Analysis of the Global Banking Network by Random Matrix Theory

Ali Namakia,b,d, Jamshid Ardalankiac,d, Reza Raeia, Leila Hedayatifard,e, Ali Hosseinyd,g, Emmanuel Havenf, G.Reza Jafarid,g,h

aDepartment of Finance, University of Tehran, Tehran, IranbIran Finance Association, Tehran, Iran

cDepartment of Financial Management, Shahid Beheshti University, G.C., Evin, Tehran, 19839, IrandCenter for Complex Networks and Social Datascience, Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran

eNew England Complex Systems Institute, NECSI HQ 277 Broadway, Cambridge, MA — 02139, United StatesfFaculty of Business Administration, Memorial University, St. John’s, Canada and IQSCS, UK

gDepartment of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, IranhDepartment 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 shuffledone, 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 different 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 different 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 effects [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: alinamaki@ut.ac.ir (Ali Namaki),ehaven@mun.ca (Emmanuel Haven), gjafari@gmail.com (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.

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2. 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 theInverse 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 effect 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) =1∑n

l=1(ukl )

4; (1)

where l = 1, . . . , n and ukl is the lth element of kth eigenvector

(lk). To further clarify the concept, one may consider examplesbelow:

– In case all elements of a certain eigenvector are equal to1√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 effective 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

GBR

DEU

BEL

JPN

CHE

ITA

NLD

ZAF

IRL

FIN

GRC

KOR

LUX

TWN

AUS

CHL

PHL

HKG

AUT

DNK

ESP

SWE

CAN

BRA

MEX FRA

USA

GBRLUXESPKORZAFPHLMEXITAHKGGRCFINTWNCHLCANBRAAUSAUTIRLDNKSWEJPNDEUNLDBELCHEFRAUSA

040008000120001600020000

Trading Volume

1978-Q3

AUS

AUT

BEL

BRA

CAN

CHL

TWN

DNK

FIN

FRA

DEU

GRC

HKG

IRL

ITA

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CHEGBR

USA

JPN

LUX

CAN

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MEX NLD

CHE

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KOR

MAC CH

LTW

NPH

LIR

LFI

NGR

CZA

FES

PSW

EAU

TDN

KDE

UFR

ABE

LIT

AUS

AHK

GGB

R

GBRJPNUSADEUBELFRALUXESPKORZAFPHLMEXMACITAHKGGRCTWNCHLCANBRAAUSAUTSWEIRLDNKFINNLDCHE

020000400006000080000

Trading Volume

1988-Q3

AUS

AUT

BEL

BRA

CAN

CHL

TWN

DNK

FIN

FRA

DEU

GRC

HKG

IRL

ITA

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LUXMAC

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USA

USA

GBR

DEU

HKG

BEL

LUX

AUT

DNK

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ZAF

CHL

MAC

TWN

PHL

BRA

MEX AU

SKO

RES

PIR

LSW

ECA

NCH

EJP

N ITA

FRA

NLD

GBRJPNCHEUSAFRADEULUXIRLAUSFINESPKORZAFPHLMEXMACITAHKGGRCTWNCHLCANAUTBRADNKSWEBELNLD

04000080000120000160000200000

Trading Volume

1998-Q3

AUS

AUT

BEL

BRA

CAN

CHL

TWN

DNK

FIN

FRA

DEU

GRC

HKG

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ITA

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SWE

CHE

GBR

USA

USA

GBR

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BEL

CAN

GGY

JEY

AUT

SWE

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HKG FIN

IMN

ZAF

CHL

TWN

MAC PH

LKO

RBR

AM

EXDN

KGR

CDE

ULU

XIT

AES

PFR

AIR

LNL

D

GBRJEYIRLNLDBELCHELUXCANAUSGGYAUTSWEDNKGRCIMNTWNFINMACCHLESPZAFPHLHKGITAKORBRAMEXJPNUSAFRADEU

0150000300000450000600000

Trading Volume

2008-Q3

AUS

AUT

BEL

BRA

CAN

CHL

TWN

DNK

FIN

FRA

DEU

GRC

GGY

HKG

IRL IMN

ITA

JPNJEY

LUX

MAC

MEX

NLD

PHL

ZAF

KOR

ESP

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CHE

GBRUSA

USA

GBR

ITA

CAN

AUS

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ESP

SWE

DNK

FIN

AUT

GGY

JEY

BRA

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MAC GR

CCH

LZA

FIM

NPH

LLU

XNL

D IRL

CHE

JPN

FRA

DEU

CANHKGITALUXESPBELIRLSWETWNDNKFINAUTMACKORBRAMEXCHLPHLIMNGRCZAFAUSGGYJEYNLDCHEUSAFRADEUJPNGBR

0150000300000450000600000

Trading Volume

2018-Q3

AUS

AUT

BEL

BRA

CAN

CHL

TWN

DNK

FIN

FRA

DEU

GRC

GGY

HKGIRL

IMN

ITA

JPN

JEYLUX

MAC

MEX

NLD

PHL

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.

2

1978

-Q3

1980

-Q3

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0

1000000

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7000000maxshmax

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Trilli

on U

SD

1e7 Total Quarterly Trading Volume

Figure 2: up) The evolution of the largest eigenvalue, λmax, of the global bank-ing network and its shuffled, λshmax, 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 shuffling technique for the construction of a random ma-trix. The shuffling 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 shuffled. The Shuffled 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, λmax, is the most significant and is the

1978

-Q3

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0.20

0.25

0.30

0.35

0.40 < IPR >IPR Max

Figure 3: It is depicted that overtime < IPR > has tended to IPRλmax . 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 largest

eigenvalue, as shown in Fig. 2.By evaluating the behavior of λmax and comparing it with the

λmax of the corresponding shuffled 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 different from that of thelargest eigenvalue in the shuffled 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 effects.

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 difference between λmaxand λshmax, 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 shuffled 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 IPRλmax , one is ableto distinguish the temporal evolution of participation in the net-

3

Figure 4: shows the % participation of each country in the eigenvector – corresponding to the largest eigenvalue – versus %(volume j

ΣNi Volumei.)

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, IPRmean implies the effectiveness of thebanking system of most countries on the global network. How-ever, from the temporal behavior of IPRλmax , 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, %Participationshows 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 shuffled 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 IPRλmax . 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

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1 Introduction2 Methods3 Analysis of Global Banking Network by Random Matrix Theory4 Conclusion