# Network Analysis of the Shanghai Stock Exchange Based on Partial Mutual Information

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## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Experimental Results

**Figure 1.**Minimally-spanning tree for the Shanghai Stock Exchange based on partial mutual information between studied stocks. Stocks with the highest node degrees have been named with their ticker symbols. The size of the nodes is proportional to their node degree.

**Figure 2.**Planar maximally-filtered graph for the Shanghai Stock Exchange based on partial mutual information between studied stocks. Stocks with the highest node degrees have been named with their ticker symbols. The size of the nodes is proportional to their node degree.

**Figure 3.**Pearson’s correlation coefficients between distances associated with the three used dependency measures (δ based on Pearson’s correlation (Corr), d based on mutual information (MI) and D based on partial mutual information (PMI)) for all pairs of stocks within the studied set. It is clear that using partial mutual information changes the analysis very slightly with regards to mutual information, but both give significantly different results from the analysis using Pearson’s correlation coefficients.

**Table 1.**Percentage of links between instruments belonging to the same economic sector in all links within the studied networks. As a reference, the same is shown for an unrestricted network or a full graph. Both mutual information (d) and partial mutual information (D) reproduce the sector structure from price changes slightly more accurately than Pearson’s correlation coefficient (δ), which is in agreement with similar studies of other markets. MST, minimally-spanning tree; PMFG, maximally-filtered graph.

Distance | MST | PMFG |
---|---|---|

δ | 64.97% | 53.85% |

d | 66.24% | 56.84% |

D | 66.88% | 58.12% |

None | 13.96% | 13.96% |

**Figure 4.**Degree distributions (with fitted power law and log-normal distribution) for: (

**a**) a minimally-spanning tree based on correlation; (

**b**) a planar maximally-filtered graph based on correlation; (

**c**) MST based on mutual information; (

**d**) PMFG based on MI; (

**e**) MST based on partial mutual information; and (

**f**) PMFG based on PMI.

**Table 2.**The sum of node degrees for stocks belonging to the studied economic sectors in MST (1) and PMFG (3) based on partial mutual information, the average node degree for stocks belonging to the studied sectors in MST (2) and PMFG (4) based on partial mutual information and the average earnings per share (EPS) ratio for stocks belonging to the studied sectors (5). There is a negative correlation between the EPS ratio in a sector and the sector’s average importance in the network.

Sector | (1) | (2) | (3) | (4) | (5) |
---|---|---|---|---|---|

Communications | 7 | 2.33 | 22 | 7.33 | 0.29 |

Consumer Discretionary | 46 | 1.92 | 136 | 5.67 | 0.58 |

Consumer Staples | 14 | 1.56 | 37 | 4.11 | 2.17 |

Energy | 37 | 2.06 | 117 | 6.50 | 0.49 |

Financials | 80 | 2.11 | 228 | 6.00 | 0.92 |

Healthcare | 17 | 2.13 | 47 | 5.88 | 0.54 |

Industrials | 38 | 2.00 | 113 | 5.95 | 0.15 |

Materials | 55 | 2.12 | 171 | 6.58 | 0.33 |

Technology | 10 | 1.43 | 39 | 5.57 | 0.53 |

Utilities | 10 | 1.67 | 26 | 4.33 | 0.46 |

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- P.A. Samuelson. “Proof That Properly Anticipated Prices Fluctuate Randomly.” Ind. Manag. Rev. 6 (1965): 41–49. [Google Scholar]
- J. Tobin. “A general equilibrium approach to monetary theory.” J. Money Credit Bank. 1 (1969): 15–29. [Google Scholar] [CrossRef]
- H. Markowitz. “Portfolio Selection.” J. Financ. 7 (1952): 77–91. [Google Scholar]
- R. Mantegna. “Hierarchical structure in financial markets.” Eur. Phys. J. B 11 (1999): 193–197. [Google Scholar] [CrossRef]
- P. Cizeau, M. Potters, and J. Bouchaud. “Correlation structure of extreme stock returns.” Quant. Financ. 1 (2001): 217–222. [Google Scholar] [CrossRef]
- K. Forbes, and R. Rigobon. “No contagion, only interdependence: measuring stock market comovements.” J. Financ. 57 (2002): 2223–2261. [Google Scholar] [CrossRef]
- B. Podobnik, and H. Stanley. “Detrended Cross-Correlation Analysis: A New Method for Analyzing Two Nonstationary Time Series.” Phys. Rev. Lett. 100 (2008). [Google Scholar] [CrossRef]
- T. Aste, W. Shaw, and T.D. Matteo. “Correlation structure and dynamics in volatile markets.” New J. Phys. 12 (2010): 085009. [Google Scholar] [CrossRef]
- D. Kenett, T. Preis, G. Gur-Gershgoren, and E. Ben-Jacob. “Quantifying Meta-Correlations in Financial Markets.” Europhys. Lett. 99 (2012): 38001. [Google Scholar] [CrossRef]
- G. Bonanno, F. Lillo, and R. Mantegna. “High-frequency Cross-correlation in a Set of Stocks.” Quant. Financ. 1 (2001): 96–104. [Google Scholar] [CrossRef]
- M. Tumminello, T.D. Matteo, T. Aste, and R. Mantegna. “Correlation based networks of equity returns sampled at different time horizons.” Eur. Phys. J. B 55 (2007): 209–217. [Google Scholar] [CrossRef]
- M. Munnix, R. Schafer, and T. Guhr. “Impact of the tick-size on financial returns and correlations.” Physica A 389 (2010): 4828–4843. [Google Scholar] [CrossRef]
- G. Bonanno, N. Vandewalle, and R.N. Mantegna. “Taxonomy of stock market indices.” Phys. Rev. E 62 (2000): 7615–7618. [Google Scholar] [CrossRef]
- S. Maslov. “Measures of globalization based on cross-correlations of world financial indices.” Physica A 301 (2001): 397–406. [Google Scholar] [CrossRef]
- S. Drozdz, F. Grummer, F. Ruf, and J. Speth. “Towards identifying the world stock market cross-correlations: DAX versus Dow Jones.” Physica A 294 (2001): 226–234. [Google Scholar] [CrossRef]
- R. Coelho, C. Gilmore, B. Lucey, P. Richmond, and S. Hutzler. “The evolution of interdependence in world equity markets-Evidence from minimum spanning trees.” Physica A 376 (2007): 455–466. [Google Scholar] [CrossRef]
- C.G. Gilmore, B.M. Lucey, and M. Boscia. “An ever-closer union? Examining the evolution of linkages of European equity markets via minimum spanning trees.” Physica A 387 (2008): 6319–6329. [Google Scholar]
- M. Eryigit, and R. Eryigit. “Network structure of cross-correlations among the world market indices.” Physica A 388 (2009): 3551–3562. [Google Scholar] [CrossRef]
- D.M. Song, M. Tumminello, W.X. Zhou, and R.N. Mantegna. “Evolution of worldwide stock markets, correlation structure, and correlation-based graphs.” Phys. Rev. E 84 (2011): 026108. [Google Scholar] [CrossRef]
- L. Sandoval, and I. Franca. “Correlation of financial markets in times of crisis.” Physica A 391 (2012): 187–208. [Google Scholar] [CrossRef]
- M. McDonald, O. Suleman, S. Williams, S. Howison, and N.F. Johnson. “Detecting a currency’s dominance or dependence using foreign exchange network trees.” Phys. Rev. E 72 (2005): 046110. [Google Scholar] [CrossRef]
- G. Bonanno, G. Caldarelli, F. Lillo, and R. Mantegna. “Topology of correlation-based minimal spanning trees in real and model markets.” Phys. Rev. E 68 (2003): 046130. [Google Scholar] [CrossRef]
- R.N. Mantegna, and H.E. Stanley. Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge, United Kingdom: Cambridge University Press, 2000. [Google Scholar]
- B. Rosser. “Econophysics and Economic Complexity.” Adv. Complex Syst. 11 (2008): 745–760. [Google Scholar] [CrossRef]
- W.A. Brock, D.A. Hsieh, and B. LeBaron. Nonlinear Dynamics, Chaos, and Instability. Statistical Theory and Economic Evidence. Cambridge, UK: MIT Press, 1991. [Google Scholar]
- M. Qi. “Nonlinear Predictability of Stock Returns Using Financial and Economic Variables.” J. Bus. Econ. Stat. 17 (1999): 419–429. [Google Scholar]
- D. McMillan. “Nonlinear predictability of stock market returns: Evidence from nonparametric and threshold models.” Int. Rev. Econ. Financ. 10 (2001): 353–368. [Google Scholar] [CrossRef]
- D. Sornette, and J. Andersen. “A Nonlinear Super-Exponential Rational Model of Speculative Financial Bubbles.” Int. J. Mod. Phys. C 13 (2002): 171–188. [Google Scholar] [CrossRef]
- K. Oh, and K. Kim. “Analyzing stock market tick data using piecewise nonlinear model.” Expert Syst. Appl. 22 (2002): 249–255. [Google Scholar] [CrossRef]
- P.H. Franses, and D.V. Dijk. “Forecasting stock market volatility using (non-linear) Garch models.” J. Forecast. 15 (1996): 229–235. [Google Scholar] [CrossRef]
- A. Abhyankar, L. Copeland, and W. Wong. “Nonlinear Dynamics in Real-Time Equity Market Indices: Evidence from the United Kingdom.” Econ. J. 105 (1995): 864–880. [Google Scholar] [CrossRef]
- P. Chen. “A Random Walk or Color Chaos on the Stock Market? Time-Frequency Analysis of S&P Indexes.” Stud. Nonlinear Dyn. Econom. 1 (1996): 87–103. [Google Scholar]
- A. Abhyankar, L. Copeland, and W. Wong. “Uncovering Nonlinear Structure in Real-Time Stock Market Indices.” J. Bus. Econ. Stat. 15 (1997): 1–14. [Google Scholar]
- P.A. Ammermann, and D.M. Patterson. “The cross-sectional and cross-temporal universality of nonlinear serial dependencies: Evidence from world stock indices and the Taiwan Stock Exchange.” Pac. Bas. Financ. J. 11 (2003): 175–195. [Google Scholar] [CrossRef]
- D. Hsieh. “Testing for Nonlinear Dependence in Daily Foreign Exchange Rates.” J. Bus. 62 (1989): 339–368. [Google Scholar] [CrossRef]
- R. Meese, and A. Rose. “An Empirical Assessment of Non-Linearities in Models of Exchange Rate Determination.” Rev. Econ. Stud. 58 (1991): 603–619. [Google Scholar] [CrossRef]
- C. Brooks. “Testing for non-linearity in daily sterling exchange rates.” Appl. Financ. Econ. 6 (1996): 307–317. [Google Scholar] [CrossRef]
- M. Qi, and Y. Wu. “Nonlinear prediction of exchange rates with monetary fundamentals.” J. Empir. Financ. 10 (2003): 623–640. [Google Scholar] [CrossRef]
- M. Baghli. “A model-free characterization of causality.” Econ. Lett. 91 (2006): 380–388. [Google Scholar] [CrossRef]
- P. Fiedor. “Networks in financial markets based on the mutual information rate.” Phys. Rev. E 89 (2014): 052801. [Google Scholar] [CrossRef]
- P. Fiedor. “Information-theoretic approach to lead-lag effect on financial markets.” Eur. Phys. J. B 87 (2014): 168. [Google Scholar] [CrossRef]
- D. Kenett, M. Tumminello, A. Madi, G. Gur-Gershgoren, R. Mantegna, and E. Ben-Jacob. “Dominating clasp of the financial sector revealed by partial correlation analysis of the stock market.” PLoS ONE 5 (2010): e15032. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- D. Kenett, X. Huang, I. Vodenska, S. Havlin, and H. Stanley. “Partial correlation analysis: Applications for financial markets.” Available online: http://arxiv.org/abs/1402.1405 (accessed on 27 May 2015).
- C.K. Tse, J. Liua, and F.C. Lau. “A network perspective of the stock market.” J. Empir. Financ. 17 (2010): 659–667. [Google Scholar] [CrossRef]
- M. Tumminello, F. Lillo, and R.N. Mantegna. “Correlation, hierarchies, and networks in financial markets.” J. Econ. Behav. Organ. 75 (2010): 40–58. [Google Scholar] [CrossRef]
- G. Charness, F. Feri, M.A. Meléndez-Jiménez, and M. Sutter. “Experimental Games on Networks: Underpinnings of Behavior and Equilibrium Selection.” Econometrica 82 (2014): 1615–1670. [Google Scholar]
- B. Zheng, X.F. Jiang, and P.Y. Ni. “A mini-review on econophysics: Comparative study of Chinese and western financial markets.” Chin. Phys. B 23 (2014): 078903. [Google Scholar] [CrossRef]
- H. Han, L.Y. Wu, and N.N. Song. “Financial Networks Model Based on Random Matrix.” Act. Phys. Sin. 63 (2014): 138901. [Google Scholar]
- W.Q. Huang, X.T. Zhuang, and S. Yao. “A network analysis of the Chinese stock market.” Physica A 388 (2009): 2956–2964. [Google Scholar] [CrossRef]
- X.F. Jiang, and B. Zheng. “Anti-correlation and subsector structure in financial systems.” EPL 97 (2012): 48006. [Google Scholar] [CrossRef]
- J. Shen, and B. Zheng. “Cross-correlation in financial dynamics.” EPL 86 (2009): 48005. [Google Scholar] [CrossRef]
- Y. Ma, X. Zhuang, and L. Li. “Complex Networks Based Research on Characteristics of the Shanghai Stock Exchange Rich-Club.” J. Northeast. Univ. Nat. Sci. 32 (2011): 447–451. [Google Scholar]
- Y. Ma, X. Zhuang, and L. Li. “Community and robustness of the correlated networks of stock ownership structure.” Syst. Eng. Theory Prac. 31 (2011): 2241–2251. [Google Scholar]
- Y. Ma, X. Zhuang, and L. Li. “Study on the Effect of the Regional Development Strategies and the Rejuvenation of Old Industrial Bases in Northeastern China based on Cross-shareholding Networks of the Listed Companies.” J. Syst. Manag. 20 (2011): 715–721. [Google Scholar]
- B. Shia, and T. You. “Linkage effect and community structure of CITIC industry indices before and after the financial crisis.” J. Bus. Econ., 2014. (in print). [Google Scholar]
- W. Feller. An Introduction to Probability Theory and Its Applications. New York, NY, USA: Wiley, 1971. [Google Scholar]
- C.E. Shannon. “A Mathematical Theory of Communication.” Bell Syst. Tech. J. 27 (1948): 379–423, 623–656. [Google Scholar] [CrossRef]
- S. Frenzel, and B. Pompe. “Partial Mutual Information for Coupling Analysis of Multivariate Time Series.” Phys. Rev. Lett. 99 (2007): 204101. [Google Scholar] [CrossRef] [PubMed]
- J. Beirlant, E. Dudewicz, L. Gyorfi, and E. van der Meulen. “Nonparametric entropy estimation: An overview.” Int. J. Math. Stat. Sci. 6 (1997): 17–39. [Google Scholar]
- G. Darbellay, and I. Vajda. “Estimation of the information by an adaptive partitioning of the observation space.” IEEE Trans. Inform. Theory 45 (1999): 1315–1321. [Google Scholar] [CrossRef]
- L. Paninski. “Estimation of entropy and mutual information.” Neural Comput. 15 (2003): 1191–1254. [Google Scholar] [CrossRef]
- C. Daub, R. Steuer, J. Selbig, and S. Kloska. “Estimating mutual information using b-spline functions—An improved similarity measure for analyzing gene expression data.” BCM Bioinform. 5 (2004): 118–130. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- W. Nemenman, W. Bialek, and R. de Ruyter van Steveninck. “Entropy and information in neural spike trains: Progress on the sampling problem.” Phys. Rev. E 69 (2004): 056111. [Google Scholar] [CrossRef]
- J. Bonachela, H. H, and M. Munoz. “Entropy estimates of small data sets.” J. Phys. A Math. Theor. 41 (2008): 202001. [Google Scholar] [CrossRef]
- T. Schurmann, and P. Grassberger. “Entropy estimation of symbol sequences.” Chaos 6 (1996): 414–427. [Google Scholar] [CrossRef] [PubMed]
- T. Cover, and J. Thomas. Elements of Information Theory. New York, NY, USA: John Wiley & Sons, 1991. [Google Scholar]
- A. Kraskov, H. Stogbauer, R. Andrzejak, and P. Grassberger. “Hierarchical clustering using mutual information.” Europhys. Lett. 70 (2005): 278–289. [Google Scholar] [CrossRef]
- M. Tumminello, T. Aste, T.D. Matteo, and R.N. Mantegna. “A tool for filtering information in complex systems.” PNAS 102 (2005): 10421–10426. [Google Scholar] [CrossRef] [PubMed]
- G. Miller. “An additivity theorem for the genus of a graph.” J. Comb. Theory 43 (1987): 25–47. [Google Scholar] [CrossRef]
- G. Ringel. Map Color Theorem. Berlin, Germany: Springer, 1974. [Google Scholar]
- P. Fiedor. “Frequency Effects on Predictability of Stock Returns.” In Proceedings of the IEEE Computational Intelligence for Financial Engineering & Economics 2014; Edited by A. Serguieva, D. Maringer, V. Palade and R.J. Almeida. London, UK: IEEE, 2014, pp. 247–254. [Google Scholar]
- P. Fiedor. “Sector strength and efficiency on developed and emerging financial markets.” Physica A 413 (2014): 180–188. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

You, T.; Fiedor, P.; Hołda, A.
Network Analysis of the Shanghai Stock Exchange Based on Partial Mutual Information. *J. Risk Financial Manag.* **2015**, *8*, 266-284.
https://doi.org/10.3390/jrfm8020266

**AMA Style**

You T, Fiedor P, Hołda A.
Network Analysis of the Shanghai Stock Exchange Based on Partial Mutual Information. *Journal of Risk and Financial Management*. 2015; 8(2):266-284.
https://doi.org/10.3390/jrfm8020266

**Chicago/Turabian Style**

You, Tao, Paweł Fiedor, and Artur Hołda.
2015. "Network Analysis of the Shanghai Stock Exchange Based on Partial Mutual Information" *Journal of Risk and Financial Management* 8, no. 2: 266-284.
https://doi.org/10.3390/jrfm8020266