparison confirms that the accuracy ofthe reconstruction model is acceptably good. For the reconstructed interbanknetwork, we obtain the following characteristics which are consistent with thepreviously known stylized facts: the short path length, the small clusteringcoefficient, the disassortative property, and the core and peripheral structure.Community analysis shows that the number of communities is 2-3 in the nor-mal period, 1 in the economic crisis (2003, 2008-2013). The major nodes ineach community have been the major commercial banks. Since 2013, the majorcommercial banks have lost the average PageRank and the leading regionalbanks have obtained both the average degree and the average PageRank. Theobserved changing role of banks is considered as a result of the quantitativeand qualitative easing monetary policy started by Bank of Japan in April of2013.
Y. IkedaGraduate School of Advanced Integrated Studies in Human Survivability, Kyoto University,Kyoto 606-8306, JAPANTel.: +81-75-762-2102E-mail: [email protected]. TakedaGraduate School of Advanced Integrated Studies in Human Survivability, Kyoto University,Kyoto 606-8306, JAPANTel.: +81-75-762-2168E-mail: [email protected] a r X i v : . [ ec on . GN ] J a n Yuichi Ikeda, Hidetoshi Takeda
Keywords
First keyword · Second keyword · More
Policymakers had traditionally focused on ensuring the financial soundness ofindividual financial institutions, especially deposit taking institutions. Suchapproach is known as the microprudential policy. The global financial crisis,however, made it clear that keeping individual financial institutions soundwas not enough. Based on the experience of the crisis, policy makers of manycountries understood that it was very important to analyze and assess risksin the entire financial system and to take adequate measures to limit systemicrisk. This is the background of the emergence of the macroprudential policy[1] [2] [3]. Since the global financial crisis, many countries are expanding theirtoolkits for more systemic approaches to financial regulation and supervisionbased on the macroprudential framework.From the understandings of the necessity of the macroprudential policy,we studied channels of distress propagation from the financial sector to thereal economy through the supply chain network in Japan from 1980 to 2015,using a spin dynamics model [4]. This study was conducted based on therelated econophysics studies [5], [6], [7], [8]. However, the interbank networkwas not considered in this paper, because data of interbank transaction isnot available to the public. In the framework of the macroprudential policy,particular attention is paid to consequences of the interconnectedness amongfinancial institutions, financial markets, and other components of the financialsystem. Also, feedback loops between real economies and financial systems arechecked carefully.The goal of this paper is to reconstruct the adjacency matrix of the inter-bank network. We develop a network reconstruction model based on entropymaximization considering sparsity of network. Here the reconstruction is toestimate network’s adjacency matrix from node’s local information. We recon-struct the interbank network in Japan from financial data in balance sheets ofindividual banks using the developed reconstruction model. Here the interbankmoney market works for short-term lending and borrowing between banks. Weexamine accuracy of the reconstructed interbank network by comparing theactual data and analyze the characteristics of the interbank network.This paper is organized as follows. In section 2, preceding studies for in-terbank market and network reconstruction are explained. In section 3, theridge entropy maximization model is proposed as a reconstruction model con-sidering the sparsity of the real-world networks. In section 4, data used in thereconstruction of the interbank network is explained. In section 5, the resultsof reconstruction are shown and the accuracy and characteristics of the re-constructed interbank network are discussed. Finally, section 6 concludes thepaper. econstruction of Interbank Network using Ridge Entropy Maximization Model 3
Current Deposit Accounts at Central Banks
Central banks provide safe andconvenient settlement assets in the form of deposits in current accounts thatfinancial institutions hold at central banks.Current account deposits at central banks serve three major roles: (1) Pay-ment instrument for transactions among financial institutions, the Bank, andthe government; (2) Cash reserves for financial institutions to pay individualsand firms; and (3) Reserves of financial institutions subject to the reserve re-quirement system. Most central banks - more than 90 % according to Gray [9]- have adopted reserve requirement systems (RRS). Under the RRS, deposit-taking institutions are required to put at least a certain portion of their de-posits received from their depositors at their central bank current accounts.The portion is determined by the central bank of the country. The minimumamount that deposit-taking institutions must have in their current account atcentral banks is the minimum reserve requirements.Gray [9] states three main purpose of the RRS: prudential, monetary con-trol, and liquidity management. First, the RRS serves as a means of prudentialby ensuring that banks hold a certain proportion of high quality, liquid assets.Second, the RRS serves as a means of monetary control. The uses of reserverequirements are normally described in terms of two channels: the money mul-tiplier, and the impact of reserve requirement on interest rate spreads. Last,the RRS serves as a means of liquidity management through averaging ofreserve balances, decreasing surplus reserve balances.The RRS was initially introduced for prudential purposes. Later, in the1930s, changes in reserve requirement ratios began to be used as a meansof monetary policy [10]. Currently, central banks of developed countries withadvanced financial markets conduct their monetary policies through operationsin their short-term financial markets and reserve requirement ratios are not
Yuichi Ikeda, Hidetoshi Takeda used as a means of monetary policy. For instance, Bank of Japan has notchanged reserve requirement ratio since October 1991.
Unconventional Monetary Policy and Reserves
In the 2000s (especially afterthe global financial crisis), central banks of developed countries have imple-mented so-called unconventional monetary policies and have greatly expandedtheir balance sheets. As a result, these central banks now have huge excessreserve balances (reserves that exceed required level). Under such a circum-stance, by paying interest on excess reserve balances as a practical lower boundon the interbank market interest rate, the central banks started making useof such reserve balances as a tool for market operations and also for keepingmarket functions [11], [12].
Fund Settlement System for Interbank Transactions
Many central banks pro-vide online fund settlement services as infrastructures for efficient and se-cure fund settlements among financial institutions. Such services are providedthrough large computer systems, such as, Bank of Japan Financial NetworkSystem (BoJ Net), Fedwire Funds (US), and TARGET2 (Euro area). Trans-actions in interbank markets are settled by fund transer between two accountsheld at the central bank. The smooth operation of these funds settlementsystems is a prerequisite for the full functioning of the interbank markets.Fund settlement services provided by central banks have common features.For instance, the fund settlements through their settlement systems are fi-nal. Also, transactions are settled based on the Real Time Gross Settlement(RTGS) method to reduce settlement risk. The introduction of the RTGSsystem comes from considerations to limit the systemic risk. In the deferrednet settlement (DNS) system that had been common in the past, a fail ofsettlement at the designated time by a participant may trigger consecutivefails in settlements. This comes from the fact that a participant often makesuse of the funds to be received from other participants for its own payments.Thus, there was a risk that a fail of a payment by a participant might stopwhole settlements among many financial institutions. This would be a form ofthe materialization of systemic risk. For this reason, many countries switchedfrom the DNS system to the RTGS system. With the RTGS method, fundsare settled immediately for each order based on instructions from the financialinstitution. Therefore, the direct impact of a default is limited to its instructedcounterparty [13]. According to a World Bank survey, as of 2016, 103 out of113 countries (91%) have adopted the RTGS fund settlement system [14].2.2 Network ReconstructionThe network reconstruction has been an actively studied topic in networkscience and a large number of works have been published during the year [15],[16], [17], [18], [19], [20], [21]. The following is a brief desciption of modelsclosely related to the model described in section 3. econstruction of Interbank Network using Ridge Entropy Maximization Model 5
MaxEnt algorithm
The MaxEnt algorithm maximizes the entropy S ( t ij ) bychanging t ij under the following constraints [22], [23]: s outi = (cid:88) j t ij , (1) s inj = (cid:88) i t ij , (2) G = (cid:88) i s outi = (cid:88) j s inj , (3)where s outi and s inj are assumed to be given. The analytical solution is easilyobtained as t MEij = s outi s inj G . (4)It is, however, noted that the solution in Eq. (4) gives a fully connected net-work, although the real-world networks are often known as sparse networks.
Iterative proportional fitting
Iterative proportional fitting (IPF) has been in-troduced to correct the dense property of t MEij at least partially. By minimizingthe Kullback-Leibler (KL) divergence between a generic non-negative t ij withnull diagonal entries and the MaxEnt solution t MEij in Eq. (4), we obtain t IP Fij [24]: min (cid:88) ij ( i (cid:54) = j ) t ij ln t ij t MEij = (cid:88) ij ( i (cid:54) = j ) t IP Fij ln t IP Fij t MEij . (5)The solution t IP Fij has null diagonal elements, but does show the sparsityequivalent to the real-world networks.
Drehmann & Tarashev approach
Starting from the MaxEnt matrix t MEij , asparse network is obtained in the following three steps [25]: First, choose arandom set of off-diagonal elements to be zero. Second, treat the remainingnon-zero elements as random variables distributing uniformly between zero andtwice their MaxEnt estimated value t DTij ∼ U (0 , t MEij ). This means that theexpected value of weights under this distribution coincides with the MaxEntmatrix t MEij . Third, run the IPF algorithm to correctly restore the value of themarginals in Eqs. (1) and (2). It is, however, noted that we need to specifythe set of off-diagonal non-zero elements. Therefore, an accurate sparsity doesnot emerge spontaneously in this approach.
Yuichi Ikeda, Hidetoshi Takeda
Convex Optimization
Configuration entropy S is written using bilateral trans-action t ij between banks i and j as follows, S = log (cid:16)(cid:80) ij t ij (cid:17) ! (cid:81) ij t ij ! ≈ (cid:88) ij t ij log (cid:88) ij t ij − (cid:88) ij t ij log t ij . (6)Here an approximation is applied to factorial ! using Stirling’s formula. Thefirst term of R.H.S. of Eq.(6) does not change the value of S by changing t ij because (cid:80) ij t ij is constant. Consequently, we have a convex objective function: S = − (cid:88) ij t ij log t ij . (7)Entropy S is to be maximized under the constraints given by Eqs. (1) - (3),where s outi and s inj correspond to the local information given for each node. Sparse Modeling
The accuracy of the reconstruction will be improved usingthe sparsity of the interbank network. The sparsity is characterized by theskewness of the observed in-degree and out-degree distributions. This meansthat a limited fraction of nodes have a large number of links and most nodeshave a small number of links and consequently the adjacency matrix of inter-national trade is sparse.To take into account the sparsity, the objective function (7) is modified byapplying the concept of Lasso (least absolute shrinkage and selection operator)or ridge regularization [26] [27] [28] to our convex optimization problem.
Lasso or Ridge Regularizations
In the case of Lasso or L1 regularization, ourproblem is formulated as the maximization of objective function z : z ( t ij ) = S − (cid:88) ij | t ij | = − (cid:88) ij t ij log t ij − β (cid:88) ij | t ij | (8)with local constraints. Here the second term of R.H.S. of Eq. (8) is L1 regular-ization. β is a control parameter. However the L1 regularization term (cid:80) ij | t ij | is constant in our problem. Therefore, we need a variant of the Lasso concept,e.g. ridge or L2 regularization.By considering this fact, our problem is reformulated as the maximizationof objective function z : z ( t ij ) = S − (cid:88) ij t ij = − (cid:88) ij t ij log t ij − β (cid:88) ij t ij (9)with local constraints. Here the second term of R.H.S. of Eq. (9) is L2 regu-larization.In the Lasso regularization, many parameters (variables) become zero andonly small number of variables are finite values. On the other hand, in theridge regularization, relatively small values are obtained for many parameters(variables). econstruction of Interbank Network using Ridge Entropy Maximization Model 7 Fig. 1
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Reconstruction Model considering Sparsity
In the theory of thermodynamics,the equilibrium of a system is obtained by minimizing thermodynamic poten-tial F : F = E − T S (10)where E , T , and S are internal energy, temperature, and entropy, respectively.Eq. (10) is rewritten as a maximization problem as follows, z ≡ − T F = S − T E. (11)We note that Eq. (11) has the same structure to Eq. (9). Thus we interpretthe meaning of control parameter β and L2 regularization term as inversetemperature 1 /T and internal energy E , respectively.In summary, we obtain the ridge entropy maximization model [29]:maximize z ( p ij ) = − (cid:80) ij p ij log p ij − β (cid:80) ij p ij subject to G = (cid:80) ij t ijs outi G = (cid:80) j t ij G = (cid:80) j p ijs inj G = (cid:80) i t ij G = (cid:80) i p ij t ij ≥ Lending and Borrowing in Interbank Market
Imakubo and Soemjima studiedthe transaction in Japanese interbank market. Table 4 in paper [30] shows that:(1) Lending from the major banks to to the major banks, the trust banks, the
Yuichi Ikeda, Hidetoshi Takeda regional banks, and the 2nd tier regional banks are 7 . × , 2 . × ,2 . × , 0 . × in JPY, respectively. (2) Lending from the trust banks tothe major banks, the trust banks, the regional banks, and the 2nd tier regionalbanks are 4 . × , 0 . × , 0 . × , and 0 . × in JPY, respectively.(3) Lending from the regional banks to the major banks, the trust banks, theregional banks, and the 2nd tier regional banks are 7 . × , 1 . × ,1 . × , and 0 . × in JPY, respectively. (4) Lending from the 2nd-tierregional banks to the major banks, the trust banks, the regional banks, andthe 2nd tier regional banks are 3 . × , 0 . × , 0 . × , and 0 . × in JPY, respectively. This study shows almost no transaction among the trustbanks, the leading regional banks, and the second-tier regional banks at theend of December, 2005. We consider this sparsity to reconstruct the interbanknetwork in Japan. Comparison of Interbank Market and Balance Sheet
Table 4 in paper [30]shows that aggregated call loan (lending) from the major banks, the trustbanks, the regional banks, the 2nd-tier regional banks are 11 . × , 5 . × ,9 . × , and 3 . × in JPY, respectively. Aggregated call loan of the majorbanks, the trust banks, the regional banks, and the 2nd-tier regional banksrecorded in balance sheets in 2005 are 15 . × , 5 . × , 27 . × , and3 . × in JPY, respectively. The aggregated call loan (lending) in balancesheet is consistent with the interbank market data, except for the transactionof the regional banks.Table 4 in paper [30] also shows that aggregated call money (borrowing)of the major banks, the trust banks, the regional banks, the 2nd-tier regionalbanks are 21 . × , 4 . × , 4 . × , and 0 . × in JPY, respec-tively. Aggregated call money of the major banks, the trust banks, the regionalbanks, and the 2nd-tier regional banks recorded in balance sheets in 2005 are94 . × , 6 . × , 14 . × , and 0 . × in JPY, respectively. Theaggregated call money (borrowing) in balance sheet is consistent with the in-terbank market data, except for the transaction of the major banks. Call Loan and Call Money in Balance Sheet
Call loan (lending) and call money(borrowing) of 98 banks are recorded in balance sheets. The banks are cat-egorized as the major commercial bank, the leading regional bank, the trustbank, and the second-tier regional bank. Temporal change of call loan andcall money for each bank are shown in Fig. 2. The leading regional banks havelarge amount of call loan and the major commercial banks and the trust bankshave large amount of call money. This implies that the leading regional bankslend money to the major commercial banks and the trust banks. It is howevernoted that the amount of both call loan and call money has decreased since theearly 2000s. This coincides with the recent increase of purchasing governmentbond.Distribution of log-transformed call loan in 2000 and 2016 are shown in Fig.3. The distributions are regarded as an unimodal distribution approximately.Distribution of log-transformed call money in 2000 and 2016 are shown in econstruction of Interbank Network using Ridge Entropy Maximization Model 9
Fig. 2
Temporal change of call loan and call money for each bank log(Call Loan) F r equen cy log(Call Loan) F r equen cy Fig. 3
Distribution of log-transformed call loan in 2000 and 2016 log(Call Money) F r equen cy log(Call Money) F r equen cy Fig. 4
Distribution of log-transformed call money in 2000 and 2016
Fig.4. The distributions are considered as an unimodal distribution approx-imately. The amount of aggregate call loan is not equal to the amount ofaggregate call money. Therefore, node “other” is needed as a slack variable inthe ridge entropy maximization model formulated in Eqs. (12).
Macroeconomic Variables
By considering capital-adequacy rule, monetary pol-icy, and banks asset, we focus on the following monthly macro-economic vari-ables from Oct 1985 to Dec 2015: (1) Monetary base “mb”, (2) Short-termgovernment bond (1 year) “nbond1yr”, (3) Long-term government bond (10 T i m e mb T i m e nbond_1year T i m e nbond_10year T i m e callrate
80 100 140 T i m e exchgrate T i m e m2cd T i m e banklend T i m e banketc T i m e govdebt
100 200 300 T i m e landprice
100 150 200 250 T i m e houseprice
400 800 1400 T i m e bankruptcy Fig. 5
Macroeconomic Variables year) “nbond10yr”, (4) Call rate “callrate”, (5) Exchange rate “exchgrate”,(6) Money stock “m2cd”, (7) Bank lending “banklend”, (8) Other bank as-sets “banketc”, (9) Government debt “govdebt”, (10) Residential land price“landprice”, (11) Residential house price “houseprice”, and (12) Number ofbankruptcy “bnkrpt”. Temporal change of these macroeconomic variables areshown in Fig. 5. Correlation coefficients were estimated among these macro-scopic variables. High correlation was obtained for the following combinations:“nbond1yr” and “callrate” (0.991), “houseprice” and “landprice” (0.946), “gov-debt” and “m2cd” (0.962), “banketc” and “m2cd” (0.945), “nbond10yr” and“callrate” (0.938), where the numbers in parenthesis are the correlation coef-ficients.The decrease of the aggregated call loan and call money since the early2000s was observed as described above. Fig. 5 shows that government debt“govdebt” increased significantly since the early 2000s. This implies that theleading regional banks cease from lending money to the major commercialbanks and the trust banks and instead start up purchasing government bond. econstruction of Interbank Network using Ridge Entropy Maximization Model 11
Actual between categories R e c o n s t r u c t e d b e t w ee n c a t e g o r i e s Fig. 6
Reconstructed Interbank Network. The left panel shows the distribution of weight t ij for the obtained interbank network. The right panel shows the comparison with thetransaction between 4 categories of banks for the reconstructed interbank network and theactual values The interbank network in Japan was reconstructed using the ridge entropymaximization model in Eqs. (12). The number of banks in each category are5, 59, 3, and 31 are the major commercial bank, the leading regional bank,the trust bank, and the second-tier regional bank, respectively. Call loan s outi of bank i and call money s inj of bank j are taken from balance sheet of eachbank and are given as constraints of the model. In addition to the banks, aslack variable is incorporated in the model to balance of the aggregated callloan and the aggregated call money. In the objective function in Eqs. (12),we assumed β = 15. The optimization was done using package software forconvex programming problem CVXR in R [31]. Reconstruction of Interbank Network
The distribution of transaction t ij forthe reconstructed interbank network in 2005 is shown in the left panel of Fig.6. The leftmost peak in the distribution is regarded as zero and therefore thiscorresponds to spurious links.Transaction t ij for the reconstructed interbank network were used to calcu-late the transaction between 4 categories of banks and compared to the actualvalues taken from Table 4 in paper [30]. The comparison with the transactionbetween 4 categories of banks for the reconstructed interbank network and theactual values is shown in the right panel of Fig. 6. The comparison confirmedthat the accuracy of the reconstruction model is acceptably good. Binary Interbank Network by truncating Links
Binary interbank network a ij was obtained by truncated links whose weights t ij are smaller than 1 /
100 ofthe maximum transaction: a ij = (cid:40) , t ij > max t ij , t ij ≤ max t ij . (13) Fig. 7
The distributions of reconstructed transaction t ij are shown for 2000 in the leftpanel and for 2016 in the right panel. The distributions of reconstructed transaction t ij are shown in Fig. 7. The leftpanel is the distribution in 2000 and the right panel in 2016. The red verticalline shows 1 /
100 of the maximum transaction and therefore the links smallerthan this threshold are ignored in the binary interbank network.
Centralities and Community Structure
Three centralities: the clustering coef-ficient, the path length, and the assortativity were calculated for the recon-structed binary network in the period from 2000 to 2016. The communitystructures were identified by maximizing the modularity for the binary net-works. The temporal change of the centralities: the clustering coefficient, thepath length, and the assortativity, and the modularity are shown in Fig. 8 be-tween 2000 and 2016. The identified community structures are shown in Fig.9 for the binary network reconstructed between 2000 and 2016.The observed average shortest path in Fig. 8 is approximately 2 which isconsistent with the previous study [32]. Fig. 8 also shows that small cluster-ing coefficient and disassortative property. The strong disassortativity impliesthat a few large banks are linked to a large number of small banks. Theseare also consistent with the previous studies [33] and [34]. Therefore, the re-constructed interbank network reproduced the stylized facts known for theinterbank networks.Figure 8 shows that the modularity is approximately 0 . Comparison with Degree-preserved Randomized Network econstruction of Interbank Network using Ridge Entropy Maximization Model 13 -1.500-1.000-0.5000.0000.5001.0001.5002.0002.500 modularity cluster eoe ff path length assorta � vity G r a p h i n d i c e s Fig. 8
The temporal change of the centralities: the clustering coefficient, the shortest pathlength, and the assortativity, and the modularity are shown for the binary network recon-structed from 2000 to 2016. N u m b e r o f N o d e s Fig. 9
The community structures were identified by maximizing the modularity the binarynetwork reconstructed from 2000 to 2016. The temporal change of the modularity is shownin Fig. 8. network during the normal period and is equivalent during the economic crisisperiod. The clustering coefficient is equivalent during the entire period. Thedisassortativity becomes weaker in the randomized network during the entireperiod. The comparison of the shortest path implies that the randomizationdoes not affect on the network during the economic crisis period. The compar-ison of the disassortativity means that the strong disassortativity emerged inthe reconstructed interbank network. The rest properties are expected by thebasic concept of the reconstruction model.
Core and Peripheral Structure
The core and peripheral structure has beenknown in the interbank network [30]. The network structure is depicted inFig. 11 for the binary interbank network in 2016. The central part shadedby red region in Fig. 11 are interlinked densely each other and forms threemajor communities. In this central part, a few large banks are linked to alarge number of banks located in the peripheral region. In addition to this, wenote that many isolated banks surrounding the central part. In the originalreconstructed network, these banks are linked weakly to the banks located in A v e r a g e S h o r t e s t P a t h original randomized -0.0500.0000.0500.1000.1500.2000.2500.3000.3500.4000.4502000 2002 2004 2006 2008 2010 2012 2014 2016 C l u s t e r i n g c o e ffi c i e n t original randomized -1.200-1.000-0.800-0.600-0.400-0.2000.0002000 2002 2004 2006 2008 2010 2012 2014 2016 A ss o r t a � v i t y original randomized Fig. 10
The clustering coefficient, the shortest path length, and the assortativity in degree-preserved randomized network
Fig. 11
Core and peripheral structure in 2016 the central part, and are disconnected by the truncation in Eq. (13). Therefore,these isolated banks should be regarded as a part of the peripheral banks.Cumulative degree distribution and Pagerank distribution in 2016 are shownin Figure 12. These distributions implies that the heterogeniety of nodes isstrong and thus major nodes could be identified using the value of degree andPageRank.
Major Nodes in Each Community
In 2004, major nodes in community 1 areMUFG bank (17 , . , . , . econstruction of Interbank Network using Ridge Entropy Maximization Model 15 d e g r ee Cumulative distribution P a g e r a n k Cumulative distribution
Fig. 12
Heterogeniety of nodes in degree distribution and PageRank distribution in 2016 where the values in the parenthesis are degree and Pagarank, respectively.Major nodes in community 2 are Aozora bank (62 , .
33) and Shinsei bank(1 , . , . , . , . , . , . , . , . , . , . , . , . Changing Role of Bank Categories
The average degree and average PageRankin each bank categories are summarized in Fig. 13. Since 2013, the majorcommercial banks have lost the average PageRank and the leading regionalbanks have obtained both the average degree and the average PageRank. The A v e r a g e D e g r ee Year
Major Regional_L Regional_S Trust A v e r a g e P a g e R a n k Year
Major Regional_L Regional_S Trust
Fig. 13
Changing role of banks observed in average degree and average PageRank observed changing role of banks is considered as a result of the quantitativeand qualitative monetary easing policy started by Bank of Japan in April of2013. The ridge entropy maximization model in Eqs. (12) captured the changeof the interbank network caused by the quantitative and qualitative monetaryeasing policy.
We developed a network reconstruction model based on entropy maximizationconsidering the sparsity of networks. Here the reconstruction is to estimatenetwork’s adjacency matrix from node’s local information. We reconstructedthe interbank network in Japan from financial data in balance sheets of in-dividual banks using the developed reconstruction model in the period from2000 to 2016. The observed sparsity of the interbank network was successfullyreproduced.We examined accuracy of the reconstructed interbank network by compar-ing the actual data and analyze the characteristics of the interbank network.The comparison with the transaction between four categories of banks for thereconstructed interbank network and the actual values confirmed that the ac-curacy of the reconstruction model is acceptably good. For the reconstructedinterbank network, we obtained the following characteristics which are consis-tent with the previously known stylized facts: the short path length [32], thesmall clustering coefficient [33], the disassortative property [34], and the coreand peripheral structure [30].Community analysis showed that the number of communities is 2-3 inthe normal period, 1 in the economic crisis (2003, 2008-2013). Major nodesin each community were identified using the value of degree and PageRank.This identified that the major nodes in each community have been the majorcommercial banks. Recently, role of the leading regional banks is increasing.Since 2013, the major commercial banks have lost the average PageRank andthe leading regional banks have obtained both the average degree and theaverage PageRank. The observed changing role of banks is considered as a econstruction of Interbank Network using Ridge Entropy Maximization Model 17 result of the quantitative and qualitative monetary easing policy started byBank of Japan in April of 2013.
Acknowledgements
This study was funded by the Ministry of Education, Science, Sports,and Culture (Grants-in-Aid for Scientific Research (B), Grant No. 17KT0034 (2017-2019).
Conflict of interest
On behalf of all authors, the corresponding author states that there is noconflict of interest.
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