Analysis of the Global Banking Network by Random Matrix Theory
Ali Namaki, Jamshid Ardalankia, Reza Raei, Leila Hedayatifar, Ali Hosseiny, Emmanuel Haven, G.Reza Jafari
AAnalysis of the Global Banking Network by Random Matrix Theory
Ali Namaki a,b,d , Jamshid Ardalankia c,d , Reza Raei a , Leila Hedayatifar d,e , Ali Hosseiny d,g , Emmanuel Haven f , G.Reza Jafari d,g,h a Department of Finance, University of Tehran, Tehran, Iran b Iran Finance Association, Tehran, Iran c Department of Financial Management, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran d Center for Complex Networks and Social Datascience, Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran e New England Complex Systems Institute, NECSI HQ 277 Broadway, Cambridge, MA — 02139, United States f Faculty of Business Administration, Memorial University, St. John’s, Canada and IQSCS, UK g Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran h Department of Network and Data Science, Central European University, 1051 Budapest, Hungary
Abstract
Since 2008, the network analysis of financial systems is one of the most important subjects in economics. In this paper, we haveused the complexity approach and Random Matrix Theory (RMT) for analyzing the global banking network. By applying thismethod on a cross border lending network, it is shown that the network has been denser and the connectivity between peripheralnodes and the central section has risen. Also, by considering the collective behavior of the system and comparing it with the shu ffl edone, we can see that this network obtains a specific structure. By using the inverse participation ratio concept, we can see that after2000, the participation of di ff erent modes to the network has increased and tends to the market mode of the system. Although noimportant change in the total market share of trading occurs, through the passage of time, the contribution of some countries inthe network structure has increased. The technique proposed in the paper can be useful for analyzing di ff erent types of interactionnetworks between countries. Keywords:
Global Banking Network, Complex Systems, Random Matrix Theory, Financial Contagion
1. Introduction
Since the recent global financial crisis, cross-border lend-ing and financial contagions have gained importance. This im-portance stems from the propagated e ff ects [1, 2] of financialcrises on political and economic situations [3, 4]. This fact hasprompted a lot of research on the systemic dependence of theinternational banking sector [5–11].One of the most recent approaches for analyzing this situa-tion comes from the notion of complexity [5, 12]. The purposeof complexity science in finance focuses on the analysis of thestructure and the dynamics of entangled systems. Many schol-ars have applied complexity techniques for analyzing financialcontagion [6, 9, 10, 13, 14]. Their findings suggested that con-nectivity of financial institutions is the source of potential con-tagions. Random Matrix Theory is one of the useful methods for ana-lyzing the behavior of complex systems [12, 15–23]. This the-ory was developed by researchers to describe the situation ofenergy levels of quantum systems [24, 25].The universality regime of the eigenvalue statistics is the suc-cess factor of Random Matrix Theory [26–28]. Based on pre-vious studies, it is shown that when the size of the matrix isvery large, the eigenvalue distribution tends towards a specificdistribution [28].
Email addresses: [email protected] (Ali Namaki), [email protected] (Emmanuel Haven), [email protected] (G.Reza Jafari)
Random Matrix Theory has been applied to analyze the be-havior of coupling matrices [12]. This technique divides thecontents of the coupling matrix into noise and informationparts. The noise part of the coupling matrix conforms to theRandom Matrix Theory findings and the information part devi-ates from them. This concept stems from the idea of solving theproblem of non-stationary cross correlation and measurementnoise, as a result of market conditions and the finite length oftime series [26, 28].It is shown that the majority of their eigenvalues agree withthe random matrix predictions, but the largest eigenvalue hasdeviations from those estimations [22, 26, 27, 29]. In essence,this eigenvalue develops an energy gap that separates it fromthe other eigenvalues [17]. The largest eigenvalue is related toa strongly delocalized eigenvector that presents the collectiveevolution of the system, and this is called the market mode.From this perspective, the largest eigenvalue’s magnitude re-flects the coupling strength of the system [17].One of the systems which can be analyzed by the complex-ity approach, is the global banking network [30]. In this paper,by applying Random Matrix Theory as a useful technique fromcomplexity science, we want to analyze the global banking net-work.Our paper is organized as follows. In Section 2 we presentour methods and, in section 3 we apply Random Matrix Theoryon the global banking network and present our findings. Then,in section 4 we conclude.
Preprint submitted to Elsevier July 30, 2020 a r X i v : . [ q -f i n . S T ] J u l . Methods Primarily Random Matrix Theory has been presented bysome scholars in nuclear physics such as Mehta [24, 25], foranalyzing the energy levels of complex quantum systems. Sub-sequently, the mentioned method helped to address specific is-sues in other fields, such as finance [17, 26–28].Based on the perception from random matrix theory, theeigenvalues –in the real matrix– which deviate from the rangeof the eigenvalues –in the random matrix– possess relativelymore complete information from the system [23, 27, 28].In Random Matrix Theory, there is a parameter named as the
Inverse Participation Ratio IPR which is based on the theoryof Anderson’s localization [31], and it computes the number ofcomponents which significantly participate in each eigenvector.This notion shows the e ff ect of components of each eigenvector,and specifically how the largest eigenvalues deviate from thebulk region which is densely occupied by eigenvalues of therandom matrix. Based on the previous papers [17, 32], IPR canbe applied as an indicator for measuring the collective behaviorof the networks. The formula of this concept is as follows:
IPR ( k ) = (cid:80) nl = ( u kl ) ; (1)where l = , . . . , n and u kl is the l th element of k th eigenvector( lk ). To further clarify the concept, one may consider examplesbelow:– In case all elements of a certain eigenvector are equal to √ N , IPR will be equal to N . This implies that whole elementsare significantly influential on the systems’ behavior.– On the other hand, if just a single element is equal to 1 andthe others are equal to 0, IPR would be equal to 1. This im-plies that only this component is e ff ective in the correspondingeigenvector.Hence, one can perceive that IPR clarifies the number of in-fluential elements in a certain eigenvector.
3. Analysis of Global Banking Network by Random MatrixTheory
The banking industry is one of the most important sectors infinance. In this regard, one of the significant aspects of financialcontagion is the emergence and transmission of crisis through-out the banking network. In Fig. 1, the evolution of the globalbanking network in 5 snapshots (1978-Q3, 1988-Q3, 1998-Q3,2008-Q3 and 2018-Q3) has been depicted. The left panel inFig. 1 shows the dendrogram structure of communities for trad-ing weighted matrices. Also, the right column shows the evo-lution of the network topology. As depicted, the network hasbeen denser over time. Not only the contributions have risen,but also the peripheral nodes are arranged closer and connectedto the central section.In this study, we apply Random Matrix Theory for the dataof BIS bilateral locational statistics provided by the
Bank forInternational Settlements (BIS) [33] from 1978 until 2019. Thisdata includes all ‘core’ countries (the qualifier ‘core’ is used by G B R D E U B E L J P N C H E I T A N L D Z A F I R L F I N G R C K O R L U X T W N A U S C H L P H L H K G A U T D N K E S P S W E C A N B R A M E X F R A U S A GBRLUXESPKORZAFPHLMEXITAHKGGRCFINTWNCHLCANBRAAUSAUTIRLDNKSWEJPNDEUNLDBELCHEFRAUSA040008000120001600020000
Trading Volume 1978-Q3
AUS
AUT
BEL
BRA
CAN
CHLTWN
DNK
FIN
FRA DEU
GRC
HKG
IRL
ITA
JPN
LUX
MEX
NLD
PHL ZAF
KOR
ESP
SWE
CHE
GBR
USA J P N L U X C A N B R A M E X N L D C H E A U S K O R M A CC H L T W N P H L I R L F I N G R C Z A F E S P S W E A U T D N K D E U F R A B E L I T A U S A H K GG B R GBRJPNUSADEUBELFRALUXESPKORZAFPHLMEXMACITAHKGGRCTWNCHLCANBRAAUSAUTSWEIRLDNKFINNLDCHE020000400006000080000
Trading Volume 1988-Q3
AUSAUT
BEL
BRA
CAN
CHL TWN
DNK
FIN FRA DEU
GRCHKG
IRL
ITA
JPN
LUX
MAC
MEX
NLD
PHL ZAF
KOR
ESP
SWE
CHEGBR
USA U S A G B R D E U H K G B E LL U X A U T D N K G R C F I N Z A F C H L M A C T W N P H L B R A M E X A U S K O R E S P I R L S W E C A N C H E J P N I T A F R A N L D GBRJPNCHEUSAFRADEULUXIRLAUSFINESPKORZAFPHLMEXMACITAHKGGRCTWNCHLCANAUTBRADNKSWEBELNLD04000080000120000160000200000
Trading Volume 1998-Q3
AUS
AUT
BEL
BRA
CAN
CHL TWN
DNK
FIN
FRA
DEU
GRC
HKG
IRL
ITA
JPN
LUX
MAC
MEX
NLD
PHLZAF
KOR
ESP
SWECHE
GBR
USA U S A G B R J P N C H E B E L C A N GG Y J E Y A U T S W E A U S H K G F I N I M N Z A F C H L T W N M A C P H L K O R B R A M E X D N K G R C D E U L U X I T A E S P F R A I R L N L D GBRJEYIRLNLDBELCHELUXCANAUSGGYAUTSWEDNKGRCIMNTWNFINMACCHLESPZAFPHLHKGITAKORBRAMEXJPNUSAFRADEU0150000300000450000600000
Trading Volume 2008-Q3
AUS
AUT
BEL
BRA
CAN CHL
TWN
DNK
FIN
FRADEU
GRC
GGY
HKG
IRL
IMN
ITA
JPN
JEY
LUX
MAC
MEX
NLD
PHL
ZAF
KOR
ESP
SWE
CHE GBR
USA U S A G B R I T A C A N A U S H K G B E L E S P S W E D N K F I N A U T GG Y J E Y B R A M E X T W N K O R M A C G R CC H L Z A F I M N P H LL U X N L D I R L C H E J P N F R A D E U CANHKGITALUXESPBELIRLSWETWNDNKFINAUTMACKORBRAMEXCHLPHLIMNGRCZAFAUSGGYJEYNLDCHEUSAFRADEUJPNGBR0150000300000450000600000
Trading Volume 2018-Q3
AUS
AUT
BEL
BRA
CAN
CHL
TWN
DNK
FIN
FRA DEU
GRC
GGY
HKG
IRL
IMN
ITA
JPN
JEY
LUX
MAC
MEX
NLDPHL
ZAF
KOR
ESP
SWE
CHEGBR
USA
Figure 1: The evolution of global banking network is demonstrated for the 5snapshots of 1978-Q3, 1988-Q3, 1998-Q3, 2008-Q3 and 2018-Q3. Left col-umn; shows the evolution of trading matrices between countries. In order toextract the structure of their communities, we have applied the dendrogramweighted matrices. Right column; shows the evolution of network topology. - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q maxshmax - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q T r illi o n U S D Total Quarterly Trading Volume
Figure 2: up) The evolution of the largest eigenvalue, λ max , of the global bank-ing network and its shu ffl ed, λ shmax , are depicted. down) The evolution of totaltrading volume is demonstrated. many researchers such as [30], for 31 countries which regularlyreport their financial data to BIS).We create a weighted and directed financial transaction net-work corresponding to each quarter from 1978 until 2019. Eachlink corresponds to a loan given by a certain country to anotherone. Previous studies specifically shed light onto countries’ de-pendency network and showed an increase in the dependencystructure of the network of those countries during the passageof time [30]. As already discussed, Random Matrix Theory is apowerful approach for analyzing complex systems. In this pa-per we apply this concept for the analysis of the global bankingnetwork as a complex network. For this purpose, we choosethe shu ffl ing technique for the construction of a random ma-trix. The shu ffl ing method which is applied in this research israndomization of bilateral trading volume (or links) in the net-work. It means that the PDF remains unchanged and the bilat-eral trading relations will be shu ffl ed. The Shu ffl ed matrix is anindication of no information in the system.The global banking network possesses an adjacency matrix.This matrix can intrinsically be explained by the eigenvalue de-composition methods [34]. The eigenvector corresponding tothe largest eigenvalue, λ max , is the most significant and is the - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q - Q IPR > IPR
Max
Figure 3: It is depicted that overtime < IPR > has tended to IPR λ max . It impliesthat the contribution of countries has generally increased. market mode of the network [17, 26, 27, 35].In this regard, we assess the temporal behavior of the largesteigenvalue, as shown in Fig. 2.By evaluating the behavior of λ max and comparing it with the λ max of the corresponding shu ffl ed matrix in Fig. 2, one can ob-serve the information content of the market mode. As depictedin Fig. 2, the temporal behavior of the largest eigenvalue in thebanking interaction matrix, is totally di ff erent from that of thelargest eigenvalue in the shu ffl ed matrix. This phenomenon de-termines the existence of information content embedded in thelargest eigenvalue of the banking interaction matrix.When it comes to Fig. 2, the behavior of the maximum eigen-value has been ascending. This issue – as stated before— hasbilateral e ff ects.The reasoning behind this is that on the one side, it causesmore strength and stability in the network, whilst on the otherside, it yields to a more agile contagion throughout the net-work [7]. In the post-crisis era after 2008, simultaneous to adecrease in the maximum eigenvalue, the collective behavior ofthe system has reduced and accordingly, local identities havebeen more significant.Since the so-obtained eigenvalue does not describe all thedetails and properties of the collective behavior, one should in-vestigate other quantities in the network.It is observable that during the global financial crisis, a struc-tural emergence with an increase in the di ff erence between λ max and λ shmax , Fig. 2, has occurred.However, after the crisis, a significant decrease in the behav-ior of the largest eigenvalue of the banking matrix relating tothat of the shu ffl ed matrix has emerged.Based on the above concepts, one of the best approaches foranalyzing the global banking network is the Random MatrixTheory technique.As already discussed, one should keep in mind that IPR pos-sesses the ability of information extraction from the collectivebehaviors of the systems.In Fig. 3, By comparing < IPR > and IPR λ max , one is ableto distinguish the temporal evolution of participation in the net-3 igure 4: shows the % participation of each country in the eigenvector – corresponding to the largest eigenvalue – versus %( volume j Σ Ni Volume i .) work.In Fig. 3, we investigate the inverse participation ratio (IPR) in a temporal process. In this context, by focusing on the meaninverse participation ratio , < IPR > , and also, the inverse par-ticipation ratio of the largest eigenvalue corresponding to thelargest eigenvector , we investigate banking behaviors of thecountries and their influences on the network structure and themarket trend. In Fig. 3, IPR mean implies the e ff ectiveness of thebanking system of most countries on the global network. How-ever, from the temporal behavior of IPR λ max , we observe thatover time, less participation from those countries on the largesteigenvector emerges.In Fig. 4, % Participation stands for the contribution per-centage of each country in the eigenvector corresponding to thelargest eigenvalue. %
Volume is the trading volume of a coun-try divided by the total trading volume. Hence, %
Participation shows the contribution in the structure, and, %
Volume showsthe contribution in the total trading volume. Thereby, Fig. 4visualizes the contributions in the structure versus the contribu-tion in trading volume within each year. In 2018-Q3, for theUS, while the percentage of contribution in the structure hasbeen approximately constant, the percentage of contribution intrading volume decreased.
4. Conclusion
In this paper, by applying Random Matrix Theory, the globalbanking network is investigated. For this purpose, we computethe matrix of interaction of banking sectors of BIS countries,and then by using the Random Matrix Theory approach, thebehavior of the largest eigenvalue and Inverse Participation Ra-tio of this eigenvalue, as the market mode of the system overtime, has been analyzed. The value of the largest eigenvalue in-creases during the passage of time. By observing the behaviorof trading volume, it is shown that these increases stem fromthe expansion of the network to some extent. Also, by com-paring with the shu ffl ed one, we can deduce that the systemgets a specific structure. Generally speaking, the global bank-ing network, today, is more dense and interconnected. Also, wecan see that after the year 2000 the value of the mean IPR hasdropped and converged to
IPR λ max . It means that more countrieshave become more influential on the global banking network.Furthermore, despite small changes in the share of total trad-ing volume, some countries such as the UK, have become moreimportant in the network structure.As a concluding remark, the identities of banking systemsof BIS countries stems from two parts, i.e. i) from their ownidentities individually and, ii) from their interactions in theglobal banking network. As a suggestion for further work, onecan construct the interaction matrices of the countries based onother variables such as commercial interactions and so on.4 eferences [1] G. Iori, R. N. Mantegna, L. Marotta, S. Miccich`e, J. Porter, M. Tum-minello, Networked relationships in the e-MID interbank market: A trad-ing model with memory, Journal of Economic Dynamics and Control 50(2015) 98–116. doi:10.1016/j.jedc.2014.08.016 .[2] A. G. Haldane, R. M. May, Systemic risk in banking ecosystems, Nature469 (7330) (2011) 351–355. doi:10.1038/nature09659 .[3] K. R. Carmen M. Reinhart, This Time Is Di ff erent: Eight Centuries ofFinancial Folly, 1st Edition, Princeton University Press, 2009.[4] M. G. A. Contreras, G. Fagiolo, Propagation of economic shocks in input-output networks: A cross-country analysis, Physical Review E 90 (6). doi:10.1103/physreve.90.062812 .[5] A. S. E. B. e. Cont, Rama; Moussa, Network structure and systemicrisk in banking systems, SSRN Electronic Journal doi:10.2139/ssrn.1733528 .[6] J. Etesami, A. Habibnia, N. Kiyavash, Econometric modeling of systemicrisk: going beyond pairwise comparison and allowing for nonlinearity,LSE Research Online Documents on Economics 70769, London Schoolof Economics and Political Science, LSE Library (Mar. 2017).URL https://ideas.repec.org/p/ehl/lserod/70769.html [7] S. Battiston, D. D. Gatti, M. Gallegati, B. Greenwald, J. E. Stiglitz, Li-aisons dangereuses: Increasing connectivity, risk sharing, and systemicrisk, Journal of Economic Dynamics and Control 36 (8) (2012) 1121–1141. doi:10.1016/j.jedc.2012.04.001 .[8] F. Betz, N. Hautsch, T. A. Peltonen, M. Schienle, Systemic risk spilloversin the european banking and sovereign network, Journal of Financial Sta-bility 25 (2016) 206–224. doi:10.1016/j.jfs.2015.10.006 .URL https://doi.org/10.1016/j.jfs.2015.10.006 [9] M. D’Errico, S. Battiston, T. Peltonen, M. Scheicher, How does risk flowin the credit default swap market?, ECB Working Paper 2041 (2017). doi:10.2866/086521 .[10] S. Battiston, G. Caldarelli, Systemic risk in financial networks, Journal ofFinancial Management, Markets and Institutions, Rivista semestrale online (2 / doi:10.12831/75568 .URL [11] F. Atyabi, O. Buchel, L. Hedayatifar, Driver countries in global bankingnetwork.[12] G. Jafari, A. H. Shirazi, A. Namaki, R. Raei, Coupled time series analysis:Methods and applications, Computing in Science & Engineering 13 (6)(2011) 84–89. doi:10.1109/mcse.2011.102 .[13] P. Glasserman, H. P. Young, Contagion in financial networks, Journalof Economic Literature 54 (3) (2016) 779–831. doi:10.1257/jel.20151228 .[14] M. Bardoscia, S. Battiston, F. Caccioli, G. Caldarelli, Pathways towardsinstability in financial networks, Nature Communications 8 (1). doi:10.1038/ncomms14416 .[15] M. POTTERS, J.-P. BOUCHAUD, Financial applications of random ma-trix theory: a short review, arXiv preprint arXiv:0910.1205.[16] X. F. Jiang, T. T. Chen, B. Zheng, Structure of local interactions incomplex financial dynamics, Scientific Reports 4 (1). doi:10.1038/srep05321 .[17] A. Namaki, A. Shirazi, R. Raei, G. Jafari, Network analysis of a finan-cial market based on genuine correlation and threshold method, PhysicaA: Statistical Mechanics and its Applications 390 (21-22) (2011) 3835–3841. doi:10.1016/j.physa.2011.06.033 .[18] M. MacMahon, D. Garlaschelli, Community detection for correlation ma-trices, Physical Review X 5 (2). doi:10.1103/physrevx.5.021006 .[19] L. Sandoval, I. D. P. Franca, Correlation of financial markets in timesof crisis, Physica A: Statistical Mechanics and its Applications 391 (1-2)(2012) 187–208. doi:10.1016/j.physa.2011.07.023 .[20] A. H. Shirazi, G. R. Jafari, J. Davoudi, J. Peinke, M. R. R. Tabar,M. Sahimi, Mapping stochastic processes onto complex networks, Jour-nal of Statistical Mechanics: Theory and Experiment 2009 (07) (2009)P07046. doi:10.1088/1742-5468/2009/07/p07046 .[21] J. Jurczyk, T. Rehberg, A. Eckrot, I. Morgenstern, Measuring criticaltransitions in financial markets, Scientific Reports 7 (1). doi:10.1038/s41598-017-11854-1 .[22] J. Kwapie´n, S. Dro˙zd˙z, Physical approach to complex systems, PhysicsReports 515 (3-4) (2012) 115–226. doi:10.1016/j.physrep.2012.01.007 . [23] A. Namaki, G. Jafari, R. Raei, Comparing the structure of an emergingmarket with a mature one under global perturbation, Physica A: StatisticalMechanics and its Applications 390 (17) (2011) 3020–3025. doi:10.1016/j.physa.2011.04.004 .[24] M. L. MEHTA, Random Matrices, Elsevier, 1991. doi:10.1016/c2009-0-22297-5 .[25] M. L. MEHTA, Preface to the third edition, in: Random Matrices, Else-vier, 2004, pp. xiii–xiv. doi:10.1016/s0079-8169(04)80088-6 .[26] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, T. Guhr, H. E.Stanley, Random matrix approach to cross correlations in financial data,Physical Review E 65 (6). doi:10.1103/physreve.65.066126 .[27] L. LALOUX, P. CIZEAU, M. POTTERS, J.-P. BOUCHAUD, RANDOMMATRIX THEORY AND FINANCIAL CORRELATIONS, InternationalJournal of Theoretical and Applied Finance 03 (03) (2000) 391–397. doi:10.1142/s0219024900000255 .[28] A. NAMAKI, R. RAEI, G. R. JAFARI, COMPARING TEHRAN STOCKEXCHANGE AS AN EMERGING MARKET WITH a MATUREMARKET BY RANDOM MATRIX APPROACH, International Jour-nal of Modern Physics C 22 (04) (2011) 371–383. doi:10.1142/s0129183111016300 .[29] G.-J. Wang, C. Xie, S. Chen, J.-J. Yang, M.-Y. Yang, Random matrixtheory analysis of cross-correlations in the US stock market: Evidencefrom pearson’s correlation coe ffi cient and detrended cross-correlation co-e ffi cient, Physica A: Statistical Mechanics and its Applications 392 (17)(2013) 3715–3730. doi:10.1016/j.physa.2013.04.027 .[30] J. A. Reyes, C. Minoiu, and, A network analysis of global banking:1978-2009, IMF Working Papers 11 (74) (2011) 1. doi:10.5089/9781455227051.001 .[31] G. Lim, S. Kim, J. Kim, P. Kim, Y. Kang, S. Park, I. Park, S.-B. Park,K. Kim, Structure of a financial cross-correlation matrix under attack,Physica A: Statistical Mechanics and its Applications 388 (18) (2009)3851–3858. doi:10.1016/j.physa.2009.05.018 .[32] M. Saeedian, T. Jamali, M. Kamali, H. Bayani, T. Yasseri, G. Jafari,Emergence of world-stock-market network, Physica A: Statistical Me-chanics and its Applications 526 (2019) 120792. doi:10.1016/j.physa.2019.04.028 .[33] Bank for international settlements (bis).URL https://stats.bis.org/ [34] P. Barucca, M. Kieburg, A. Ossipov, Eigenvalue and eigenvector statisticsin time series analysis, arXiv preprint arXiv:1904.05079.[35] A. Utsugi, K. Ino, M. Oshikawa, Random matrix theory analysis of crosscorrelations in financial markets, Physical Review E 70 (2). doi:10.1103/physreve.70.026110 ..