Network-centric indicators for fragility in global financial indices
Areejit Samal, Sunil Kumar, Yasharth Yadav, Anirban Chakraborti
NNetwork-centric indicators for fragility in global financial indices
Areejit Samal, Sunil Kumar, Yasharth Yadav, and Anirban Chakraborti
4, 5, 6 The Institute of Mathematical Sciences (IMSc),Homi Bhabha National Institute (HBNI), Chennai 600113 India Department of Physics, Ramjas College, University of Delhi, New Delhi 110007 India Indian Institute of Science Education and Research (IISER), Pune 411008 India School of Computational and Integrative Sciences,Jawaharlal Nehru University, New Delhi 110067, India Centre for Complexity Economics, Applied Spirituality and Public Policy (CEASP),Jindal School of Government and Public Policy, O.P. Jindal Global University, Sonipat 131001, India Centro Internacional de Ciencias, Cuernavaca 62210, M´exico
Over the last two decades, financial systems have been studied and analysed from the perspectiveof complex networks, where the nodes and edges in the network represent the various financialcomponents and the strengths of correlations between them. Here, we adopt a similar network-basedapproach to analyse the daily closing prices of 69 global financial market indices across 65 countriesover a period of 2000-2014. We study the correlations among the indices by constructing thresholdnetworks superimposed over minimum spanning trees at different time frames. We investigate theeffect of critical events in financial markets (crashes and bubbles) on the interactions among theindices by performing both static and dynamic analyses of the correlations. We compare and contrastthe structures of these networks during periods of crashes and bubbles, with respect to the normalperiods in the market. In addition, we study the temporal evolution of traditional market indicators,various global network measures and the recently developed edge-based curvature measures. Weshow that network-centric measures can be extremely useful in monitoring the fragility in the globalfinancial market indices.
1. INTRODUCTION
It is possible to describe a financial market using the framework of complex networks such that the nodes in anetwork represent the financial components and an edge between any two components indicates an interaction betweenthem. A correlation matrix constructed using the cross-correlations of fluctuations in prices can be utilized to identifysuch interactions. However, a network resulting from the correlation matrix contains densely connected structures.A growing amount of research is focused on methods devised to extract relevant correlations from the correlationmatrix and study the topological, hierarchical and clustering properties of the resulting networks. Mantegna et al.[1, 2] introduced the minimum spanning tree (MST) to extract networks from the correlation matrices computedfrom the asset returns. Dynamic asset trees, introduced by Onnela et al. [3, 4], were analysed to monitor theevolution of financial stock markets using the hierarchical clustering properties of such trees. Boginski et al. [5]constructed threshold networks by extracting the edges with correlation values exceeding a chosen threshold andanalyzed degree distribution, cliques and independent sets on the threshold network. Tumminello et al. [6] introducedplanar maximally filtered graph (PMFG) as a tool to extract important edges from the correlation matrix, whichcontains more information than the MST, while also preserving the hierarchical structure induced by MST. Triangularloops and four-element cliques in PMFG could provide considerable insights into the structure of financial markets.Network-based analysis has been widely used to study not only particular stock market structures but also thecomplex networks of correlations among different financial market indices across the globe. For example, MSThas been used on stock markets to detect underlying hierarchical organization [7–9]. Bonanno et al. [10] studiedthe correlations of 51 global financial indices and showed that the corresponding MST was clustered according tothe geographical locations of the indices. In addition, the changes in the topological structure of MST could helpunderstand the evolution of financial systems [11–13]. MST and threshold networks have been used to analyse theindices during the global financial crisis of 2008 [14–16]. It has also been shown that geography is one of the majorfactors which govern the hierarchy of the global market [17, 18]. Also, Eryˇgit and Eryˇgit [19] had investigated thetemporal evolution of clustering networks (MST and PMFG) of 143 financial indices corresponding to 59 countriesacross the world from the period 1995-2008, and once again found that the clustering in the networks of financialindices was according to their geographical locations. From the time dependent network and centrality measures theyshowed that the integration of the global financial indices has increased with time. Further, Chen et al. [20] analyzeddynamics of threshold networks of regional and global financial markets from the period 2012-2018, proposed a modelfor the measurement of systemic risk based on network topology and then concluded that network-based methodsprovide a more accurate measurement of systemic risk compared to the traditional absorption technique. Silva etal. [21] studied the average criticality of countries during different periods in the crisis and found that the USA is a r X i v : . [ q -f i n . S T ] J a n the most critical country, followed by European countries, Oceanian and Asian countries, and finally Latin Americancountries and Canada . They also found a decrease in the network fragility after the global financial crisis. It hasbeen also shown that financial crises can be captured using networks of volatility spillovers [22, 23]. Wang et al. [24]constructed and analysed dynamical structure of MSTs and hierarchical trees computed from the Pearson correlationsas well as partial correlations, among 57 global financial markets from the period 2005-2014, and concluded that MSTbased on partial correlations provided more information when compared to MST based on Pearson correlations. Themarket indices from different stock markets across the globe comprise assets that are very different – apart from stocksof the big multinational companies that are traded across markets, the stock markets would have little in common,and hence would be expected to behave independently. However, all the aforementioned studies suggest in contrary.In this brief research report, we study the evolution of correlation structures among 69 global financial indicesthrough the years 2000 to 2014. To ensure that we consider only the most relevant correlations, we constructthe network by creating an MST (which connects all the nodes) and then add extra edges from the correlationmatrix exceeding a certain threshold, which gives modular structures. Our findings corroborate the earlier results ofgeographical clustering [17, 25]. We then study the changes occurring in the market by analysing the fluctuationsin various global network measures and the recently developed edge-based geometric measures. Since there arecomplex interactions that occur among groups of three or more nodes, which cannot be described simply by pairwiseinteractions, the higher-order architecture of complex financial systems captured by the geometrical measures canhelp us in the betterment of systemic risk estimation and give us an indication of the global market efficiency. Tothe best of our knowledge, the present work is the first investigation of discrete Ricci curvatures in networks of globalmarket indices. Thus, we find that this approach along with all these network measures can be used to monitor thefragility of the global financial network and as indicators of crashes and bubbles occurring in the markets. This couldin turn relate the health of the financial markets with the development or downturn of the global economy, as well asgauge the impact of certain market crises in the multi-level financial-economic phenomena.
2. METHODS2.1. Data description
This study is based on a dataset collected from Bloomberg which comprises the daily closing prices of 69 globalfinancial market indices from 65 countries, and this information was compiled for a period of T = 3513 days over 14years from 11 January 2000 to 24 June 2014. Note that the working days for different markets are not same due todifferences in holidays across countries. To overcome any inconsistencies due to this difference in working days, wefiltered the data by removing days on which >
30% of the markets were not operative. Conversely, if <
30% of themarkets were not operative on a day, we used the closing price of such markets on the previous day to complete thedataset. Supplementary Table S1 lists the 69 global market indices considered here, along with their countries andgeographical regions.
Given the daily closing price g j ( t ) for market index j on day t , wherein j = 1 , , . . . , N with N = 69 indices, weconstruct a time series of logarithmic returns as r j ( t ) = ln g j ( t ) − ln g j ( t − C τij ( t ) = (cid:104) r i r j (cid:105) − (cid:104) r i (cid:105) (cid:104) r j (cid:105) σ i σ j , (1)where the mean and standard deviation are computed over a period of τ = 80 days with end date as t . We alsoconstruct the ultrametric distance matrix with elements D τij ( t ) = (cid:113) − C τij ( t )) that take values between 0 and 2.To study the temporal dynamics of the global market indices, we computed the correlation matrices for overlappingwindows of τ = 80 days with a rolling shift of ∆ τ = 20 days. Thence, we obtained 172 correlation frames between 11January 2000 to 24 June 2014.We have computed three market indicators from these correlation matrices C τ ( t ). Firstly, the mean correlationgives the average of the correlations in the matrix C τ ( t ). Secondly, we have computed the eigen-entropy [26] whichinvolves calculation of the Shannon entropy using the eigenvector centralities of the correlation matrix C τ ( t ) of marketindices. Both mean correlation and eigen-entropy has been shown to detect critical events in financial markets [26–28].Thirdly, we have computed the risk corresponding to the Markowitz portfolio of the market indices, which is a proxyfor the fragility or systemic risk of the global financial network [29]. A detailed description of the Markowitz portfoliooptimization is given in the Supplementary Material. The distance matrix for the time frame ending on t can be viewed as a complete, undirected and weighted graph D τ ( t ) where the element D τij ( t ) is the weight of the edge between market indices i and j . To extract the importantedges from D τ ( t ), we first construct its minimum spanning tree (MST) M τ ( t ) using Prim’s algorithm [30]. As MSTis an over-simplified network without cycles, it may lose crucial information on clusters or cliques. To overcome this,we add edges with correlation C τij ≥ .
65 in D τ ( t ) to M τ ( t ) and obtain the threshold graph S τ ( t ). Thereafter, westudy the temporal evolution of different network measures in S τ ( t ).Firstly, we have computed standard global network measures such as the number of edges, edge density, averagedegree, average weighted degree [31], average shortest path length, diameter, average clustering coefficient [32], mod-ularity [33, 34], communication efficiency [35], global reaching centrality (GRC) [36], network entropy [37], globalassortativity [38, 39] and clique number. Note that the chosen set of global network measures studied here are by nomeans exhaustive and also depend very much on the specific questions of interest, see for example, Wang et al. [40]for several gravitational centrality measures. Secondly, we have also computed four edge-centric curvature measures,namely, Ollivier-Ricci (OR) curvature [29, 41, 42], Forman-Ricci (FR) curvature [42–45], Menger-Ricci (MR) curva-ture [46, 47] and Haantjes-Ricci (HR) curvature [46, 47]. A detailed description of these network measures along withthe appropriate natural weight, strength or distance, to use in each case is included in the Supplementary Material. The multidimensional scaling (MDS) technique tries to embed N objects in high-dimensional space into a low-dimensional space (typically, 2- or 3-dimensions), while preserving the relative distance between pairs of objects [48].Here, we construct the (average) correlation matrix C T between the 69 market indices for the complete period of T = 3513 days between 11 January 2000 to 24 June 2014 using Eq. 1. Then, we compute the distance matrix D T from C T for the complete period. Thereafter, we use MDS to map the 69 market indices into a 2-dimensional spacesuch that the distances between pairs of indices in D T are preserved. To create the MDS plot, we used the in-builtfunction cmdscale.m in MATLAB . Moreover, we also construct the MST M T starting from the distance matrix D T , andthen, the threshold network S T for the complete period from 2000 to 2014 by adding edges with C Tij ≥ .
65 to M T .
3. RESULTS AND DISCUSSION
The primary goal of this investigation is to evaluate different network measures for their potential to serve asindicators of fragility or systemic risk and monitor the health of the global financial system. For this purpose, wecompiled a dataset of the daily closing prices of 69 global financial market indices from 65 different countries for a14-year period from 2000 to 2014 (Methods). Thereafter, we use the time-series of the logarithmic returns of the dailyclosing prices for 69 global market indices to compute the Pearson cross-correlation matrices C τ ( t ) with window sizeof τ = 80 days with overlapping shift of ∆ τ = 20 days, and ending on trading days t (Methods). Subsequently, weemploy a minimum spanning tree (MST) based approach to construct 172 threshold networks S τ ( t ) corresponding tothe cross-correlation matrices C τ ( t ) spanning the 14-year period (Methods). Here, we study the temporal evolutionof the structure of these correlation-based threshold networks S τ ( t ) of global market indices using several networkmeasures, and moreover, contrast the evolution of network properties with generic market indicators such as meancorrelation and minimum risk obtained using Markowitz framework.We reiterate that the threshold networks S τ ( t ) are constructed by computing the MST of the cross-correlationmatrices C τ ( t ) followed by addition of edges with correlation C τij ≥ .
65 (Methods). Intuitively, this network con-struction procedure ensures that each threshold network is a connected graph and captures the most relevant edges(correlations) between market indices. Since the obtained results may depend on the choice of the threshold (0 . .
65 asthreshold in Main text, and in networks constructed using 0 .
75 or 0 .
85 as threshold in Supplementary Material. Inthe sequel, we will show that the qualitative nature of the obtained results are not very sensitive to the choice of 0 . .
75 or 0 .
85 as thresholds to construct the networks of global market indices.In Figures 1, 2 and Supplementary Figure S1, we show the temporal evolution of generic indicators and networkmeasures in the threshold networks of global market indices over the 14-year period (2000-2014). Moreover, the four
Threshold Network(MST + edges with C ij ≥ 0.65) M ean c o rr e l a t i on E i genen t r op y M i n i m u m r i sk N u m be r o f edge s A v e r age w e i gh t eddeg r ee G RC D i a m e t e r C l u s t e r i ng c oe ff i c i en t M odu l a r i t y C o mm un i c a t i one ff i c i en cy N e t w o r k en t r op y - - - - - - - - - - - - - - - - - - - - - - - - - - - - G l oba l a ss o r t a t i v i t y U S H o u s i n g B u b b l e L e h m a n B r o t h e r s C r a s h D o w J o n e s F l a s h C r a s h A u g u s t S t o c k M a r k e t s F a ll FIG. 1. Evolution of generic indicators and network characteristics for the global market indices networks S τ ( t ), constructedfrom the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over a periodof 14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ .
65 tothe minimum spanning trees (MST). From top to bottom, we compare the plots of mean correlation among market indices,minimum risk corresponding to the Markowitz portfolio optimization, eigen-entropy, number of edges, average weighted degree,diameter, clustering coefficient, modularity, communication efficiency, global reaching centrality (GRC), network entropy andglobal assortativity. The four shaded regions correspond to the epochs around the four important market events, namely, UShousing bubble, Lehman brothers crash, Dow Jones flash crash, and August 2011 stock markets fall. O lli v i e r F o r m an -60-40-200 M enge r - - - - - - - - - - - - - - - - - - - - - - - - - - - - H aan t j e s -10004000900014000 M ean c o rr e l a t i on C li quenu m be r Number of edges: 109Modularity: 0.508Number of communities: 8
Number of edges: 246Modularity: 0.418Number of communities: 7
Number of edges: 390Modularity: 0.232Number of communities: 5
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USA1USA2 USA3USA4 CANMEX PANARG BRACHL PERCRI BMUJAMJPNHKG TWN AUSPAKLKATHA IDN IND1IND2SGPMYS PHLMNGUKDEUFRAESPCHEITAPRT IRL ISLABWBELLUX ALA BVTSWEAUTGRC POLCZE RUSHUN ROU UKR SVKESTLVALTUTUR MLTZAF EGYTUNBWANGA ISR LBNSAUJOROMNQATMUS
USA1USA2USA3USA4CAN MEX PANARG BRACHL PERCRIBMUJAM JPNHKG TWN AUSPAKLKA THAIDNIND1IND2 SGP MYSPHLMNGUKDEU FRAESPCHEITA PRTIRL ISLABW BEL LUXALA BVTSWEAUT GRCPOLCZERUS HUN ROUUKR SVK ESTLVALTUTURMLTZAF EGYTUNBWA NGAISRLBN SAU JOROMN QAT MUS
Threshold network(MST + edges with C ij ≥ 0.65) FIG. 2. Evolution of network characteristics and visualization of the threshold networks S τ ( t ) of market indices with windowsize τ = 80 days and an overlapping shift of ∆ τ = 20 days, constructed by adding edges with correlation C τij ( t ) ≥ .
65 to theMST. (Lower panel) Comparison of the plots of mean correlation among market indices, clique number, average of Ollivier-Ricci (OR), Forman-Ricci (FR), Menger-Ricci (MR), and Haantjes-Ricci (HR) curvature of edges in threshold networks overthe 14-year period. (Upper panel) Visualization of the threshold networks at three distinct epochs of τ = 80 days endingon trading days t equal to 04-08-2005 (normal), 14-08-2006 (US housing bubble) and 04-06-2010 (Dow Jones flash crash).Threshold networks show higher number of edges and lower number of communities during crisis. Correspondingly, there isan increase in mean correlation, clique number, average OR, MR and HR curvature, and decrease in average FR curvature ofthreshold networks during financial crisis. Node colours and labels are based on geographical region and country, respectively,of the indices and edge colours are based on the community determined by Louvain method. The four USA market indices,NASDAQ, NYSE, RUSSELL1000 and SPX, are labelled as USA1, USA2, USA3 and USA4, respectively, while the two Indianindices, namely, NIFTY and SENSEX30, are labelled as IND1 and IND2, respectively. Threshold network(MST + edges with C ij ≥ 0.65) −1−0.8−0.6−0.4−0.200.20.40.60.81 M ean c o rr e l a t i on M i n i m u m r i sk E i gen en t r op y N u m be r o f edge s A v e r age s t r eng t h D i a m e t e r C l u s t e r i ng c oe ff i c i en t M odu l a r i t y C o mm un i c a t i on e ff i c i en cy G RC N e t w o r k en t r op y G l oba l a ss o r t a t i v i t y C li que nu m be r O lli v i e r F o r m an M enge r H aan t j e s Mean correlationMinimum riskEigen entropyNumber of edgesAverage strengthDiameterClustering coefficientModularityCommunication efficiency GRCNetwork entropyGlobal assortativityClique numberOllivierFormanMengerHaantjes −0.6−0.16−0.57−0.55−0.56 −0.78−0.19−0.48−0.84−0.840.4−0.77 −0.150−0.11−0.13−0.120.1100.31−0.17−0.31−0.3 −0.91−0.25−0.65−0.98−0.980.52−0.930.92−0.97−0.72−0.850.18−0.95−0.86 FIG. 3. Correlations between generic indicators and network characteristics of the global market indices networks S τ ( t ),constructed from the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over aperiod of 14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ . shaded regions in Figure 1 highlight four periods of financial crisis, namely, US housing bubble, Lehman brotherscrash, Dow Jones flash crash, and August 2011 stock markets fall. From Figure 1, it is seen that the mean correlationbetween market indices increases during periods of financial crisis. Also, the eigen-entropy which is directly computedfrom the correlation matrix C τ ( t ) increases during crisis. Earlier works have shown that mean correlation and eigen-entropy are indicators of instabilities in stock market network [26, 28], and we show here that these measures can alsoserve as indicators of crisis in network of global financial indices. In Figure 1, we also show the temporal evolutionof the minimum risk corresponding to the portfolio comprising the market indices using the Markowitz framework.Moving on to widely-used network properties, it is seen that the number of edges, edge density, average degree, averageweighted degree, clustering coefficient, communication efficiency and network entropy increase while diameter, averageshortest path length and modularity decrease during periods of financial crisis (Figure 1; Supplementary Figure S1).In Figure 1, we also show evolution of two other network measures, global reaching centrality (GRC) and globalassortativity. In Figure 2, we also visualize the threshold network at three distinct time windows of τ = 80 daysending on trading days t corresponding to 04-08-2005 (normal period), 14-08-2006 (US housing bubble crisis) and04-06-2010 (Dow Jones flash crash) where the node colours are based on geographical regions of the market indicesand edge colours are based on modules determined by Louvain method [34] for community detection. The identifiedcommunities in the three networks corresponding to normal period, US housing bubble and Dow Jones flash crashtypically reflect the geographical proximity of financial market indices. For example, the indices of USA, Canada, Threshold network(MST + edges with C ij ≥ 0.65) (c) (b)(a) Number of edges: 125Modularity: 0.511Number of communities: 6
Multidimesional scaling analysisCommunities in the threshold network
MLT QATABW JORCHE LBN NGABWATUNEGYZAFALA TUR LTU LVAEST SVKUKR ROUHUNRUS CZE POLGRCAUTBVTISL MNGUSA4DEUFRA OMNMUSITA SAUIRL PHLESP BEL LUXUK MYSSWE SGPIND2IND1 IDN THALKA PAK AUSTWNHKG JPNJAMBMU CRIPERCHL BRAARG PANISR MEXCANPRT USA3USA2 USA1
LBN NGABWAMUS PHLMYSSGPIND2IND1 IDN THAPAK AUSTWNHKG JPNCRI PANPRT ISRSWEUK LUXBELESP IRLITA FRADEUALACHEABW PERISL BVTAUT GRCPOLMLT CZERUS HUNROUUKRZAFTUR JAM USA1USA2USA4 USA3CAN MEX ARGBRACHL SVK QAT SAUOMN MNGEST LVA TUNLTUEGY JORBMULKA X -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Y -0.6-0.4-0.20.00.20.40.6 USA1 USA2USA3USA4CANMEXPAN ARG BRACHLPERCRI BMUJAM JPN HKGTWN AUSPAK LKA THAIDN IND1 IND2SGPMYSPHLMNG UKDEUFRAESPCHE ITAPRTIRLISL ABWBELLUX ALABVT SWEAUTGRC POLCZERUS HUNROUUKRSVK ESTLVA LTU TURMLT ZAFEGYTUNBWANGA ISRLBNSAUJOROMN QAT MUS
FIG. 4. The average correlation structure between 69 market indices over the 14-year period is visualized based on thecorrelation matrix C T for the complete period of T = 3513 days between 2000 to 2014. (a) Visualization of the overallthreshold network S T corresponding to C T obtained by combining MST plus edges with correlation ≥ .
65. The node coloursare based on geographical regions of the market indices and edge colours are based on communities obtained from Louvainmethod. (b)
Visualization of the communities in the overall threshold network S T after removing the inter-module edges. It isevident that the market indices form communities in this network based on their geographical proximity. (c) Multidimensionalscaling (MDS) map in 2-dimensions of the 69 market indices. In this figure, the indices are labelled in different colours basedon their geographical region and country, respectively. The four USA market indices, NASDAQ, NYSE, RUSSELL1000 andSPX, are labelled as USA1, USA2, USA3 and USA4, respectively, while the two Indian indices, NIFTY and SENSEX30, arelabelled as IND1 and IND2, respectively.
Mexico, Argentina, Brazil and Chile form a single community in the threshold network for the normal period (Figure2). It is evident that the number of edges in threshold networks correspoding to US housing bubble (246 edges) orDow Jones flash crash (390 edges) are much higher in comparison to that for normal period (109 edges). In contrast,the modularity of threshold networks corresponding to the crisis periods, US housing bubble (0.418) or Dow Jonesflash crash (0.232) are lower in comparison to that for normal period (0.508). In Figure 2, it is clearly seen thatthe clique number or size of the largest clique in threshold networks increases during financial crisis, and this is alsoevident from the network visualizations for normal period, US housing bubble and Dow Jones flash crash. Note thatbubbles are not easy to detect. In fact, our proposition is that holistic approaches with network measures, both node-and edge-based measures, including geometric curvatures, may help us to better detect and distinguish the bubblesfrom market crashes, as also pointed out in recent contributions [26, 49]. In sum, we find that during a normal periodthe network of global market indices is less connected, very modular and heterogeneous, whereas during a fragileperiod the network is highly connected, less modular and more homogeneous.In addition to the node-centric global network measures described in the preceding paragraph, we have also studiededge-centric network measures, specifically, four discrete Ricci curvatures [Olivier-Ricci (OR), Forman-Ricci (OR),Menger-Ricci (MR) and Haantjes-Ricci (HR)] in threshold networks of global market indices. From Figure 2, it is seenthat the average OR, MR or HR curvature of edges increase during crisis periods in comparison to normal periods.In contrast, the average FR curvature of edges decreases during crisis periods in comparison to normal periods.Notably, Sandhu et al. [29] have shown that OR curvature can serve as indicator of fragility in stock market networks.However, to our knowledge, the present work is the first investigation of discrete Ricci curvatures in networks of globalmarket indices. Note that different discretizations of Ricci curvature do not capture the entire features of the classicaldefinition for continuous spaces, and thus, the four discrete Ricci curvatures studied here can capture different aspectsof analyzed networks [42]. Overall, our results suggest that discrete Ricci curvatures can serve as indicators of fragilityand monitor the health of the global financial system.In Figure 3, we show the correlation between generic market indicators and different characteristics of the thresholdnetworks S τ ( t ) of global market indices computed across the 14-year period from 2000 to 2014. From this figure,it is seen that eigen-entropy and several network measures have a very high (absolute) Pearson correlation ( ≈ . .
65, we have shown in Supplementary Figures S2-S9 that the qualitativeconclusions remain unchanged even when networks with threshold of 0 .
75 and 0 .
85 are considered. In other words,our results are robust to the choice of threshold used to construct the networks of global market indices.In previous works, the econophysics community has employed either minimum spanning tree (MST) [7, 9–13, 15, 19]or planar maximally fitered subgraph (PMFG) [12, 19] or threshold networks [11, 14, 20] to study the correlationstructure between global financial market indices. As far as we know, this work is the first to use threshold networksof MST plus edges with correlation higher than a specified threshold, to study the temporal evolution of relationshipsbetween global financial market indices. In contrast, such threshold networks based on MST have been used earlierto study the structure of stock market networks [29, 49]. While MST has a tree structure without loops or cycles,PMFG or threshold network permit loops or cycles. In Supplementary Text and Figures S10-S13, we also displaythe temporal evolution and correlation between generic market indicators and network measures in PMFG of globalmarket indices constructed from cross-correlation matrices C τ ( t ). While the construction of PMFG unlike thresholdnetworks is independent of any specific choice of the threshold, the number of edges (thus, edge density and averagedegree) is fixed in case of PMFG (Supplemetary Figures S10 and S11). Due to this reason, we find that most ofthe network measures studied here are not correlated with the generic market indicator, mean correlation of marketindices, in PMFG case (Supplementary Figure S13). Still, we find that average weighted degree (strength), clusteringcoefficient and communication efficiency have very high correlation with mean correlation of market indices in PMFGbased networks (Supplementary Figure S13). Based on these results, the threshold network construction based onMST plus edges with high correlation seems a better framework to monitor the state of the global financial system.Finally, we have also studied the average correlation structure between global market indices over the 14-year periodby computing the correlation matrix C T between the 69 market indices by taking window size as the complete periodof T days between 2000 to 2014 (Methods). Subsequently, we have constructed a threshold network S T correspondingto C T by combining MST plus edges with correlation above the chosen threshold of 0 .
65 (Methods). In Figure 4(a), wevisualize this overall threshold network S T of market indices for the complete 14-year period of T days. In this figure,the node colours are based on geographical regions of the market indices and edge colours are based on communitiesobtained from Louvain method. In Figure 4(b), we have separated the communities in this overall threshold network S T of market indices by removing the inter-module edges in the visualization. From Figure 4(a,b), it is clear that themarket indices form communities in this overall threshold network based on their geographical proximity. Moreover,we have also employed multidimensional scaling (MDS) technique to map the 69 market indices into a 2-dimensionalspace such that the distances between pairs of indices are preserved (Figure 4(c); Methods). It can be seen that theMDS map is able to partition the 69 market indices into groups based on their geographical proximity, and further,the structure in the MDS map has close resemblance to the community structure of the overall threshold network(Figure 4). For example, the grouping of indices from USA, Canada, Mexico, Argentina, Brazil and Chile can be seenin both the threshold network and MDS map (Figure 4). Interestingly, when we plotted in Supplementary Figure S14,the evolution of the eigenvector centralities of the nodes (market indices), as well as their OR and FR curvature, wefound that there exist certain periods of time, when some of the countries in close geographical proximity display high(absolute) values and others display low values, indicative of the changes in the complex interactions and communitystructures.
4. SUMMARY AND CONCLUDING REMARKS
In summary, we have investigated the daily closing prices of 69 global financial indices over a 14-year periodusing various techniques of cross-correlations based network analysis. We have been able to continuously monitorthe complex interactions among the global market indices by using a variety of network-centric measures, including,recently developed edge-centric discrete Ricci curvatures. In the present study of the global market indices, the noveltylies in: (i) Construction of the threshold network S τ ( t ), as superposition of the MST of the cross-correlation matrixand the network of edges with correlations C τij ≥ .
65, which ensures that each threshold network is a connectedgraph and captures the most relevant edges (correlations) between market indices. In Supplementary Material, wehave also reported the results for networks constructed using MST and two other threshold values, i.e., C τij ≥ . C τij ≥ .
85. Besides, we have also reported results for networks constructed using PMFG method. (ii) The usageof discrete Ricci curvatures in networks of global market indices, which capture the higher-order architecture of thecomplex financial system. To the best of our knowledge, this is the first study employing edge-based discrete Riccicurvatures to networks of global financial indices. Our recent work underscores the utility of edge-based curvaturemeasures in analysis of networks of stocks [49] or global financial indices. In future, curvature measures may also findapplication in other financial networks including Banking networks [50]. (iii) The largest yet by no means exhaustivesurvey of network measures to identify potential network-centric indicators of fragility and systemic risk in the systemof global financial market indices.The global financial system has become increasingly complex and interdependent, and thus prone to sudden unpre-dictable changes like market crises. Our results, compared to the traditional market indicators, do provide a deeperunderstanding of the system of global financial markets. Specially, we find that the four discrete Ricci curvatures canbe effectively used as indicators of fragility in global financial markets. We reiterate that the methods used in thiswork can detect instabilities in the market, and can be used as early warning signals so that policies can be made inorder to prevent the occurrence of such events in the future.
DATA AVAILABILITY STATEMENT
The codes used to construct the networks from correlation matrices and compute the different network measuresare publicly available via the GitHub repository: https://github.com/asamallab/FinNetIndicators . AUTHOR CONTRIBUTIONS
A.S., S.K. and A.C. conceived the project. A.S., S.K., Y.Y. and A.C. performed the computations. S.K. compiledthe dataset. Y.Y. and A.S. prepared the figures and tables. A.S. and A.C. analyzed the results. A.S., S.K., Y.Y. andA.C. wrote the manuscript. All authors have read and approved the manuscript.0
CONFLICT OF INTEREST
The authors declare that the research was conducted in the absence of any commercial or financial relationshipsthat could be construed as a potential conflict of interest.
ACKNOWLEDGMENTS
A.C. acknowledges support from the project UNAM-DGAPA-PAPIIT AG 100819 and CONACyT Project FRON-TERAS 201. A.S. acknowledges financial support from Max Planck Society Germany through the award of a MaxPlanck Partner Group in Mathematical Biology and a Ramanujan fellowship (SB/S2/RJN-006/2014) from the Scienceand Engineering Research Board (SERB), India.
Correspondence to:
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We computed the risk corresponding to the portfolio comprising the market indices using the Markowitz frameworkas an indicator of the market risk for an investor who wishes to maximize the expected returns with the constraintof minimum variance. That is, the scheme minimizes w (cid:48) Σw − φR (cid:48) w with respect to the normalized weight vectorw, where Σ is the covariance matrix calculated from the logarithmic returns of the market indices, φ is the measureof risk appetite of investor and R (cid:48) is the expected return of the market indices. We specify short-selling constraint, φ = 0 and w i ≥
0, such that we get a convex combination of returns of market indices for finding the minimum riskportfolio. These computations were performed using the in-built function in
MATLAB Portfolio package ( https://in.mathworks.com/help/finance/portfolio.html ). STANDARD NETWORK MEASURES
Each network investigated in this work can be represented as a weighted and undirected graph G ( V, E ) where V is the set of vertices (or nodes) and E is the set of edges (or links) in the graph. Also, an edge in a weightedgraph has weight assigned to it, and this weight in real networks typically represents the distance or strength betweenvertices forming the edge. Depending on the network measure employed for characterizing the weighted graph, eitherthe strength or the distance between two vertices could be the appropriate natural weight to use in the associatedcomputation (Supplementary Table S2). Recall that while the strength represents similarity between two vertices,the distance reflects dissimilarity between them.Here, we have studied weighted networks constructed from cross-correlation among global financial market indices(see Methods section in main text). For these networks of global market indices, we use the absolute value ofthe correlation C τij ( t ), that is | C τij ( t ) | , between two market indices i and j as the strength of the edge betweenvertices i and j in the threshold network for epoch ending at t for computations, and the ultrametric distance D τij ( t ) = (cid:113) − C τij ( t )) as the distance of the edge between vertices i and j for computations (Supplementary TableS2). Note that the strength of an edge given by | C τij ( t ) | in the network of global financial market indices can take avalue between 0 and 1, while the distance of an edge given by D τij ( t ) can take a value between 0 and 2.In this work, we have characterized the structure of the global market indices network represented as a weightedand undirected graph using the following measures. • The number of edges m is given by m = | E | and the number of vertices n is given by n = | V | , where | | denotescardinality of the set. • The number of edges incident on a given vertex gives its degree. The average degree of vertices can be expressedas (cid:104) k (cid:105) = mn , where n is the number of vertices and m is the number of edges in graph G . • The weighted degree (or strength) of a vertex is defined as the sum of weights of the edges incident on thevertex [31]. Consequently, the average weighted degree of vertices can be defined as (cid:104) k w (cid:105) = m w n , where m w isthe sum of weights assigned to all edges in graph G . We remark that the strength is the natural edge weightwhile computing the average weighted degree. In the main text, we also sometimes refer to average weighteddegree as average strength . • The edge density is defined as the ratio of the number of edges and the number of possible edges in graph G .Since a total of n ( n − edges are possible in an undirected graph G ignoring self-edges, the edge density is givenby mn ( n − . • The shortest path between any two vertices i and j in a graph is defined as a path wherein the sum of thedistance along all the edges in the path is the minimum among all possible paths connecting the two vertices.The shortest path length , denoted by d ( i, j ), is the sum of distances along edges in the shortest path betweenvertices i and j in the graph. The average (shortest) path length is an average of the shortest path lengthsbetween every pair of vertices in the graph, that is, (cid:104) L (cid:105) = 1 n ( n − (cid:88) i (cid:54) = j ∈ V d ( i, j ) . (2)3We remark that distance is the natural edge weight while computing the average shortest path length in weightedgraphs. • The diameter of a graph is defined as the maximum of the shortest paths between all pairs of vertices, that is, max { d ( i, j ) ∀ i, j ∈ V } . • Communcation efficiency characterizes the global information flow or ability to exchange information in anetwork [35]. The communication efficiency µ c is defined as µ c = 1 n ( n − (cid:88) i (cid:54) = j ∈ V d ( i, j ) . (3)We remark that distance is the natural edge weight while computing the communication efficiency in weightedgraphs. • The clustering coefficient of a vertex gives a measure of its tendency to form triads with its neighbouring vertices.Onnela [32] has proposed an approach to measure the clustering coefficient in weighted networks. For a vertex i in weighted graph G , clustering coefficient is defined as C i = 2 k i ( k i − (cid:88) j,k ( a ij a ik a jk ) / , (4)where j and k are the neighbours of vertex i and the summation runs over all such pairs of neighbours. Thequantity in the summation is the intensity of the triangle attached to vertex i , and it takes the value 0 if atriangle is not formed. The average clustering coefficent of a graph G is the average of the clustering coefficientsacross all vertices in G . We remark that the strength is the natural edge weight while computing the clusteringcoefficient in weighted graphs. • A network is said to exhibit community structure if it is possible to divide the vertices into distinct groups ofdensely connected vertices. Modularity measures edge density within a community in comparison to the edgesbetween communities. Modularity of a weighted graph G is defined as [33, 34] Q = 12 m w (cid:88) i (cid:54) = j ∈ V [ a ij − s i s j m w ] δ ( c i , c j ) (5)where s i and s j give the sum of weights of edges attached to vertices i and j , respectively, c i and c j are thecommunities of i and j , respectively, and m w is the sum of weights of all edges in G . We remark that thestrength is the natural edge weight while computing the modularity in weighted graphs. • Assortative mixing refers to the tendency of a vertex to attach to other vertices with similar properties in thenetwork. A network is said to be assortative if high degree vertices tend to link with other high degree vertices.The assortativity coefficient was introduced by Newman [38] to measure degree correlations between vertices inan unweighted network. It is possible to extend this definition to weighted graphs by measuring how stronglyany two vertices with similar degree tend to link with each other [39]. The global assortativity of a weightedgraph G is defined as: r w = m w (cid:80) e a ij k i k j − (cid:104) m w (cid:80) e a ij ( k i + k j ) (cid:105) m w (cid:80) e a ij ( k i + k j ) − (cid:104) m w (cid:80) e a ij ( k i + k j ) (cid:105) , (6)where m w is the sum of weights of all edges, k i is the degree of the vertex i , a ij is the weight of edge betweenvertices i and j , and the summation runs over all edges e in weighted graph G . We remark that the strength isthe natural edge weight while computing the global assortativity in weighted graphs. • Measures for assortative mixing are limited since they quantify only the linear dependence.
Network entropy was introduced to measure a network’s heterogeneity [37], which follows a more general information-theoreticapproach. The remaining (excess) degree of a vertex is defined as the number of edges leaving the vertex otherthan the one used to reach the vertex. The probability that a randomly chosen vertex has an excess degree k is given by the remaining degree distribution q k = ( k +1) p k +1
EDGE-BASED CURVATURE MEASURES
The Ricci curvature in differential geometry is applicable to smooth manifolds [55]. As the classical definition ofRicci curvature is not directly applicable to discrete objects including graphs or networks, multiple discrete notionsof Ricci curvature have been proposed to date [42]. While the classical definition of Ricci curvature is associated tovectors in smooth manifolds, in the case of discrete networks, the Ricci curvature is naturally associated to edges inthe graph [42]. Thus, the discrete Ricci curvatures are associated to edges rather than vertices or nodes in a graph.In other words, the discrete Ricci curvatures can be employed for edge-based analysis in contrast to commonly usedmeasures such as degree and clustering coefficient which are suited for node-based analysis of networks [42, 44].Recall that the classical notion of Ricci curvature captures two essential geometric properties of the manifold,namely, volume growth and dispersion of geodesics. However, the discretizations of Ricci curvature which have beenemployed to characterize the structure of networks cannot capture the entire spectrum of geometric properties of theclassical notion [42]. Thus, different notions of discrete Ricci curvatures may capture different aspects of the structureof complex networks. In this section, we describe four notions of discrete Ricci curvature that we have used to studythe networks of global market indices.
Ollivier-Ricci curvature [41, 56] has proposed a discrete notion of Ricci curvature which captures the volume growth property of the classicaldefinition. Olivier’s proposal is based on the observation that in spaces of positive (negative) curvature, balls are closer5(farther) to each other on the average than their centres. The
Ollivier-Ricci curvature (OR) of an edge e betweenvertices i and j in undirected graph G is defined as O ( e ) = 1 − W ( m i , m j ) d ( i, j ) , (11)where m i and m j are discrete probability measures assigned to vertices i and j , respectively, d ( i, j ) is the distancebetween i and j , as defined in the previous section, and W denotes the Wasserstein distance [57], which is thetransportation distance between m i and m j , given by W ( m i , m j ) = inf µ i,j ∈ (cid:81) ( m i ,m j ) (cid:88) ( i (cid:48) ,j (cid:48) ) ∈ V × V d ( i (cid:48) , j (cid:48) ) µ i,j ( i (cid:48) , j (cid:48) ) , (12)where (cid:81) ( m i , m j ) is the set of probability measures µ i,j that satisfy (cid:88) j (cid:48) ∈ V µ i,j ( i (cid:48) , j (cid:48) ) = m i ( i (cid:48) ) , (cid:88) i (cid:48) ∈ V µ i,j ( i (cid:48) , j (cid:48) ) = m j ( j (cid:48) ) . (13)The probability distribution m i is taken to be uniform over the the neighbouring vertices of i [58]. We have computedthe average OR curvature of edges in networks of global market indices in this work. While computing the ORcurvature in networks of global market indices, the weight of each edge is taken to be the distance. Given the ORcurvature of edges in the graph, it is straightforward to define the OR curvature of a vertex v as O ( v ) = (cid:88) e ∼ v O ( e ) (14)where e ∼ v is the set of edges e incident on vertex v . The above definition of OR curvature of a vertex is analogousto scalar curvature in Riemannian geometry [42]. Forman-Ricci curvature
Forman’s discretization [43] captures the geodesic dispersal property of the classical notion of Ricci curvature [44]. It isbased on the relation between the Riemann-Laplace operator and Ricci curvature. Recently,
Forman-Ricci curvature (FR) was adapted for the analysis of unweighted and weighted networks [44, 59]. Intuitively, FR curvature quantifiesthe information spread at the ends of an edge in the network. High negative FR value for an edge indicates morespread of information at its ends. For an edge e between vertices i and j in an undirected graph G , FR is defined as F ( e ) = w e w i w e + w j w e − (cid:88) e i ∼ e, e j ∼ e (cid:34) w i √ w e w e j + w j √ w e w e j (cid:35) (15)where w e denotes the weight of the edge e , w i and w j denote the weights associated with the vertices i and j ,respectively, e i ∼ e and e j ∼ e denote the set of edges incident on vertices i and j , respectively, after excluding theedge e . We have computed the average FR curvature of edges in networks of global market indices in this work. Whilecomputing the FR curvature in networks of global market indices, the weight of each edge is taken to be the distancewhile the weight of each vertex is taken to be 1.Given the FR curvature of edges in the graph, it is straightforward to define the FR curvature of a vertex v as F ( v ) = (cid:88) e ∼ v F ( e ) (16)where e ∼ v is the set of edges e incident on vertex v . The above definition of FR curvature of a vertex is analogousto scalar curvature in Riemannian geometry [59]. Menger-Ricci curvature
Menger defined the curvature of a metric triangle T [60] formed by three points in space as the reciprocal R ( T ) of theradius R ( T ) of the circumcircle of that triangle. Given a triangle T = T ( a, b, c ) with sides a, b, c in a metric space6( M, d ), the Menger curvature of T is given by K M ( T ) = (cid:112) p ( p − a )( p − b )( p − c ) a · b · c (17)where p = ( a + b + c ) /
2. It is possible to extend the above definition to unweighted and undirected networks [46, 47],where one considers combinatorial triangles with length of each side equal to 1, and this gives K M ( T ) = √ /
2. Thenthe
Menger-Ricci curvature (MR) of an edge e in the graph G can be defined as κ M ( e ) = (cid:88) T e ∼ e κ M ( T e ) , (18)where T e ∼ e denote the triangles adjacent to the edge e . An edge will have high positive value of MR curvature ifit is part of many triangles in the network. In this work, we have computed the average MR curvature of edges innetworks of global market indices. Haantjes-Ricci curvature
Haantjes [61] defined the curvature of a metric curve as the ratio of the length of the arc of the curve and that of thechord it subtends. More precisely, given three points p , q and r on a curve in a metric space such that p lies between q and r , the Haantjes curvature at the point p is defined as κ H ( p ) = 24 lim q,r → p l ( (cid:98) qr ) − d ( q, r ) (cid:0) d ( q, r ) (cid:1) , (19)where l ( (cid:98) qr ) denotes the length of the arc (cid:98) qr . The above definition can be extended to networks by replacing the arc (cid:98) qr with a path between the two vertices and the subtending chord by the edge between the two vertices [46, 47]. Givena simple path π = i, . . . , j between the two vertices i and j connected by an edge e in the unweighted graph G , theHaantjes curvature of the path takes the value κ H ( π ) = √ n − , (20)where n is the number of edges appearing in the path π . Then the Haantjes-Ricci curvature (HR) of the edge e canbe defined as [46, 47] κ H ( e ) = (cid:88) π ∼ e κ H ( π ) , (21)where the summation runs over all the paths between vertices i and j . In this work, we have computed the average HRcurvature of edges in networks of global market indices by ignoring edge weights. Further, computational constraintspermitted only consideration of paths π of length ≤ PLANAR MAXIMALLY FILTERED GRAPH (PMFG) CONSTRUCTION AND CHARACTERISTICS
Here, we describe an alternate network construction framework, namely, the planar maximally filtered graph(PMFG) [6], which has been widely-used to study the relationship between global financial market indices. Briefly, thePMFG P τ ( t ) of market indices can be constructed for the time-series of cross-correlation matrices C τ ( t ) of windowsize τ = 80 days and an overlapping shift of ∆ τ = 20 days over the 14-year period as follows (see Methods sectionin main text for the computation of cross-correlation matrices C τ ( t ) starting from the logarithmic returns of dailyclosing prices of 69 global market indices). Firstly, a sorted list of edges is created based on the decreasing order ofcorrelation in the matrix C τ ( t ). Next, each edge in the sorted list is considered for inclusion in the PMFG based onthe decreasing order of correlation. An edge between vertices i and j is added to PMFG, if and only if the resultinggraph can be embedded on a sphere, i.e., it is a planar graph. Following this scheme, the final network obtained is aPMFG with 3( N −
1) edges where N is the number of vertices in the graph. Note that the minimum spanning tree(MST) in contrast to PMFG contains N − g = k where k is a postive integer. In case of PMFG, the graph is embedded on asurface of genus g = 0 which is a sphere.In Supplementary Figures S10-S13, we show the temporal evolution and correlation between generic market indi-cators and network measures in PMFG P τ ( t ) of market indices constructed from cross-correlation matrices C τ ( t ) asdescribed above. Threshold Network(MST + edges with C ij ≥ 0.65) U S H o u s i n g B u b b l e L e h m a n B r o t h e r s C r a s h D o w J o n e s F l a s h C r a s h A u g u s t S t o c k M a r k e t s F a ll M ean c o rr e l a t i on E dgeden s i t y N u m be r o f edge s A v e r agedeg r ee - - - - - - - - - - - - - - - - - - - - - - - - - - - - A v e r agepa t h l eng t h FIG. S1. Evolution of generic indicators and network characteristics for the global market indices networks S τ ( t ), constructedfrom the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over a period of14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ .
65 to theminimum spanning tree (MST). From top to bottom, we compare the plot of mean correlation among market indices, numberof edges, edge density, average degree and average path length. The four shaded regions correspond to the epochs around thefour important market events, namely, US housing bubble, Lehman brothers crash, Dow Jones flash crash, and August 2011stock markets fall. Threshold Network(MST + edges with C ij ≥ 0.75) M ean c o rr e l a t i on E i genen t r op y M i n i m u m r i sk N u m be r o f edge s A v e r age w e i gh t eddeg r ee G RC D i a m e t e r C l u s t e r i ng c oe ff i c i en t M odu l a r i t y C o mm un i c a t i one ff i c i en cy N e t w o r k en t r op y - - - - - - - - - - - - - - - - - - - - - - - - - - - - G l oba l a ss o r t a t i v i t y -0.30.00.30.60.9 U S H o u s i n g B u b b l e L e h m a n B r o t h e r s C r a s h D o w J o n e s F l a s h C r a s h A u g u s t S t o c k M a r k e t s F a ll FIG. S2. Evolution of generic indicators and network characteristics for the global market indices networks S τ ( t ), constructedfrom the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over a periodof 14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ .
75 tothe minimum spanning tree (MST). From top to bottom, we compare the plot of mean correlation among market indices,minimum risk corresponding to the Markowitz portfolio optimization, eigen-entropy, number of edges, average weighted degree,diameter, clustering coefficient, modularity, communication efficiency, global reaching centrality (GRC), network entropy andglobal assortativity. The four shaded regions correspond to the epochs around the four important market events, namely, UShousing bubble, Lehman brothers crash, Dow Jones flash crash, and August 2011 stock markets fall. Threshold Network(MST + edges with C ij ≥ 0.75) M ean c o rr e l a t i on E dgeden s i t y N u m be r o f edge s A v e r agedeg r ee - - - - - - - - - - - - - - - - - - - - - - - - - - - - A v e r agepa t h l eng t h U S H o u s i n g B u b b l e L e h m a n B r o t h e r s C r a s h D o w J o n e s F l a s h C r a s h A u g u s t S t o c k M a r k e t s F a ll FIG. S3. Evolution of generic indicators and network characteristics for the global market indices networks S τ ( t ), constructedfrom the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over a period of14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ .
75 to theminimum spanning tree (MST). From top to bottom, we compare the plot of mean correlation among market indices, numberof edges, edge density, average degree and average path length. The four shaded regions correspond to the epochs around thefour important market events, namely, US housing bubble, Lehman brothers crash, Dow Jones flash crash, and August 2011stock markets fall. Threshold Network(MST + edges with C ij ≥ 0.85) M ean c o rr e l a t i on E i genen t r op y M i n i m u m r i sk N u m be r o f edge s A v e r age w e i gh t eddeg r ee G RC D i a m e t e r C l u s t e r i ng c oe ff i c i en t M odu l a r i t y C o mm un i c a t i one ff i c i en cy N e t w o r k en t r op y - - - - - - - - - - - - - - - - - - - - - - - - - - - - G l oba l a ss o r t a t i v i t y -0.30.00.30.60.9 U S H o u s i n g B u b b l e L e h m a n B r o t h e r s C r a s h D o w J o n e s F l a s h C r a s h A u g u s t S t o c k M a r k e t s F a ll FIG. S4. Evolution of generic indicators and network characteristics for the global market indices networks S τ ( t ), constructedfrom the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over a periodof 14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ .
85 tothe minimum spanning tree (MST). From top to bottom, we compare the plot of mean correlation among market indices,minimum risk corresponding to the Markowitz portfolio optimization, eigen-entropy, number of edges, average weighted degree,diameter, clustering coefficient, modularity, communication efficiency, global reaching centrality (GRC), network entropy andglobal assortativity. The four shaded regions correspond to the epochs around the four important market events, namely, UShousing bubble, Lehman brothers crash, Dow Jones flash crash, and August 2011 stock markets fall. Threshold Network(MST + edges with C ij ≥ 0.85) M ean c o rr e l a t i on E dgeden s i t y N u m be r o f edge s A v e r agedeg r ee - - - - - - - - - - - - - - - - - - - - - - - - - - - - A v e r agepa t h l eng t h U S H o u s i n g B u b b l e L e h m a n B r o t h e r s C r a s h D o w J o n e s F l a s h C r a s h A u g u s t S t o c k M a r k e t s F a ll FIG. S5. Evolution of generic indicators and network characteristics for the global market indices networks S τ ( t ), constructedfrom the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over a period of14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ .
85 to theminimum spanning tree (MST). From top to bottom, we compare the plot of mean correlation among market indices, numberof edges, edge density, average degree and average path length. The four shaded regions correspond to the epochs around thefour important market events, namely, US housing bubble, Lehman brothers crash, Dow Jones flash crash, and August 2011stock markets fall. Number of edges: 83Modularity: 0.639Number of communities: 8
Number of edges: 128Modularity: 0.545Number of communities: 7
Number of edges: 127Modularity: 0.332Number of communities: 6 F o r m an -25-20-15-10-50 M enge r - - - - - - - - - - - - - - - - - - - - - - - - - - - - H aan t j e s -500500150025003500 M ean c o rr e l a t i on O lli v i e r -0.20.00.20.40.60.8 C li quenu m be r USA1USA2USA3 USA4CANMEXPAN ARGBRACHLPERCRI BMUJAMJPNHKGTWN AUS PAKLKATHAIDN IND1IND2SGP MYSPHLMNG UKDEUFRA ESPCHE ITAPRTIRLISL ABWBELLUX ALABVTSWEAUTGRCPOL CZERUSHUN ROUUKR SVK ESTLVA LTUTUR MLT ZAFEGYTUNBWANGAISR LBNSAU JOROMNQATMUS
USA1 USA2USA3USA4 CANMEXPAN ARGBRA CHLPERCRIBMUJAM JPNHKG TWNAUS PAKLKA THAIDN IND1IND2SGPMYSPHLMNG UKDEU FRAESPCHE ITAPRT IRLISL ABWBELLUX ALABVT SWEAUTGRCPOL CZE RUSHUNROU UKR SVKESTLVALTU TURMLT ZAFEGYTUN BWANGA ISR LBNSAU JOROMNQAT MUS
USA1USA2USA3 USA4CANMEXPAN ARGBRACHL PERCRIBMU JAMJPN HKGTWN AUSPAK LKATHA IDN IND1IND2SGP MYSPHL MNGUKDEUFRA ESPCHE ITAPRTIRL ISLABWBELLUX ALABVT SWE AUT GRCPOLCZERUS HUNROU UKRSVKESTLVA LTUTUR MLTZAF EGYTUNBWA NGAISRLBN SAUJOR OMNQAT MUS
Threshold network(MST + edges with C ij ≥ 0.75) FIG. S6. Evolution of network characteristics and visualization of the threshold networks S τ ( t ) of market indices with windowsize τ = 80 days and an overlapping shift of ∆ τ = 20 days, constructed by adding edges with correlation C τij ( t ) ≥ .
75 to theMST. (Lower panel) Comparison of the plots of mean correlation among market indices, clique number, average of Ollivier-Ricci (OR), Forman-Ricci (FR), Menger-Ricci (MR), and Haantjes-Ricci (HR) curvature of edges in threshold networks overthe 14-year period. (Upper panel) Visualization of the threshold networks at three distinct epochs of τ = 80 days endingon trading days t equal to 04-08-2005 (normal), 14-08-2006 (US housing bubble) and 04-06-2010 (Dow Jones flash crash).Threshold networks show higher number of edges and lower number of communities during crisis. Correspondingly, there isan increase in mean correlation, clique number, average OR, MR and HR curvature, and decrease in average FR curvature ofthreshold networks during financial crisis. Node colours and labels are based on geographical region and country, respectively,of the indices and edge colours are based on the community determined by Louvain method. The four USA market indices,NASDAQ, NYSE, RUSSELL1000 and SPX, are labelled as USA1, USA2, USA3 and USA4, respectively, while the two Indianindices, NIFTY and SENSEX30, are labelled as IND1 and IND2, respectively. O lli v i e r -0.4-0.20.00.20.4 F o r m an -15-10-50 M enge r - - - - - - - - - - - - - - - - - - - - - - - - - - - - H aan t j e s -200100400700 M ean c o rr e l a t i on C li quenu m be r Number of edges: 75Modularity: 0.685Number of communities: 10
Number of edges: 104Modularity: 0.591Number of communities: 8
Number of edges: 128Modularity: 0.445Number of communities: 7
USA1USA2USA3 USA4CANMEX PANARG BRACHL PERCRIBMU JAM JPNHKG TWNAUSPAK LKATHA IDNIND1 IND2SGP MYS PHLMNGUKDEU FRAESPCHE ITAPRTIRL ISLABWBELLUX ALA BVTSWE AUTGRC POLCZERUSHUN ROU UKRSVKESTLVA LTUTUR MLTZAF EGYTUN BWANGAISR LBNSAU JOROMNQATMUS
USA1USA2USA3 USA4CANMEXPAN ARGBRA CHLPER CRI BMUJAM JPNHKG TWNAUSPAK LKA THAIDNIND1 IND2 SGPMYS PHLMNGUKDEU FRAESPCHEITA PRTIRL ISL ABWBEL LUXALABVTSWEAUTGRCPOLCZERUSHUN ROUUKRSVKEST LVALTU TURMLT ZAFEGY TUNBWANGA ISRLBNSAUJOROMN QAT MUS
USA1 USA2USA3USA4CANMEXPAN ARG BRACHLPERCRIBMU JAMJPNHKG TWNAUSPAK LKATHAIDN IND1IND2SGPMYS PHL MNG UKDEUFRA ESPCHE ITAPRTIRLISL ABWBELLUX ALABVT SWEAUT GRCPOLCZERUSHUNROUUKRSVKESTLVALTU TURMLT ZAFEGYTUNBWANGA ISRLBNSAUJOR OMNQATMUS
Threshold network(MST + edges with C ij ≥ 0.85) FIG. S7. Evolution of network characteristics and visualization of the threshold networks S τ ( t ) of market indices with windowsize τ = 80 days and an overlapping shift of ∆ τ = 20 days, constructed by adding edges with correlation C τij ( t ) ≥ .
85 to theMST. (Lower panel) Comparison of the plots of mean correlation among market indices, clique number, average of Ollivier-Ricci (OR), Forman-Ricci (FR), Menger-Ricci (MR), and Haantjes-Ricci (HR) curvature of edges in threshold networks overthe 14-year period. (Upper panel) Visualization of the threshold networks at three distinct epochs of τ = 80 days endingon trading days t equal to 04-08-2005 (normal), 14-08-2006 (US housing bubble) and 04-06-2010 (Dow Jones flash crash).Threshold networks show higher number of edges and lower number of communities during crisis. Correspondingly, there isan increase in mean correlation, clique number, average OR, MR and HR curvature, and decrease in average FR curvature ofthreshold networks during financial crisis. Node colours and labels are based on geographical region and country, respectively,of the indices and edge colours are based on the community determined by Louvain method. The four USA market indices,NASDAQ, NYSE, RUSSELL1000 and SPX, are labelled as USA1, USA2, USA3 and USA4, respectively, while the two Indianindices, NIFTY and SENSEX30, are labelled as IND1 and IND2, respectively. Threshold network(MST + edges with C ij ≥ 0.75) −1−0.8−0.6−0.4−0.200.20.40.60.81 M ean c o rr e l a t i on M i n i m u m r i sk E i gen en t r op y N u m be r o f edge s A v e r age s t r eng t h D i a m e t e r C l u s t e r i ng c oe ff i c i en t M odu l a r i t y C o mm un i c a t i on e ff i c i en cy G RC N e t w o r k en t r op y G l oba l a ss o r t a t i v i t y C li que nu m be r O lli v i e r F o r m an M enge r H aan t j e s Mean correlationMinimum riskEigen entropyNumber of edgesAverage strengthDiameterClustering coefficientModularityCommunication efficiency GRCNetwork entropyGlobal assortativityClique numberOllivierFormanMengerHaantjes −0.52−0.12−0.49−0.47−0.48 −0.7−0.1−0.4−0.84−0.840.39−0.76 −0.86−0.21−0.59−0.99−0.990.46−0.930.91−0.96−0.68−0.81−0.54−0.93−0.82 FIG. S8. Correlation between generic indicators and network characteristics of the global market indices networks S τ ( t ),constructed from the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over aperiod of 14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ . Threshold network(MST + edges with C ij ≥ 0.85) −1−0.8−0.6−0.4−0.200.20.40.60.81 M ean c o rr e l a t i on M i n i m u m r i sk E i gen en t r op y N u m be r o f edge s A v e r age s t r eng t h D i a m e t e r C l u s t e r i ng c oe ff i c i en t M odu l a r i t y C o mm un i c a t i on e ff i c i en cy G RC N e t w o r k en t r op y G l oba l a ss o r t a t i v i t y C li que nu m be r O lli v i e r F o r m an M enge r H aan t j e s Mean correlationMinimum riskEigen entropyNumber of edgesAverage strengthDiameterClustering coefficientModularityCommunication efficiency GRCNetwork entropyGlobal assortativityClique numberOllivierFormanMengerHaantjes −0.52−0.12−0.49−0.47−0.48 −0.7−0.1−0.4−0.84−0.840.39−0.76 −0.86−0.21−0.59−0.99−0.990.46−0.930.91−0.96−0.68−0.81−0.54−0.93−0.82 FIG. S9. Correlation between generic indicators and network characteristics of the global market indices networks S τ ( t ),constructed from the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over aperiod of 14 years (2000-2014). The threshold networks S τ ( t ) were constructed by adding edges with correlation C τij ( t ) ≥ . Planar Maximally Filtered Graph(PMFG) M ean c o rr e l a t i on E i genen t r op y M i n i m u m r i sk N u m be r o f edge s A v e r age w e i gh t eddeg r ee G RC D i a m e t e r C l u s t e r i ng c oe ff i c i en t M odu l a r i t y C o mm un i c a t i one ff i c i en cy N e t w o r k en t r op y - - - - - - - - - - - - - - - - - - - - - - - - - - - - G l oba l a ss o r t a t i v i t y -0.3-0.2-0.10.00.1 U S H o u s i n g B u b b l e L e h m a n B r o t h e r s C r a s h D o w J o n e s F l a s h C r a s h A u g u s t S t o c k M a r k e t s F a ll FIG. S10. Evolution of generic indicators and network characteristics for PMFG P τ ( t ) of global market indices, constructedfrom the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over a period of14 years (2000-2014). From top to bottom, we compare the plot of mean correlation among market indices, minimum riskcorresponding to the Markowitz portfolio optimization, eigen-entropy, number of edges, average weighted degree, diameter,clustering coefficient, modularity, communication efficiency, global reaching centrality (GRC), network entropy and globalassortativity. The four shaded regions correspond to the epochs around the four important market events, namely, US housingbubble, Lehman brothers crash, Dow Jones flash crash, and August 2011 stock markets fall. Planar Maximally Filtered Graph(PMFG) M ean c o rr e l a t i on E dgeden s i t y N u m be r o f edge s A v e r agedeg r ee - - - - - - - - - - - - - - - - - - - - - - - - - - - - A v e r agepa t h l eng t h U S H o u s i n g B u b b l e L e h m a n B r o t h e r s C r a s h D o w J o n e s F l a s h C r a s h A u g u s t S t o c k M a r k e t s F a ll FIG. S11. Evolution of generic indicators and network characteristics for PMFG P τ ( t ) of global market indices, constructedfrom the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over a period of 14years (2000-2014). From top to bottom, we compare the plot of mean correlation among market indices, number of edges, edgedensity, average degree and average path length. The four shaded regions correspond to the epochs around the four importantmarket events, namely, US housing bubble, Lehman brothers crash, Dow Jones flash crash, and August 2011 stock marketsfall. O lli v i e r -0.20.00.20.4 F o r m an -14-13-12-11-10 M enge r - - - - - - - - - - - - - - - - - - - - - - - - - - - - H aan t j e s M ean c o rr e l a t i on C li quenu m be r Planar Maximally Filtered Graph(PMFG)
FIG. S12. Evolution of network characteristics for PMFG P τ ( t ) of global market indices with window size τ = 80 days and anoverlapping shift of ∆ τ = 20 days. Comparison of the plots of mean correlation among market indices, clique number, averageof Ollivier-Ricci (OR), Forman-Ricci (FR), Menger-Ricci (MR), and Haantjes-Ricci (HR) curvature of edges in PMFG over the14-year period. Planar Maximally Filtered Graph(PMFG)
UDUDUD UD UD UD UD UD UD UD UD UDUDUDUDUDUDUDUDUDUDUDUD UDUD UDUD UDUD UDUD −0.85−0.66−0.48−0.29−0.110.080.260.450.630.821 M ean c o rr e l a t i on M i n i m u m r i sk E i gen en t r op y N u m be r o f edge s A v e r age s t r eng t h D i a m e t e r C l u s t e r i ng c oe ff i c i en t M odu l a r i t y C o mm un i c a t i on e ff i c i en cy G RC N e t w o r k en t r op y G l oba l a ss o r t a t i v i t y C li que nu m be r O lli v i e r F o r m an M enge r H aan t j e s Mean correlationMinimum riskEigen entropyNumber of edgesAverage strengthDiameterClustering coefficientModularityCommunication efficiency GRCNetwork entropyGlobal assortativityClique numberOllivierFormanMengerHaantjes −0.38−0.21−0.43−0.36 −0.36 −0.46 −0.07 −0.12 −0.1 −0.02 −0.57−0.1−0.43−0.61 −0.62−0.28−0.55−0.39−0.14 −0.26 −0.02−0.26 −0.25 −0.46−0.69−0.11 −0.08−0.01−0.09−0.05 −0.1 −0.29−0.14−0.27 −0.15 −0.11 −0.23−0.08−0.27 −0.02 −0.37 −0.39 −0.22−0.35−0.85 FIG. S13. Correlation between generic indicators and network characteristics for PMFG P τ ( t ) of global market indices,constructed from the correlation matrices C τ ( t ) of window size τ = 80 days and an overlapping shift of ∆ τ = 20 days over aperiod of 14 years (2000-2014). For some pairs of measures investigated here, the correlation cannot be computed as at leastone of the measures has zero variance, and thus, we specify undefined (‘UD’) for correlation between such pairs of measures inthis plot. U SA U SA U SA U SA C A N M EXPA N A R G B R A CH L PE RCR I B M U J A M J P NH K G T W N A U SPAK L KA T H A I DN I ND I ND S G P M YSP H L M N G U K D E U F R AESP CH E I T AP R T I R L I S L AB W BE LL U XA L ABV T S W EA U T G RC P O L C Z E RU S HUNR O UU K R SVKES T L VA L T U -5 T UR M L TZ A F E G Y T UN B W A N G A I S R L B N SA U J O R O M N Q A T M U S O lli v i e r U SA U SA U SA U SA C A N M EXPA N A R G B R A CH L PE RCR I B M U J A M J P NH K G T W N A U SPAK L KA T H A I DN I ND I ND S G P M YSP H L M N G U K D E U F R AESP CH E I T AP R T I R L I S L AB W BE LL U XA L ABV T S W EA U T G RC P O L C Z E RU S HUNR O UU K R SVKES T L VA L T U T UR M L TZ A F E G Y T UN B W A N G A I S R L B N SA U J O R O M N Q A T M U S EV C -1600-1400 -1200 -1000 U SA U SA -800 U SA U SA C A N M EXPA N A R G B R A C H L P E R C R I B M U J A M J P N H K G T W N A U S F o r m an P A K L K A T H A I D N I N D I N D S G P M Y S -600 P H L M N G U K D E U F R A E S P C H E I T A P R T I R L I S L A B W B E LL U X A L A B V T S W E A U T G R C P O L C Z E R U S H U N R O UU K R -400 S V K E S T L V A L T U T U R M L T Z A F E G Y T U N B W A N G A I S R L B N S A U J O R O M N Q A T M U S -2000200 -1500-1000-5000 (a)(c)(b) Eigenvector centrality of nodesForman-Ricci curvature of nodesOllivier-Ricci curvature of nodes
FIG. S14. Three dimensional plot showing the evolution of (a) normalized eigenvector centrality, (b)
Ollivier-Ricci (OR)curvature, and (c)
Forman-Ricci (FR) curvature for each of the 69 market indices (nodes) across the time-series of 172threshold networks S τ ( t ) of market indices computed with window size τ = 80 days and an overlapping shift of ∆ τ = 20 days,constructed by adding edges with correlation C τij ( t ) ≥ .
65 to the MST. We compute the OR and FR curvature of nodes inthe threshold networks based on the OR and FR curvature of incident edges on each node. Here, the colours of the marketindices are based on geographical region of their country. The four USA market indices, NASDAQ, NYSE, RUSSELL1000 andSPX, are labelled as USA1, USA2, USA3 and USA4, respectively, while the two Indian indices, NIFTY and SENSEX30, arelabelled as IND1 and IND2, respectively. It can be seen that there exist certain periods of time, when some of the marketindices in close geographical proximity display high (absolute) values while others display low values, indicative of the changesin the complex interactions and community structures in global financial network. TABLE S1. List of 69 global financial market indices across 65 countries considered in this work. For each financial marketindex, the table lists the country of origin, country code, index code, region code and geographical region. This dataset wasobtained from Bloomberg.
S. No. Country Countrycode Index code Region code Region
51 Ukraine UKR PFTS EME Europe Middle East52 Slovakia SVK SKSM EME Europe Middle East53 Estonia EST TALSE EME Europe Middle East54 Lativa LVA RIGSE EME Europe Middle East55 Lithuania LTU VILSE EME Europe Middle East56 Turkey TUR XU100 EME Europe Middle East57 Malta MLT MALTEX EME Europe Middle East58 South Africa ZAF JALSH AME Africa/Middle East59 Egypt EGY HERMES AME Africa/Middle East60 Tunisia TUN TUSISE AME Africa/Middle East61 Botswana BWA BGSMDC AME Africa/Middle East62 Nigeria NGA NGSEINDX AME Africa/Middle East63 Israel ISR TA-25 AME Africa/Middle East64 Lebanon LBN BLOM AME Africa/Middle East65 Saudi Arabia SAU SASEIDX AME Africa/Middle East66 Jordan JOR JOSMGNFF AME Africa/Middle East67 Oman OMN MSM30 AME Africa/Middle East68 Qatar QAT DSM AME Africa/Middle East69 Mauritius MUS SEMDEX AME Africa/Middle East
TABLE S2. Classification of network measures employed to characterize the networks of global market indices into those forunweighted or weighted graphs. For each measure evaluated in the weighted graph, we list the appropriate natural weight,strength or distance, which is used in the associated computation.