Network geometry and market instability
Areejit Samal, Hirdesh K. Pharasi, Sarath Jyotsna Ramaia, Harish Kannan, Emil Saucan, Jürgen Jost, Anirban Chakraborti
NNetwork geometry and market instability
Areejit Samal, ∗ Hirdesh K. Pharasi, ∗ Sarath Jyotsna Ramaia, HarishKannan, Emil Saucan, J¨urgen Jost,
6, 7 and Anirban Chakraborti
8, 9, 10 The Institute of Mathematical Sciences (IMSc),Homi Bhabha National Institute (HBNI), Chennai 600113 India Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico, Cuernavaca 62210, M´exico Department of Applied Mathematics and Computational Sciences,PSG College of Technology, Coimbatore 641004, India Department of Mathematics, University of California San Diego, La Jolla, California 92093, USA Department of Applied Mathematics, ORT Braude College, Karmiel 2161002, Israel Max Planck Institute for Mathematics in the Sciences, Leipzig 04103 Germany The Santa Fe Institute, Santa Fe, New Mexico 87501, USA 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 Universiity, Sonipat 131001, India Centro Internacional de Ciencias, Cuernavaca 62210, M´exico
The complexity of financial markets arise from the strategic interactions among agents tradingstocks, which manifest in the form of vibrant correlation patterns among stock prices. Over thepast few decades, complex financial markets have often been represented as networks whoseinteracting pairs of nodes are stocks, connected by edges that signify the correlation strengths.However, we often have interactions that occur in groups of three or more nodes, and these can-not be described simply by pairwise interactions but we also need to take the relations betweenthese interactions into account. Only recently, researchers have started devoting attention to thehigher-order architecture of complex financial systems, that can significantly enhance our ability toestimate systemic risk as well as measure the robustness of financial systems in terms of market effi-ciency. Geometry-inspired network measures, such as the Ollivier-Ricci curvature and Forman-Riccicurvature, can be used to capture the network fragility and continuously monitor financial dynamics.Here, we explore the utility of such discrete Ricci-type curvatures in characterizing the structureof financial systems, and further, evaluate them as generic indicators of the market instability. Forthis purpose, we examine the daily returns from a set of stocks comprising the USA S&P-500 andthe Japanese Nikkei-225 over a 32-year period, and monitor the changes in the edge-centric networkcurvatures. We find that the different geometric measures capture well the system-level features ofthe market and hence we can distinguish between the normal or ‘business-as-usual’ periods and allthe major market crashes. This can be very useful in strategic designing of financial systems andregulating the markets in order to tackle financial instabilities.
1. INTRODUCTION
For centuries science had thrived on the method of reductionism– considering the units of a system in isolation, andthen trying to understand and infer about the whole system. However, the simple method of reductionism has severelimitations [1], and fails to a large extent when it comes to the understanding and modeling the collective behaviorof the components of a ‘complex system’. More and more systems are now being identified as complex systems, andhence scientists are now embracing the idea of complexity as one of the governing principles of the world we live in.Any deep understanding of a complex system has to be based on a system-level description, since a key ingredientof any complex system is the rich interplay of nonlinear interactions between the system components. The financialmarket is truly a spectacular example of such a complex system, where the agents interact strategically to determinethe best prices of the assets. So new tools and interdisciplinary approaches are needed [2, 3], and already therehas been an influx of ideas from econophysics and complexity theory [4–8] to explain and understand economic andfinancial markets.A graph or network consists of nodes connected by edges. In real-world networks, nodes represent the components orentities, while edges represent the interactions or relationships between nodes. In the context of financial markets, thenodes represent the stocks and the edges characterize the correlation strengths (or their transformations into distance ∗ These authors contributed equally to this work a r X i v : . [ q -f i n . S T ] N ov measures). Such correlation-based networks have emerged as an important tool for the modeling and analysis ofcomplex financial system [9–16]. The computed cross-correlations among stock returns allows one to construct avariety of correlation-based networks including minimum spanning tree (MST) [9, 10] and threshold network [17].Since the correlations among stocks change with time, the underlying market dynamics generates very interestingcorrelation-based networks evolving over time.Introduced long ago by Gauss and Riemann, curvature is a central concept in geometry that quantifies the extentto which a space is curved [18]. In geometry, the primary invariant is curvature in its many forms. While curvaturehas connections to several essential aspects of the underlying space, in a specific case, curvature has a connection tothe Laplacian, and hence, to the ‘heat kernel’ on a network. Curvature also has connections to the Brownian motionand entropy growth on a network. Moreover, curvature is also related to algebraic topological aspects, such as thehomology groups and Betti numbers, which are relevant, for instance, for persistent homology and topological dataanalysis [19].Recently, there has been immense interest in geometrical characterization of complex networks [20–24]. Networkgeometry can reveal higher-order correlations between nodes beyond pairwise relationships captured by edges con-necting two nodes in a graph [25–27]. From the point of view of structure and dynamics of complex networks, edgesare more important than nodes, since the nodes by themselves cannot constitute a meaningful network. Hence, it maybe more important to develop edge-centric measures rather than node-centric measures to characterize the structureof complex networks [23, 28]. Surprisingly, geometrical concepts, especially, discrete notions of Ricci curvature, haveonly very recently been used as edge-centric network measures [22, 23, 28–31]. Furthermore, curvature has deep con-nections to related evolution equations that can be used to predict the long-time evolution of networks. Although theimportance of geometric measures like curvature have been understood for quite some time, yet there has been limitednumber of applications in the context of complex financial networks. In particular, Sandhu et al. [30] studied theevolution of Ollivier-Ricci curvature [32, 33] in threshold networks for the USA S&P-500 market over a 15-year span(1998-2013) and showed that the Ollivier-Ricci curvature is negatively correlated to the increase in market networkfragility. Consequently, Sandhu et al. [30] suggested that the Ollivier-Ricci curvature can be employed as an indicatorof market fragility and study the designing of (banking) systems and framing regulation policies to combat financialinstabilities such as the sub-prime crisis of 2007-2008.In this paper, we expand the study of geometry-inspired network measures for characterizing the structure of thefinancial systems to four notions of discrete Ricci curvature, and evaluate the curvature measures as generic indicatorsof the market instability. For this purpose, we examine the daily returns from a set of stocks comprising the USAS&P-500 and the Japanese Nikkei-225 over a 32-year period, and monitor the changes in the edge-centric geometriccurvatures. We find that the discrete Ricci curvature measures, especially Forman-Ricci curvature [23, 28], capturewell the system-level features of the market and hence we can distinguish between the normal or ‘business-as-usual’periods and all the major market crises (bubbles and crashes). Our study confirms that during a normal period themarket is very modular and heterogeneous, whereas during an instability (crisis) the market is more homogeneous,highly connected and less modular [12, 15, 16, 34]. These new insights will help us to understand tipping points,systemic risk, and resilience in financial networks, and enable us to develop monitoring tools required for the highlyinterconnected financial systems and perhaps forecast future financial crises and market slowdowns.
2. RICCI-TYPE CURVATURES FOR EDGE-CENTRIC ANALYSIS OF NETWORKS
The classical notion of Ricci curvature applies to smooth manifolds, and its classical definition requires tensorsand higher-order derivatives [18]. Thus, the classical definition of Ricci curvature is not immediately applicable inthe discrete context of graphs or networks. Therefore, in order to develop any meaningful notion of Ricci curvaturefor networks, one has to inspect the essential geometric properties captured by this curvature notion, and find theirproper analogues for discrete networks. To this end, it is essential to recall that Ricci curvature quantifies two essentialgeometric properties of the manifold, namely, volume growth and dispersion of geodesics. Further, since classical Riccicurvature is associated to a vector (direction) in smooth manifolds [18], in the discrete case of networks, it is naturallyassigned to edges [28]. Thus, notions of discrete Ricci curvatures are associated to edges rather than vertices or nodesin networks [28]. Note that no discretization of Ricci curvature for networks can capture the full spectrum of propertiesof the classical Ricci curvature defined on smooth manifolds, and thus, each discretization can shed a different lighton the analyzed networks [28]. In this work, we apply four notions of discrete Ricci curvature for networks to studythe correlation-based networks of stock markets.
Ollivier-Ricci curvature
Ollivier’s discretization [32, 33] of the classical Ricci curvature has been extensively used to analyze graphs ornetworks [22, 28–31, 35–39]. Ollivier’s definition is based on the following observation. In spaces of positive curvature,balls are closer to each other on the average than their centers, while in spaces of negative curvature, balls are fartheraway on the average than their centers. Ollivier’s definition extends this observation from balls (volumes) to measures(probabilities). More precisely, the Ollivier-Ricci (OR) curvature of an edge e between nodes u and v is defined as O ( e ) = 1 − W ( m u , m v ) d ( u, v ) (1)where m u and m v represent measures concentrated at nodes u and v , respectively, W denotes the Wasserstein distance[40] (also known as the earth mover’s distance) between the discrete probability measures m u and m v , and the cost d ( u, v ) is the distance between nodes u and v , respectively. Moreover, the Wasserstein distance W ( m u , m v ) whichgives the transportation distance between the two measures m u and m v , is given by W ( m u , m v ) = inf µ u,v ∈ (cid:81) ( m u ,m v ) (cid:88) ( u (cid:48) ,v (cid:48) ) ∈ V × V d ( u (cid:48) , v (cid:48) ) µ u,v ( u (cid:48) , v (cid:48) ) , (2)with (cid:81) ( m u , m v ) being the set of probability measures µ u,v that satisfy (cid:88) v (cid:48) ∈ V µ u,v ( u (cid:48) , v (cid:48) ) = m u ( u (cid:48) ) , (cid:88) u (cid:48) ∈ V µ u,v ( u (cid:48) , v (cid:48) ) = m v ( v (cid:48) ) (3)where V is the set of nodes in the graph. The above equation represents all the transportation possibilities of themass m u to m v . W ( m u , m v ) is the minimal cost or distance to transport the mass of m u to that of m v . Note thatthe distance d ( u (cid:48) , v (cid:48) ) in Eq. 2 is taken to be the path distance in the unweighted or weighted graph. Furthermore, theprobability distribution m u for u ∈ V has to be specified, and this is chosen to be uniform over neighbouring nodesof u [36].Simply stated, to determine the OR curvature of an edge e , in Eq. 1 one compares the average distance between theneighbours of the nodes u and v anchoring the edge e in an optimal arrangement with the distance between u and v itself. Importantly, the average distance between neighbours of u and v is evaluated as an optimal transport problemwherein the neighbours of u are coupled with those of v in such a manner that the average distance is as small aspossible. In the setting of discrete graphs or networks, OR curvature by definition captures the volume growth aspectof the classical notion for smooth manifolds, see e.g. [28] for details. In this work, we have computed the average ORcurvature of edges (ORE) in undirected and weighted networks using Eq. 1. Forman-Ricci curvature
Forman’s approach to the discretization of Ricci curvature [41] is more algebraic in nature and is based on therelation between the Riemannian Laplace operator and Ricci curvature. While devised originally for a much largerclass of discrete geometric objects than graphs, an adaptation to network setting was recently introduced by some ofus [23]. The Forman-Ricci (FR) curvature F ( e ) of an edge e in an undirected network with weights assigned to bothedges and nodes is given by [23] F ( e ) = w e w v w e + w v w e − (cid:88) e v ∼ e, e v ∼ e (cid:34) w v √ w e w e v + w v √ w e w e v (cid:35) (4)where e denotes the edge under consideration between nodes v and v , w e denotes the weight of the edge e , w v and w v denote the weights associated with the nodes v and v , respectively, e v ∼ e and e v ∼ e denote the set of edgesincident on nodes v and v , respectively, after excluding the edge e under consideration which connects the two nodes v and v . Furthermore, some of us have also extended the notion of FR curvature to directed networks [42]. In caseof discrete networks, FR curvature captures the geodesic dispersal property of the classical notion [28]. In this work,we have computed the average FR curvature of edges (FRE) in undirected and weighted networks using Eq. 4.From a geometric perspective, the FR curvature quantifies the information spread at the ends of edges in a network(Figure 1). The higher the information spread at the ends of an edge, the more negative will be the value of its FRcurvature. Specifically, an edge with high negative FR curvature is likely to have several neighbouring edges connectedto both anchoring nodes, and moreover, such an edge can be seen as a funnel at both ends, connecting many othernodes. Intuitively, such an edge with high negative FR curvature can be expected to have high edge betweennesscentrality as many shortest paths between other nodes, including those quite far in the network, are also likely to passthrough this edge. Previously, some of us have empirically shown a high statistical correlation between FR curvatureand edge betweenness centrality in diverse networks [28, 43]. Menger-Ricci curvature
The remaining two curvatures studied here are adaptations of curvatures for metric spaces to discrete graphs.Indeed, both unweighted and weighted graphs can be viewed as a metric space where the distance between any twonodes can be specified by the path length between them. Among notions of metric, and indeed, discrete curvature,Menger [44] has proposed the simplest and earliest definition whereby he defines the curvature of metric triangles T formed by three points in the space as the reciprocal 1 /R ( T ) of the radius R ( T ) of the circumscribed circle of atriangle T . Recently, some of us [45, 46] have adapted Menger’s definition to networks. Let ( M, d ) be a metric spaceand T = T ( a, b, c ) be a triangle with sides a, b, c , then the Menger curvature of T is given by K M ( T ) = (cid:112) p ( p − a )( p − b )( p − c ) a · b · c (5)where p = ( a + b + c ) /
2. In the particular case of a combinatorial triangle with each side of length 1, the aboveformula gives K M ( T ) = √ /
2. Furthermore, it is clear from the above formula that Menger curvature is alwayspositive. Following the differential geometric approach, the Menger-Ricci (MR) curvature of an edge e in a networkcan be defined as [45, 46] κ M ( e ) = (cid:88) T e ∼ e κ M ( T e ) , (6)where T e ∼ e denote the triangles adjacent to the edge e . Intuitively, if an edge is part of several triangles in thenetwork, such an edge will have high positive MR curvature (Figure 1). In this work, we have computed the averageMR curvature of edges (MRE) in undirected financial networks by ignoring the edge weights and using Eq. 6. Haantjes-Ricci curvature
We have also applied another notion of metric curvature to networks which is based on the suggestion of Finslerand was developed by his student Haantjes [47]. Haantjes defined the curvature of a metric curve as the ratio betweenthe length of an arc of the curve and that of the chord it subtends. More precisely, given a curve c in a metric space( M, d ), and given three points p, q, r on c , p 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) , (7)where l ( (cid:98) qr ) denotes the length, in the intrinsic metric induced by d , of the arc (cid:98) qr . In networks, (cid:98) qr can be replaced bya path π = v , v , . . . , v n between two nodes v and v n , and the subtending chord by the edge e = ( v , v n ) betweenthe two nodes. Recently, some of us [45, 46] have defined the Haantjes curvature of such a simple path π as κ H ( π ) = l ( π ) − l ( v , v n ) l ( v , v n ) , (8)where, if the graph is a metric graph, l ( v , v n ) = d ( v , v n ), that is the shortest path distance between nodes v and v n . In particular, for the combinatorial metric (or unweighted graphs), we obtain that κ H ( π ) = √ n −
1, where π = v , v , . . . , v n is as above. Note that considering simple paths in graphs concords with the classical definition ofHaantjes curvature, since a metric arc is, by its very definition, a simple curve. Thereafter, the Haantjes-Ricci (HR)curvature of an edge e [45, 46] can be defined as κ H ( e ) = (cid:88) π ∼ e κ H ( π ) , (9)where π ∼ e denote the paths that connect the nodes anchoring the edge e . Note that while MR curvature considersonly triangles or simple paths of length 2 between two nodes anchoring an edge in unweighted graphs, the HRcurvature considers even longer paths between the same two nodes anchoring an edge (Figure 1). Moreover, fortriangles endowed with the combinatorial metric, the two notions by Menger and Haantjes coincide, up to a universalconstant. In this work, we have computed the average HR curvature of edges (HRE) in undirected financial networksby ignoring the edge weights and using Eq. 9. Moreover, due to computational constraints, we only consider simplepaths π of length ≤
3. DATA AND METHODSData description
The data was collected from the public domain of Yahoo finance database [48] for two countries: USA S&P-500index and Japanese Nikkei-225 index. The investigation in this work spans a 32-year period from 2 January 1985(02-01-1985) to 30 December 2016 (30-12-2016). We analyzed the daily closure price data of N = 194 stocks for T = 8068 days for USA S&P-500 and N = 165 stocks for T = 7998 days for Japanese Nikkei-225 markets, which arepresent in the two markets for the entire 32-year period considered here. Cross-correlation and distance matrices
We present a study of time evolution of the cross-correlation structures of return time series for N stocks (Figure1). The daily return time series is constructed as r k ( t ) = ln P k ( t ) − ln P k ( t − P k ( t ) is the adjusted closingprice of the k -th stock at time t (trading day). Then, the cross-correlation matrix is constructed using equal-timePearson cross-correlation coefficients, C ij ( 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 , where i, j = 1 , . . . , N , t indicates the end date of the epoch of size τ days, and the means (cid:104) . . . (cid:105) as well as the standarddeviations σ k are computed over that epoch.Instead of working with the correlation coefficient C ij , we use the ‘ultrametric’ distance measure: d ij ( t ) = (cid:113) − C ij ) , such that 0 ≤ d ij ≤
2, which can be used for the construction of networks [10, 12, 17, 49].Here, we computed daily return cross-correlation matrix C τ ( t ) over the short epoch of τ = 22 days and shift of therolling window by ∆ τ = 5 days, for (a) N = 194 stocks of USA S&P-500 for a return series of T = 8068 days, and (b) N = 165 stocks of Japan Nikkei-225 for T = 7998 days, during the 32 year period from 1985 to 2016. We use epochsof τ = 22 days (one trading month) to obtain a balance between choosing short epochs for detecting changes and longones for reducing fluctuations. In the main text, we show results for networks constructed from correlation matriceswith overlapping windows of ∆ τ = 5 days, while in Electronic Supplementary Material (ESM), we show results fornetworks constructed from correlation matrices with non-overlapping windows of ∆ τ = 22 days. Network construction
For a given time window of τ days ending on trading day t , the distance matrix D τ ( t ) constructed from thecorrelation matrix between the 194 stocks in USA S&P-500 index or the 165 stocks in Japan Nikkei-225 index, canbe viewed as an undirected complete graph G τ ( t ) where the weight of an edge between stocks i and j is given bythe distance d ij . For the time window of τ days ending on trading day t , we start with this edge weighted completegraph G τ ( t ) and create the minimum spanning tree (MST) T τ ( t ) using Prim’s algorithm [50]. Thereafter, we addedges in G τ ( t ) with C ij ≥ .
75 to T τ ( t ) to obtain the graph S τ ( t ) (Figure 1). We will use the graph S τ ( t ) to computedifferent discrete Ricci curvatures and other network measures. We remark that the procedure used here to constructthe graph S τ ( t ) follows previous works [12, 30] on analysis of correlation-based networks of stock markets.Intuitively, the motivation behind the above method of graph construction can be understood as follows. Firstly,the MST method gives a connected (spanning) graph between all nodes (stocks) in the specific market. Secondly,the addition of edges between nodes (stocks) with correlation C ij ≥ .
75 ensures that the important edges are alsocaptured in the graph S τ ( t ). Common network measures
Given an undirected graph G ( V, E ) with the sets of vertices or nodes V and edges E , the number of edges is givenby the cardinality of set E , that is m = | E | , and the number of nodes is given by the cardinality of set V , that is n = | V | . The edge density of such a graph is given by the ratio of the number of edges m divided by the numberof possible edges, that is, mn ( n − . The average degree (cid:104) k (cid:105) of the graph gives the average number of edges per node,that is, (cid:104) k (cid:105) = mn . In case of an edge-weighted graph where a ij denotes the weight of the edge between nodes i and j ,one can also compute its average weighted degree (cid:104) k w (cid:105) which gives the average of the sum of the weights of the edgesconnected to nodes, that is, (cid:104) k w (cid:105) = m w n where m w = (cid:80) i,j ∈ V a ij . For any pair of nodes i and j in the graph, one cancompute the shortest path length d ij between them. Thereafer, the average shortest path length (cid:104) L (cid:105) is given by theaverage of the shortest path lengths between all pairs of nodes in the graph, that is, (cid:104) L (cid:105) = 1 n ( n − (cid:88) i,j ∈ V d ij . The diameter D is given by the maximum of the shortest paths between all pairs of nodes in the graph, that is, D = max { d ij ∀ i, j ∈ V } . The communication efficiency [51] of a graph is an indicator of its global ability to exchangeinformation across the network. The communication efficiency CE of a graph is given by CE = 1 n ( n − (cid:88) i (cid:54) = j ∈ V d ij . Modularity measures the extent of community structure in the network and community detection algorithms aim topartition the graph into communities such that the modularity Q attains the maximum value [52]. The modularity Q is given by the equation [52, 53] Q = 12 m w (cid:88) i (cid:54) = j ∈ V [ a ij − k i k j m w ] δ ( c i , c j )where k i and k j give the sum of weights of edges attached to nodes i and j , respectively, c i and c j give the communitiesof i and j , respectively, and δ ( c i , c j ) is equal to 1 if c i = c j else 0. Here, we use Louvain method [53] to computethe modularity of the edge-weighted networks. Network entropy is an average measure of graph heterogeneity as itquantifies the diversity of edge distribution using the remaining degree distribution q k [54]. q k denotes the probabibilityof a node to have remaining (excess) degree k and is given by q k = ( k +1) p k +1
NetworkX [55].
GARCH( p, q ) process
The generalized ARCH process GARCH( p, q ) was introduced by Bollerslev [56]. The variable x t , a strong whitenoise process, can be written in terms of a time-dependent standard deviation σ t , such that x t ≡ η t σ t , where η t is arandom Gaussian process with zero mean and unit variance.The simplest GARCH process is the GARCH(1,1) process, with Gaussian conditional probability distributionfunction σ t = α + α x t − + β σ t − , (10)where α > α ≥ β is an additional control parameter. One can rewrite Eq. 10 as a random multiplicativeprocess σ t = α + ( α η t − + β ) σ t − . (11)For calculating this we have used an in-built function from MATLAB garch ( https://in.mathworks.com/help/econ/garch.html ). Minimum Risk Portfolio
We calculated the minimum risk portfolio in the Markowitz framework, as a measure of risk-aversion of eachinvestor with maximized expected returns and minimized variance. In this model, the variance of a portfolio showsthe importance of effective diversification of investments to minimize the total risk of a portfolio. The Markowitzmodel minimizes w (cid:48) Ωw − φR (cid:48) w with respect to the normalized weight vector w, where Ω is the covariance matrixcalculated from the stock log-returns, φ is the measure of risk appetite of investor and R (cid:48) is the expected return ofthe assets. We set short-selling constraint, φ = 0 and w i ≥ MATLAB Portfolio ( https://in.mathworks.com/help/finance/portfolio.html ).
4. RESULTS AND DISCUSSION
We analyze here the time series of the logarithmic returns of the stocks in the USA S&P-500 and Japanese Nikkei-225markets over a period of 32 years (1985-2016) by constructing the corresponding Pearson cross-correlation matrices C τ ( t ). We then use cross-correlation matrices C τ ( t ) computed over time epochs of size τ = 22 days with eitheroverlapping or non-overlapping windows (i.e. shifts of ∆ τ = 5 or 22 days, respectively) and ending on trading days t to study the evolution of the correlation-based networks S τ ( t ) and corresponding network properties, especially edge-centric geometric measures. Figure 1 gives an overview of our evaluation of discrete Ricci curvatures in correlation-based threshold networks constructed from log-returns of market stocks. Figure 1(a) shows the daily log-returns overthe 32-year period (1985-2016). An arbitrarily chosen cross-correlation matrix C τ ( t ) over time epoch of τ = 22 daysand ∆ τ = 5 days ending on 04-05-2011 and corresponding distance matrix D τ ( t ) = (cid:112) − C τ ( t )) are shown infigure 1(b) and (c), respectively. The minimum spanning tree (MST) T τ ( t ) constructed from the distance matrix D τ ( t ) is shown in figure 1(d). Thereafter, a threshold network S τ ( t ) is constructed using MST T τ ( t ) and edgeswith C ij ≥ .
75, as shown in figure 1(e). The discrete Ricci curvatures are computed from the threshold networks.In figure 1(f), we show the evolution of the discrete curvatures in threshold networks over the 32-year period. Infigure 1(g), we motivate the four discrete Ricci curvatures considered here using a simple example network.A major goal of this research is to evaluate different notions of discrete Ricci curvature for their ability to unravel thestructure of complex financial networks and serve as indicators of market instabilities. Previously, Sandhu et al. [30]have analyzed the USA S&P-500 market over a period of 15 years (1998-2013) to show that the average Ollivier-Ricci(OR) curvature of edges (ORE) in threshold networks increases during periods of financial crisis. Here, we extend theanalysis by Sandhu et al. [30] to (a) two different stock markets, namely, USA S&P-500 and Japanese Nikkei-225,(b) a span of 32 years (1985-2016), (c) four traditional market indicators (namely, index log-returns r , mean marketcorrelation µ , volatility of the market index r estimated using GARCH(1,1) process, and risk σ P corresponding to theminimum risk Markowitz portfolio of all the stocks in the market), and (d) four notions of discrete Ricci curvaturefor networks. Since discretizations of Ricci curvature are unable to capture the entire properties of the classical Riccicurvature defined on continuous spaces, the four discrete Ricci curvatures evaluated here can shed light on differentproperties of analyzed networks [28]. In particular, some of us have introduced another discretization, Forman-Ricci(FR) curvature, to the domain of networks [23]. Note that OR curvature captures the volume growth property ofclassical Ricci curvature while FR curvature captures the geodesic dispersal property [28]. Nevertheless, our empiricalanalysis has shown that the two discrete notions, OR and FR curvature, are highly correlated in model and real-worldnetworks [28]. Importantly, in large networks, computation of the OR curvature is intensive while that of the FRcurvature is simple as the later depends only on immediate neighbours of an edge [28]. Therefore, we started byinvestigating the ability of FR curvature to capture the structure of complex financial networks.Figure 2 shows the comparisons of threshold networks, as well as the behaviour of index log-returns r and averageFR curvature of edges (FRE), for (a) bubble and (b) crash periods, of the USA S&P-500 market. The upper panelof figure 2(a) shows the threshold networks near the US Housing bubble period (2006-2007) at four distinct epochsof τ = 22 days ending on trading days t equal to 23-01-2006, 10-05-2006, 29-06-2006 and 06-11-2006, with threshold (g) v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v Evaluation of discrete curvatures on a toy example O R E F R E M R E HR E −350−250−150−50 − − − − − − − − − − − − − − − − − − − − − − − Discrete Ricci Curvatures (f)
Minimum Spanning Tree (MST) (d)
Final Network (e) + edges with
CDCSEGFNHCIDITMTTCUT 21.81.61.41.210.80.60.40.20
CD C
S E G F N HC I D I T M T T
C U T Distance Matrix (c)
CDCSEGFNHCIDITMTTCUT
CD C
S E G F N HC I D I T M TT C U T Correlation Matrix (b) -0.4 -0.2 r -0.2 r - - - - - - - - - - - - - - - - - - - - - - - - - -0.2 r N .......... .......... .......... (a) Time series of log-returns of stock prices in USA S&P-500
FIG. 1. Schematic diagram describing the evaluation of discrete Ricci curvatures in correlation-based networks constructedfrom log-returns of USA S&P-500 market stocks. (a)
Time series of log-returns over a 32-year period (1985-2016). (b)
Anarbitrarily chosen cross-correlation matrix C τ ( t ) for epoch ending on 04-05-2011. (c) Corresponding distance matrix D τ ( t ) = (cid:112) − C τ ( t )) used for the construction of the threshold network. (d) Minimum spanning tree (MST) T τ ( t ) constructed usingthe distance matrix D τ ( t ). (e) Threshold network S τ ( t ) constructed by adding edges with C ij ≥ .
75 to the MST T τ ( t ). (f ) Evolution of the average of four discrete Ricci curvatures for edges, namely, Ollivier-Ricci (ORE), Forman-Ricci (FRE),Menger-Ricci (MRE) and Haantjes-Ricci (HRE), computed using the threshold networks S τ ( t ) constructed from correlationmatrices over time epochs of τ = 22 days and overlapping shift of ∆ τ = 5 days. In this figure, C τ ( t ), D τ ( t ), T τ ( t ) and S τ ( t )shown in (b)-(e) correspond to the correlation frame denoted by vertical dashed line in (a). (g) Evaluation of discrete Riccicurvatures on a toy example network which is undirected and unweighted. Here, the edge between v and v has a highlynegative FR curvature as it depends on the degree of the two nodes or number of neighbouring edges. However, the edgebetween v and v has MR and HR curvature equal to zero as the edge under consideration is not part of any trianglesor cycles, respectively. Moreover, the edge between v and v also has a highly negative FR curvature as the degree of bothanchoring vertices is 4. In contrast, the edge between v and v has positive MR and HR curvature as the edge is part of atriangle which contributes to MR curvature and the edge is part of a triangle, a pentagon and a hexagon which contribute toHR curvature. For both the edges between v and v and between v and v , one can compute OR curvature, however, onlytriangles, quadrangles and pentagons make positive contribution to the OR curvature in unweighted and undirected networks.Specifically, the edge between v and v is part of a triangle, a pentagon and a hexagon, however, only the triangle and pentagonmake positive contribution to OR curvature. FIG. 2. (a) (Upper panel) Visualization of threshold networks for USA S&P-500 market around the US Housing bubble period(2006-2007) at four distinct epochs of τ = 22 days ending on trading days 23-01-2006, 10-05-2006, 29-06-2006, and 06-11-2006,with threshold C ij ≥ .
75. Here, the colour of the nodes correspond to the different communities determined by Louvainmethod for community detection. Threshold networks show higher number of edges and lower number of communities duringa bubble. (Lower panel) Plot shows the evolution of log-returns r of S&P-500 index (blue color line) and average Forman-Riccicurvature of edges (FRE) (sienna color line) for the period around the US Housing bubble. The FRE measure, constructedfrom threshold networks, is sensitive to both local (sectoral) and global fluctuations of the market, and shows a local minimum(more negative) during the bubble, whereas not much variation is seen in r (low volatility). (b) (Upper panel) Visualizationof threshold networks for USA S&P-500 market around the August 2011 stock markets fall at four distinct epochs of τ = 22days ending on 07-01-2011, 04-05-2011, 02-09-2011, and 03-02-2012 with threshold C ij ≥ .
75. Here, the threshold networkshows significantly higher number of edges and lower number of communities during the crash. (Lower panel) Plot shows theevolution of log-returns r of S&P 500 index (blue color line) and FRE (sienna color line) for the period around the August 2011stock markets fall. During the crash r has high fluctuations (high volatility) and FRE decreases significantly (local minima). C ij ≥ .
75. Number of edges and communities in these four threshold networks are 251 , , ,
220 and 13 , , , r (blue color line) and FRE (sienna color line) aroundthe US Housing bubble period are shown in the lower panel of figure 2(a). Threshold networks show higher number(996) of edges and lower number (11) of communities for high (negative) values of FRE, but there is not much variationof r . In ESM figure S1, we show that the FRE captures the same features for three other thresholds C ij ≥ . C ij ≥ .
65, and C ij ≥ .
85, and the numbers of edges and communities for each threshold is listed in ESM table S1.The measure FRE is sensitive to both local (sectoral) and global (market) fluctuations, and shows a local minimumduring bubble. Note that during a bubble, only a few sectors of the market perform well compared to the others (thestocks within the well-performing sectors are highly correlated, but the inter-sectoral correlations are low). It is hardto identify bubble by only monitoring the market index as the returns do not show much volatility. Figure 2(b) showsthe same for the period around the August 2011 stock markets fall at four distinct epochs of τ = 22 days ending ontrading days t equal to 07-01-2011, 04-05-2011, 02-09-2011 and 03-02-2012, with threshold C ij ≥ .
75. Number ofedges and communities in these four threshold networks are 197 , , ,
198 and 14 , , ,
15, respectively. Duringthe crash, the threshold network shows sufficiently higher number of edges and extremely low number of communities.In ESM figure S2, we show that the FRE captures the same features for three other thresholds C ij ≥ . C ij ≥ . C ij ≥ .
85, and the numbers of edges and communities for each threshold is listed in ESM table S1. The plots oflog-returns r of S&P-500 index (blue color line) and FRE (sienna color line) are shown around the August 2011 stockmarkets fall period in the lower panel of figure 2(b). Note that during a market crash r displays high volatility andFRE shows a significant decrease (local minimum). Earlier Sandhu et al. [30] had focussed on OR curvature as anindicator of crashes. Here, we additionally show that discrete Ricci curvatures, especially FR curvature, are sensitiveand can detect both crash (market volatility high) and bubble (market volatility low).It is often difficult to gauge the state of the market by simply monitoring the market index or its log-returns.There exist no simple definitions of a market crash or a market bubble. The market becomes extremely correlatedand volatile during a crash, but a bubble is even harder to detect as the volatility is relatively low and only certainsectors perform very well (stocks show high correlation) but the rest of the market behaves like normal or ‘business-as-usual’. Traditionally, the volatility of the market captures the ‘fear’ and the evaluated risk captures the ‘fragility’of the market. Some of us showed in our earlier papers that the mean market correlation and the spectral propertiesof the cross-correlation matrices can be used to study the market states [14] and identify the precursors of marketinstabilities [16]. A goal of this study is to show that the state of the market can be continuously monitored withcertain network-based measures. Thus, we next performed a comparative investigation of several network measures,especially, the four discrete notions of Ricci curvature.Figures 3 and 4 show for USA S&P-500 market and Japanese Nikkei-225 market, respectively, the temporal evolutionof the market indicators and network measures, mainly edge-centric Ricci curvatures computed from the correlationmatrices C τ ( t ) of epoch size τ = 22 days and overlapping shift of ∆ τ = 5 days, over a 32-year period (1985-2016).From top to bottom, the plots represent index log-returns r , mean market correlation µ , volatility of the marketindex r estimated using GARCH(1,1) process, risk σ P corresponding to the minimum risk Markowitz portfolio of allthe stocks in the market, network entropy (NP), communication efficiency (CE), average of OR, FR, MR and HRcurvature of edges. We find that the four Ricci-type curvatures, namely, ORE, FRE, MRE and HRE, along withthe other important indicators of the markets, viz., the log-returns r , volatility, minimum risk σ P and mean marketcorrelation µ , are excellent indicators of market instabilities (bubbles and crashes). We highlight that the four discreteRicci curvatures can capture important crashes and bubbles listed in table 1 in the two markets during the 32-yearperiod studied here.In ESM figure S3, we show the temporal evolution of the four discrete Ricci curvatures computed in thresholdnetworks S τ ( t ) obtained using three different thresholds, C ij ≥ .
65 (cyan color), C ij ≥ .
75 (dark blue color) and C ij ≥ .
85 (sienna color), for the two markets. It is seen that the absolute value of ORE, FRE, MRE and HREdecreases with the increase in the threshold C ij used to construct S τ ( t ). Regardless of the three thresholds used toconstruct the threshold networks S τ ( t ), we show that the four discrete Ricci curvatures are fine indicators of marketinstabilities.In previous work, Sandhu et al. [30] had contrasted the temporal evolution of ORE in threshold networks for USAS&P-500 market with NE, graph diameter and average shortest path length. Here, we have studied the temporalevolution of a larger set of network measures in threshold networks for USA S&P-500 and Japanese Nikkei-225 marketscomputed from the correlation matrices C τ ( t ) of epoch size τ = 22 days and overlapping shift of ∆ τ = 5 days, overa 32-year period (1985-2016). From figures 3 and 4, it is seen that NE and CE are also excellent indicators of marketinstabilities. In fact, we find that common network measures such as number of edges, edge density, average degree,average shortest path length, graph diameter, average clustering coefficient and modularity are also good indicatorsof market instabilities (ESM Figure S4).In ESM figures S5 and S6, we show the temporal evolution of the market indicators and several network measures1 r µ V o l a t ili t y σ P N E C E O R E F R E M R E HR E USA S&P-500( τ = 22 days ∆τ = 5 days) −0.3−0.10.10.20.40.60.80.050.100.150.020.060.102460.51.01.50.00.40.8−350−250−150−500408002000040000 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − FIG. 3. Evolution of the market indicators and edge-centric geometric curvatures for the USA S&P-500 market. From topto bottom, we plot the index log-returns r , mean market correlation µ , volatility of the market index r estimated usingGARCH(1,1) process, risk σ P corresponding to the minimum risk Markowitz portfolio of all the stocks in the market, networkentropy (NE), communication efficiency (CE), average of Ollivier-Ricci (ORE), Forman-Ricci (FRE), Menger-Ricci (MRE),and Haantjes-Ricci (HRE) curvature of edges evaluated from the correlation matrices C τ ( t ) of window size τ = 22 days andan overlapping shift of ∆ τ = 5 days. Four vertical dashed lines indicate the epochs of four important crashes– Black Monday1987, Lehman Brothers crash 2008, DJ Flash crash 2010, and August 2011 stock markets fall (see table 1). (including edge-centric Ricci curvatures) computed from the correlation matrices C τ ( t ) of epoch size τ = 22 daysand non-overlapping shift of ∆ τ = 22 days, over a 32-year period (1985-2016) in the two markets. It can be seenthat our results are also not dependent on the choice of overlapping or non-overlapping shift used to construct thecross-correlation matrices and threshold networks.Figure 5 shows the correlogram plots of (a) USA S&P-500 and (b) Japanese Nikkei-225 markets, for the traditionalmarket indicators (index returns r , mean market correlation µ , volatility, and minimum portfolio risk σ P ), networkproperties (NE and CE) and discrete Ricci curvatures (ORE, FRE, MRE and HRE), computed for epoch size τ = 22days and overlapping shift of ∆ τ = 5 days. In ESM figure S7, we show the correlogram plots for the traditionalmarket indicators and network properties including discrete Ricci curvatures computed for epoch size τ = 22 daysand non-overlapping shift of ∆ τ = 22 days in the two markets. Notably, FRE shows the highest correlation amongthe four discrete Ricci curvatures with the traditional market indicators in the two markets, and thus, FRE is anexcellent indicator for market risk that captures local to global system-level fragility of the markets. Furthermore,both NE and CE also have high correlation with the traditional market indicators. Therefore, these measures can beused to monitor the health of the financial system and forecast market crashes or downturns. Overall, we show thatFRE is a simple yet powerful tool for capturing the correlation structure of a dynamically changing network.2 r µ V o l a t ili t y σ P N E C E O R E F R E M R E HR E Japan Nikkei-225 ( τ = 22 days ∆τ = 5 days) −0.050.050.150.20.40.60.030.050.070.000.040.082460.51.01.50.00.40.8−200−10000204060050001000015000 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − FIG. 4. Evolution of the market indicators and edge-centric geometric curvatures for the Japanese Nikkei-225 market. Fromtop to bottom, we plot the index log-returns r , mean market correlation µ , volatility of the market index r estimated usingGARCH(1,1) process, risk σ P corresponding to the minimum risk Markowitz portfolio of all the stocks in the market, networkentropy (NE), communication efficiency (CE), average of Ollivier-Ricci (ORE), Forman-Ricci (FRE), Menger-Ricci (MRE),and Haantjes-Ricci (HRE) curvature of edges evaluated from the correlation matrices C τ ( t ) of window size τ = 22 days andan overlapping shift of ∆ τ = 5 days. Four vertical dashed lines indicate the epochs of four important crashes– Black Monday1987, Lehman Brothers crash 2008, DJ Flash crash 2010, and August 2011 stock markets fall (see table 1).
5. CONCLUSION
In this paper, we have employed geometry-inspired network curvature measures to characterize the correlationstructures of the financial systems and used them as generic indicators for detecting market instabilities (bubbles andcrashes). We reiterate here that it is often difficult to gauge the state of the market by simply monitoring the marketindex or its log-returns. There exist no simple definitions of a market crash or a market bubble. The market becomesextremely correlated and volatile during a crash, but a bubble is even harder to detect as the volatility is relativelylow and only certain sectors perform very well (stocks show high correlation) but the rest of the market behaveslike normal or ‘business-as-usual’. We have examined the daily returns from a set of stocks comprising the USAS&P-500 and the Japanese Nikkei-225 over a 32-year period, and monitored the changes in the edge-centric geometriccurvatures. Our results are very robust as we have studied two very different markets, and for a very long period of32 years with several interesting market events (bubbles and crashes; see table 1). We showed that the results are notvery sensitive to the choice of overlapping or non-overlapping windows used to construct the cross-correlation matricesand threshold networks (Figures 3-4; ESM Figures S4-S6). Further, the the choice of the thresholds for constructingnetworks also has little influence on their behaviour as indicators (ESM Figures S1-S3). We found that the four3 -0.070.03-0.13-0.06-0.07-0.020.05-0.04-0.03 0.470.650.750.740.56 0.640.430.510.43-0.50.460.36 0.590.660.540.550.41 0.890.780.650.41 0.870.90.75 0.870.71 10.930.94-0.83 1 1-0.6 1-0.75 1-0.96 1-0.91 1-0.98-0.87 10.94 1 -0.030.05-0.11-0.02-0.03-0.020.03-0.03-0.04 0.50.730.670.710.53 0.50.450.440.31-0.370.320.24 0.660.740.570.640.55 0.910.740.690.46 0.860.90.75 0.90.741 10.920.92-0.79 1 1-0.68 1-0.79 1-0.96 1-0.93 1-0.98-0.87 10.94 1
USA S&P-500 ( τ = 22 days ∆τ = 5 days) Japan Nikkei-225 ( τ = 22 days ∆τ = 5 days) (a) (b) rVolatilityNECEOREFREMREHRE µσ PrVolatilityNECEOREFREMREHRE µσ P r V o l a t ili t y N E C E O R E F R E M R E HR E µ σ P r V o l a t ili t y N E C E O R E F R E M R E HR E µ σ P -1-0.8-0.6-0.4-0.200.20.40.60.81 FIG. 5. Correlogram plots of (a)
USA S&P-500 and (b)
Japan Nikkei-225 markets, for the traditional market indicators(index returns r , mean market correlation µ , volatility, and minimum risk portfolio σ P ), global network properties (networkentropy NE and communication efficiency CE) and discrete Ricci curvatures for edges (Ollivier-Ricci ORE, Forman-Ricci FRE,Menger-Ricci MRE, and Haantjes-Ricci HRE), computed for epochs of size τ = 22 days and overlapping shift ∆ τ = 5 days.TABLE 1. List of major crashes and bubbles in stock markets of USA and Japan between 1985-2016 [57–62]. Serial number Major crashes and bubbles Period Affected region th mini crash 13-10-1989 USA3 Early 90s recession 1990 USA4 Mini crash due to Asian financial crisis 27-10-1997 USA5 Lost decade 2001-2010 Japan6 9/11 financial crisis 11-09-2001 USA, Japan7 Stock market downturn of 2002 09-10-2002 USA, Japan8 US Housing bubble 2005-2007 USA9 Lehman Brothers crash 16-09-2008 USA, Japan10 Dow Jones (DJ) Flash crash 06-05-2010 USA, Japan11 Tsunami and Fukushima disaster 11-03-2011 Japan12 August 2011 stock markets fall 08-08-2011 USA, Japan13 Chinese Black Monday and 2015-2016 sell off 24-08-2015 USA different notions of discrete Ricci curvature captured well the system-level features of the market and hence we wereable to distinguish between the normal or ‘business-as-usual’ periods and all the major market crises (bubbles andcrashes) using the network-centric indicators. Our studies confirmed that during a normal period the market is verymodular and heterogeneous, whereas during an instability (crisis) the market is more homogeneous, highly connectedand less modular. Also, we find from these geometric measures that there are succinct and inherent differences inthe two markets, USA S&P-500 and Japan Nikkei-225. Importantly, among four Ricci-type curvature measures,the Forman-Ricci curvature of edges (FRE) correlates highest with the traditional market indicators and acts as anexcellent indicator for the system-level fear (volatility) and fragility (risk) for both the markets. These new insightsmay help us in future to better understand tipping points, systemic risk, and resilience in financial networks, andenable us to develop monitoring tools required for the highly interconnected financial systems and perhaps forecastfuture financial crises and market slowdowns. These can be further generalized to study other economic systems, andmay thus enable us to understand the highly complex and interconnected economic-financial systems.4 Author contributions
A.S. and A.C. designed research; A.S., H.K.P., S.J.R., H.K., E.S., J.J. and A.C. performed research and analyzeddata; A.S., H.K.P. and S.J.R. prepared the figures; A.S. and A.C. supervised the research; A.S., E.S., J.J. and A.C.wrote the manuscript with input from the other authors. All authors have read and approved the manuscript.
Acknowledgement
A.S. acknowledges financial support from Max Planck Society Germany through the award of a Max Planck PartnerGroup in Mathematical Biology. H.K.P. is grateful for financial support provided by UNAM-DGAPA and CONACYTProyecto Fronteras 952. E.S. and J.J. acknowledge support from the German-Israeli Foundation (GIF) Grant I-1514-304.6/2019. A.C. and H.K.P. acknowledge support from the projects UNAM-DGAPA-PAPIIT AG100819 andIN113620, and CONACyT Project Fronteras 201.
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TABLE S1. Number of edges ( S τ ( t ) for USA S&P-500market at eight distinct epochs of τ = 22 days ending on trading days 23-01-2006, 10-05-2006, 29-06-2006, 06-11-2006, 07-01-2011, 04-05-2011, 02-09-2011, and 03-02-2012 (around the US Housing bubble period (2006-2007) and August 2011 stockmarkets fall crisis), constructed using four different thresholds, C ij ≥ . C ij ≥ . C ij ≥ .
75, and C ij ≥ . US Housing Bubble Networks August 2011 Fall Crash NetworksEnd Date 23-01-2006 07-01-2011
Threshold 0.55 0.65 0.75 0.85 0.55 0.65 0.75 0.85
End date 10-05-2006 04-05-2011
Threshold 0.55 0.65 0.75 0.85 0.55 0.65 0.75 0.85
End date 29-06-2006 02-09-2011
Threshold 0.55 0.65 0.75 0.85 0.55 0.65 0.75 0.85
End date 06-11-2006 03-02-2012
Threshold 0.55 0.65 0.75 0.85 0.55 0.65 0.75 0.85 (a)(b)(d)(c) . . . USA S&P-500 ( τ = 22 days ∆τ = 5 days) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -0.1-0.0500.050.1 r -50050 F R E FIG. S1. Visualization of threshold networks for USA S&P-500 market around the US Housing bubble period (2006-2007) atfour distinct epochs of τ = 22 days ending on trading days 23-01-2006, 10-05-2006, 29-06-2006, and 06-11-2006, with thresholds( a ) C ij ≥ .
55, ( b ) C ij ≥ .
65, and ( c ) C ij ≥ .
85. Here, the colour of the nodes correspond to the different communitiesdetermined by Louvain method for community detection. The number of edges and communities in S τ ( t ) for different thresholdsare shown in table S3. ( d ) Plot shows the evolution of log-returns r of S&P-500 index (blue color line) and average Forman-Riccicurvature of edges (FRE) (sienna color line) computed using threshold networks with C ij ≥ .
75 for the period around the USHousing bubble. ( τ = 22 days ∆τ = 5 days) . . . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -0.1-0.0500.050.1 r -400-300-200-1000100 F R E (a)(b)(d)(c) FIG. S2. Visualization of threshold networks for USA S&P-500 market around the August 2011 stock markets fall at fourdistinct epochs of τ = 22 days ending on 07-01-2011, 04-05-2011, 02-09-2011, and 03-02-2012 with thresholds ( a ) C ij ≥ . b ) C ij ≥ .
65, and ( c ) C ij ≥ .
85. Here, the colour of the nodes correspond to the different communities determined byLouvain method for community detection. The number of edges and communities in S τ ( t ) for different thresholds are shown intable S3. ( d ) Plot shows the evolution of log-returns r of S&P-500 index (blue color line) and average Forman-Ricci curvatureof edges (FRE) (sienna color line) computed using threshold networks with C ij ≥ .
75 for the period around the August 2011stock markets fall crisis. − . . . . O R E − − F R E M R E HR E − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − USA S&P-500( τ = 22 days ∆τ = 5 days) (a) − . . . − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − O R E F R E M R E HR E Japan Nikkei-225( τ = 22 days ∆τ = 5 days) (b) FIG. S3. Comparison plots for the four edge-centric geometric curvatures, namely, Ollivier-Ricci (ORE), Forman-Ricci (FRE),Menger-Ricci (MRE) and Haantjes-Ricci (HRE) in threshold networks S τ ( t ) obtained using three different thresholds C ij ≥ . C ij ≥ .
75 (dark blue color), and C ij ≥ .
85 (sienna color) for ( a ) USA S&P-500 and ( b ) Japan Nikkei-225 markets.The curvature measures are calculated for time epochs of τ = 22 days and overlapping shift of ∆ τ = 5 days over the period(1985-2016). The absolute value of ORE, FRE, MRE and HRE decreases with the increase in the threshold C ij used toconstruct S τ ( t ). Four vertical dashed lines correspond to the epochs of four important crashes (Black Monday 1987, LehmanBrothers crash 2008, DJ Flash crash 2010, and August 2011 stock markets fall) listed in the table 1 of the main text. rNumberof edgesEdgedensityAveragedegreeAverageweighteddegreeAveragepathlengthDiameterClusteringcoefficientModularity −0.3−0.2−0.10.00.10500010000150000.00.20.40.60.805010015002040608026101410203040500.00.20.40.60.80.00.20.40.60.8 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − USA S&P-500( τ = 22 days ∆τ = 5 days) (a) Japan Nikkei-225( τ = 22 days ∆τ = 5 days) −0.050.050.150400080000.00.20.40.620601000204060 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − rNumberof edgesEdgedensityAveragedegreeAverageweighteddegreeAveragepathlengthDiameterClusteringcoefficientModularity (b) FIG. S4. Evolution of network properties for ( a ) USA S&P-500 and ( b ) Japanese Nikkei-225 markets evaluated from thecorrelation matrices C τ ( t ) of window size τ = 22 days and an overlapping shift of ∆ τ = 5 days over the period (1985-2016).From top to bottom, we compare the plot of index log-returns r with common network measures, namely, number of edges,edge density, average degree, average weighted degree, average path length, diameter, clustering coefficient and modularity. −0.2−0.10.00.1 r µ V o l a t ili t y σ P N E C E O R E −200−1000 F R E M R E HR E − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − USA S&P-500( τ = 22 days ∆τ = 22 days) (a) −0.20.00.2 r µ V o l a t ili t y σ P N E C E O R E −200−1000 F R E M R E HR E − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Japan Nikkei-225 ( τ = 22 days ∆τ = 22 days) (b) FIG. S5. Evolution of the market indicators and edge-centric geometric curvatures for ( a ) USA S&P-500 and ( b ) JapaneseNikkei-225 markets. From top to bottom, we plot the index log-returns r , mean market correlation µ , volatility of the marketindex r estimated using GARCH(1,1) process, risk σ P corresponding to the minimum risk Markowitz portfolio of all the stocksin the market, network entropy (NE), communication efficiency (CE), average of Ollivier-Ricci (ORE), Forman-Ricci (FRE),Menger-Ricci (MRE), and Haantjes-Ricci (HRE) curvature of edges evaluated from the correlation matrices C τ ( t ) of windowsize τ = 22 days and an non-overlapping shift of ∆ τ = 22 days. Four vertical dashed lines indicate the epochs of four importantcrashes– Black Monday 1987, Lehman Brothers crash 2008, DJ Flash crash 2010, and August 2011 stock markets fall. rNumberof edgesEdgedensityAveragedegreeAverageweighteddegreeAveragepathlengthDiameterClusteringcoefficientModularity −0.2−0.10.00.120006000100000.00.20.40.60408012002040604 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − USA S&P-500( τ = 22 days ∆τ = 22 days) (a) −0.20.00.20400080000.00.20.40.62060100103050 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − rNumberof edgesEdgedensityAveragedegreeAverageweighteddegreeAveragepathlengthDiameterClusteringcoefficientModularity Japan Nikkei-225( τ = 22 days ∆τ = 22 days) (b) FIG. S6. Evolution of network properties for ( a ) USA S&P-500 and ( b ) Japanese Nikkei-225 markets evaluated from thecorrelation matrices C τ ( t ) of window size τ = 22 days and an non-overlapping shift of ∆ τ = 22 days over the period (1985-2016). From top to bottom, we compare the plot of index log-returns r with common network measures, namely, numberof edges, edge density, average degree, average weighted degree, average path length, diameter, clustering coefficient andmodularity. (a) (b) rVolatilityNECEOREFREMREHRE µ rVolatilityNECEOREFREMREHRE µ r V o l a t ili t y N E C E O R E F R E M R E HR E µ r V o l a t ili t y N E C E O R E F R E M R E HR E µ -1-0.8-0.6-0.4-0.200.20.40.60.81 -0.30.03-0.51-0.27-0.3-0.250.28-0.26-0.21 0.280.660.760.750.59 0.350.280.270.21-0.250.210.14 0.620.660.550.530.4 0.90.780.660.45 0.880.90.76 0.880.741 10.930.95-0.83 1 1-0.59 1-0.76 1-0.95 1-0.91 1-0.99-0.89 10.95 1 -0.250.01-0.44-0.24-0.3-0.290.31-0.31-0.28 0.380.710.910.910.670.70.52 0.380.340.320.24-0.280.240.18 0.630.720.580.640.57 0.90.750.670.44 0.870.90.75 0.90.741 1-0.79 1 1-0.67 1-0.78 1-0.96 1-0.94 1-0.98-0.87 10.94 1 σ P σ P σ P σ P USA S&P-500 ( τ = 22 days ∆τ = 22 days) Japan Nikkei-225 ( τ = 22 days ∆τ = 22 days) FIG. S7. Correlogram plots of (a)
USA S&P-500 and (b)
Japan Nikkei-225 markets, for the traditional market indicators(index returns r , mean market correlation µ , volatility, and minimum portfolio risk σ P ), network properties (network entropy(NE) and communication efficiency (CE)) and discrete Ricci curvatures for edges (Ollivier-Ricci ORE, Forman-Ricci FRE,Menger-Ricci MRE, and Haantjes-Ricci HRE), computed for epochs of size τ = 22 days and non-overlapping shift of ∆ ττ