A perspective on correlation-based financial networks and entropy measures
Vishwas Kukreti, Hirdesh K. Pharasi, Priya Gupta, Sunil Kumar
AA perspective on correlation-based financialnetworks and entropy measures
Vishwas Kukreti , Hirdesh K. Pharasi , Priya Gupta , and Sunil Kumar School of Computational and Integrative Sciences, Jawaharlal Nehru University, New Delhi-110067, India Instituto de Ciencias F´ısicas, Universidad Nacional Aut ´onoma de M ´exico, Cuernavaca-62210, M ´exico Atal Bihari Vajpayee School of Management & Entrepreneurship, Jawaharlal Nehru University, New Delhi, India Department of Physics, Ramjas College, University of Delhi, New Delhi, India ** [email protected], [email protected] * [email protected] ABSTRACT
In this brief review, we critically examine the recent work done on correlation-based networks in financial systems. The structureof empirical correlation matrices constructed from the financial market data changes as the individual stock prices fluctuatewith time, showing interesting evolutionary patterns, especially during critical events such as market crashes, bubbles, etc. Weshow that the study of correlation-based networks and their evolution with time is useful for extracting important information ofthe underlying market dynamics. We, also, present our perspective on the use of recently developed entropy measures suchas structural entropy and eigen-entropy for continuous monitoring of correlation-based networks.
INTRODUCTION
There has been a growing interest in understanding the dynamics of complex systems in the real world. Network sciencehas emerged as an important tool and convenient framework for analyzing a wide variety of social, financial, biologicaland informative complex systems . Network science began with the seminal papers of Erd˝os and R´enyi , who proposedrandom graphs in 1959-60. Random graphs have been used to compare real-world complex networks, since the late 1990s,when a number of scientists started using networks in physical, social, and biological domains. Watts and Strogatz renewedthe modeling of networks with “small world” properties – random graphs with small diameter but highly clustered likeregular lattices. Barab´asi and Albert investigated the properties of vertex connectivity of large networks with “scale-free”power-law distributions . These were followed by a flood of papers (see, e.g., ). Thus, network science emerged as animportant tool for studying different phenomena – spread of infectious diseases , economic development , detection,characterisation, identification of long-term precursors to financial crashes , construction of robust sustainable infrastructureand technological networks , etc.Here, we briefly review the role of network science in understanding complex financial markets. Firstly, for uncovering thestructure of complex interactions among stocks at a particular instant of time (static picture). For this purpose, one starts withthe cross-correlations among stocks returns and then uses various methods of network analysis, such as threshold networks,Minimum Spanning Tree (MST) , Planar Maximally Filtered Graph (PMFG) , etc. Using these methods, one can identifystocks (or sectors) that are strongly or weakly correlated and also study their hierarchy in the network structures. Correlationsamong stocks change with time, and the underlying dynamics of the market becomes very intriguing. Secondly, a continuousmonitoring of financial market becomes very useful and necessary , since there are sizable fluctuations during crashes andbubbles. Thus, we discuss here the role of entropy measures in continuous monitoring of the financial market (dynamic picture). CORRELATION-BASED NETWORKS
Mantegna studied the hierarchical structures of correlation-based networks in financial markets . Later similar studies ofcorrelation-based networks were made (see, e.g., ). These correlation-based networks provide easy visual representation ofmultivariate time series and extract meaningful information about the complex market dynamics. The analysis of evolutionof correlation-based networks provides a deep understanding of the underlying market trends, especially during periods ofcrisis . We briefly discuss a few methods to construct correlation-based networks from empirical correlation matrix (ECM):MST, threshold network and PMFG. a r X i v : . [ q -f i n . C P ] A p r inimum Spanning Tree MST is constructed by using the distances d i j = (cid:112) ( − C i j ) , where C i j s are the elements of ECM (correlations betweenpairs of stocks i , j = , . . . , N in a market for a specific time window), such that all N vertices (stocks) are connected withexactly N − , currencyexchange rates , global foreign exchange dynamics . Among disadvantages, there is the fact that the order and classificationof nodes in a cluster of MST is not robust, and often sensitive to minor changes in correlations or spurious correlations.Therefore, for improvement of results, either noise suppression techniques like Random Matrix Theory (RMT) and powermapping have been used, or alternative algorithms such as PMFG, Triangulated Maximally Filtered Graph (TMFG), AverageLinkage Minimum Spanning Tree (ALMST), Directed Bubble Hierarchical Tree (DBHT) have been proposed. Instead ofusing pair-wise Pearson correlations, partial correlations and mutual information have also been computed for some studies .MST is useful for studying the taxonomy or the sector classification , with potential applications in portfolio optimization.Researchers have also carried out analysis of dynamical correlations using MST . This type of dynamical studies has thepotential of catching important changes and continuous monitoring of the market. By calculating correlation using rollingwindow of different lengths, one could construct and analyze the temporal networks. From such analyses, it has been found thatconfiguration of MST structure changes during crisis and there exist strong correlations between normalized tree length and theinvestment diversification potential .Figure 1 shows the analysis for three periods ending at: (first column) 23/07/1985, (second column) 08/01/2007, and(third column) 17/06/2010. Figure 1A-C show the heat-map of correlation matrices in three different periods, where wehave analyzed the S&P 500 market (consisting of 194 stocks) with an epoch of 40 days which is shifted by 20 days from1985 − Figure 1D-F , which have been generated using the Prim’s algorithm.Different colors in MSTs correspond to different sectors in the market. One can easily view the changes in the structures ofMSTs in different periods of market evolution.
Threshold Networks
In this approach, an adjacency matrix is constructed by applying a threshold value in the correlation or fixing the number ofedges of the network . It filters out the strongest correlations by putting a certain value of threshold and discard all remainingcorrelations below the value of this threshold. A small threshold value gives rise to a completely connected graph, whileincreasing value of threshold makes the connections less. Thus, one can tune the threshold in order to get the weakly orstrongly connected nodes. For a particular value of threshold, as correlation matrices change with time, the threshold networksalso change, as shown in Figure 1G-I corresponding to the ECMs shown in
Figure 1A-C . Here the Fruchterman- Reingold(forced-based) layout has been used to visualize the threshold networks.One drawback of the threshold networks is that there is a loss of information; when we put a threshold value to the correlationmatrix we discard some nodes and edges. Also, threshold networks are very sensitive to the noise (random fluctuations).However, threshold networks have been constructed and applied in different areas of finance . Planar Maximally Filtered Graph
PMFG is a network drawn in a plane, such that there are no intersecting links . If N is total number of stocks, then itcontains 3 ( N − ) links. The PMFG has the advantage that it retains the structure of MST (which contains N − . However, PMFG has a disadvantage that there exists a certainarbitrariness in its results, as there is an embedding of data from higher dimension to lower dimension with a zero genus. Figure 1J-L show the planar maximally filtered graph of matrices shown in
Figure 1A-C . We find significant changes in thestructures of PMFGs in different periods of analysis.Recently, PMFG and threshold network have been combined to produce PMFG-based threshold networks . Thresholdnetworks of the financial market are constructed over multi-scale and at multi-threshold . Robustness: Noise suppression and community detection
We have seen that many of the correlation-based networks have shown clustering with communities of stocks. Thus, communitydetection in network science serves as an important technique for extraction of the clustering information from ECM of amultivariate time series. Several community detection algorithms have been proposed . The problem is that differentcommunity detection algorithms yield different results for the same ECM. So, often domain knowledge is required to determinewhat is a sensible or meaningful community.Further, we have seen that many of the networks are sensitive to noise or spurious correlations. Properties of randommatrices have turned out to be useful in reducing noise and thus understanding dynamics of complex systems . An B CD E FG H IJ K L
Figure 1.
Static correlation-based networks: Analysis of S&P 500 market with 194 stocks (epoch of 40 days) for threedifferent periods: first, second, and third columns are corresponding to 23/07/1985 (normal period), 08/01/2007 (bubbleperiod), and 17/06/2010 (crash period) respectively. (A-C) are heat maps of correlation matrices of different periods. Minimumspanning trees using Prim’s algorithm are shown in (D-F) . From (G-I) , correlation based threshold networks at a particularvalue of threshold. Planar maximally filtered graphs (J-L) for three different periods.There are significant changes in structureof networks in different periods of analysis. nsemble of random matrices, also known as stationary or standard random (Gaussian) matrix ensemble , introduced byWigner , have been applied to many studies in physics, biology, finance, etc. (see Refs. and references therein).The probability distribution of eigenvalues of Wishart orthogonal ensemble (WOE) follows Marcenko-Pastur distribution .The ECM of a complex system is normally compared with WOE . It has been observed from eigenvalues statistics ofempirical correlation matrices that the few largest eigenvalues show deviations from the Wishart ensemble. Note that Pearsoncross-correlation assumes that the time series are stationary, which are valid for shorter lengths of time series. However, ifthe number of time series are greater than the lengths of time series, then corresponding ECMs are noisy and highly singular.For such short time series, there is a great need of noise suppression in correlation matrix to extract actual correlations. Thereare different techniques for suppressing the noise in correlation matrix . Notably, any ECM of financial market can bedecomposed into partial correlations, consisting of market C M , group C G and random C R modes, respectively . It enablesus to identify the dominant stocks, sectors and inherent structures of the market. Recently, detailed analyses of ECMs usingthese approaches have been carried out to understand the complexity in dynamics of stock market . It has been found thatduring the crisis, the eigenvalue spectrum behaves very differently from one corresponding to a normal period. ENTROPY MEASURES
Entropy measures provide an easy way for continuous monitoring of the financial market, and also prove useful in various otherapplications in finance, as summarized below.Phillippatos and Wilson had used entropy in selection of possible efficient portfolios by applying a mean-entropy approachon a randomly selected 50 securities over 14 years . Using a hybrid entropy model, Xu et al. have evaluated the asset risk dueto the randomness of the system . In 1996, Buchen and Kelly used the principle of maximum entropy for option pricing toestimate the distribution , which fitted accurately with a known probability density function. The principle of the minimumcross-entropy principle (MCEP) has been very useful in finance, which was developed by Kullback and Leibler . Later,Frittelli discovered sufficient conditions to give a interpretation of the minimal entropy martingale measure .Entropy has also been used to understand the financial hazards as well as to construct an early warning indicator forpredicting systematic risks . Maasoumi and Racine examined the predictability of the market returns using entropy measureand found that it is capable to detect the nonlinear dependence within the time series of market returns as well as betweenreturns and other prediction variables obtained from other models . Recently, Ricci curvature and entropy have been used toconstruct an economic indicator for market fragility and systemic risk . Very recently, Almog et al. presented a perspectiveon the use of entropy measures such as structural entropy , which is computed from the communities in correlation-basednetworks. Chakraborti et al. computed the eigen-entropy from the eigen-vector centrality of the stocks in the correlation-basednetwork . Below, we compare the structural entropy and eigen-entropy . Structural Entropy
Recently, the concept of Structural Entropy (SE) has been used in monitoring the dynamical correlation based networks offinancial market . The SE resolved the problem of choosing different period of crisis and extracting substantial informationfrom the large network of stock market. The SE measures the amount of heterogeneity of the network nodes with an assumptionthat more connected nodes share common attributes than others. The authors assume the nature of clusters as independentsub-units of the network. The process of calculating the structural entropy involves two steps: (i) Calculation of an optimalpartition function which places every node in a certain cluster using a community detection algorithm. (ii) Analysing thepartition function and extracting the representative value of the diversity level. Consider a network G with N nodes. Thecommunity detection algorithm partitions G nodes in M communities. Let σ denote the N -dimensional vector where the i -th component denotes the community assigned to node i . Calculate M -dimensional probability vector P ≡ (cid:2) c N , c N , . . . , c M N (cid:3) ,where c i is the size of community i . It is proportional size of the cluster in the network. Then, the formula for Shannon’sentropy is S ≡ H ( P ) ≡ − ∑ Mi = P i ln ( P i ) in terms of probability vector P . Structural entropy S of the network provides a wayto continuously monitor the state of the network. However, it is sensitive to the choice of community detection algorithmemployed in detecting communities. This arbitrariness makes the calculation of entropy dependent on the choice of the userand hence is not universal. Eigen-entropy
Very recently, the concept of eigen-entropy was used in studying financial markets . It is computed from eigenvector centralityof the network obtained from the short time series correlation matrices . In order to capture the global feature of thenetwork, every node is ranked by its eigenvector centrality and then entropy formula from information theory is used to computeeigen-entropy. Let graph G ( E , V ) consisting of vertices V and edges E . Let A be the adjacency matrix for G ( E , V ) with a i j = i and j are present and a i j =
0, if they are not. The sum of all the centralities connected to thevertices is proportional to centrality of the vertex. The adjacency matrix A satisfies the matrix equation Ax = λ x , where λ is BC Figure 2.
Dynamic analysis of stock market (Top to Bottom): Continuous monitoring of S&P 500 market over a period of1985 − A . Temporal evolution of a new measurement ‘eigen-entropy’ H ( τ ) ,calculated from eigen-vector centralities of filtered correlation matrices (after removal of noise using power map) is shown in B .It can be seen that eigen-entropy easily quantifies the order and disorder in the stock market. Evolution of structural entropy S ( τ ) calculated by using community detection algorithm is shown in C . The dashed vertical lines are corresponding to differentperiods (normal, bubble, and crash) whose static results are shown in Figure 1.the largest eigenvalue of A . A is a symmetric positive semi-definite matrix with all non-negative eigenvalues and orthogonaleigenvectors. According to the Perron-Frobenius theorem, any square matrix with all positive entries has a unique solutioncorresponding to the maximum eigenvalue and its eigenvector with all positive components. Then v th component of thecorresponding eigenvector gives the relative eigen-centrality score of the node v in the network. For an absolute score onemust normalize the eigenvector, i.e., ∑ Ni = p i =
1. The disorderness and randomness of the system uniquely be measured byeigen-entropy and defined as H = − ∑ Ni = p i ln p i . Higher the disorder of the system higher the eigen-entropy.Empirical correlation matrix of the market can be decomposed in two logical ways: (i) into three separated modes i.e.market mode C M , the group mode C G and the random mode C R , where it is arbitrary to chose the range of eigenvaluecorresponding to the group mode C G and the random mode C R and (ii) into a market mode C M and group-random modes C GR ,with no arbitrariness in the system. C M & C GR is the preferable decomposition and corresponding eigen-entropy H M and H GR and calculated as A = | C M | (matrix element-wise) and A = | C GR | (matrix element-wise), respectively. The eigen-entropycomputed using above method gives a simple yet robust measure to quantify the randomness of the financial market withoutusing any arbitrary thresholds. Further Charkraborti et al. investigated the relative-entropy which separates the phase spacebased on their disorder . The evolution dyanamics of these relative entropies in the phase space show phase-separation withpossible order-disorder transitions. These results are certainly of deep significance for the understanding of financial marketbehavior and designing strategies for risk management. Figure 2 shows how the entropy measures can be used for continuous monitoring of the financial markets.
Figure 2A-C show the evolution of S&P 500 market over a period of 1985 − r ( τ ) , eigen-entropies H ( τ ) , and structuralentropy S ( τ ) , respectively. Three vertical dashed line are corresponding to epochs ending at 23/07/1985, 08/01/2007, and17/06/2010. CONCLUDING REMARKS
In this review, we have discussed different methods for analysis of static and dynamic correlation-based networks of financialmarkets, and also studied how entropy measures can be used to identify normal, bubble, and crash periods. Specifically, we ave compared the recently developed concepts of structural entropy and eigen-entropy.The prediction of collapses of financial markets using traditional economic theories has been a daunting task. These new andalternate methods have the potential use of continuous monitoring and understanding of the complex structures and dynamicsof financial markets. These are a few of the attempts physicists have made for generation of early warning signals for crisis, andthese methods can be used for timely intervention.
CONFLICT OF INTEREST STATEMENT
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could beconstrued as a potential conflict of interest.
AUTHOR CONTRIBUTIONS
SK and HKP designed the idea, wrote the main manuscript text and prepared figures. VK and PG contributed to the literaturereview. All authors reviewed the manuscript.
ACKNOWLEDGEMENTS
The authors are grateful to Anirban Chakraborti, Hrishidev, Suchetana Sadhukhan, Kiran Sharma and Thomas H. Seligman fortheir critical inputs. HKP acknowledges postdoctoral fellowship provided by UNAM-DGAPA. This research was supportedin part by the International Centre for Theoretical Sciences (ICTS) during a visit of VK, PG and SK for participating inthe Summer research program on Dynamics of Complex Systems (Code: ICTS/Prog-DCS2019/07). The topic editors areacknowledged for supporting this open access publication.
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