# Transfer Entropy between Communities in Complex Financial Networks

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## Abstract

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## 1. Introduction

## 2. Correlation Function and Sector Correlation

#### Mode Decomposition of Correlation Matrix

## 3. Community Structure in Complex Networks

## 4. Transfer Entropy

#### Rényi Transfer Entropy

## 5. Information Transfer between Business Sectors in Financial Markets

#### 5.1. New York SE

#### 5.2. London SE

#### 5.3. Tokyo SE

#### 5.4. Shanghai SE

#### 5.5. Hong Kong SE

#### 5.6. Information Transfer of Rare Events and Market Complexity

## 6. Conclusions and Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

B | Basic Materials | C | Consumer Goods | F | Financial Services |
---|---|---|---|---|---|

${B}_{ch}$ | chemistry | ${C}_{ca}$ | car manufacture | ${F}_{ba}$ | banks |

${B}_{ir}$ | iron & steel | ${C}_{el}$ | Electronics | ${F}_{co}$ | consulting services |

${B}_{me}$ | metals | ${C}_{fo}$ | food | ${F}_{hi}$ | health insurance |

${B}_{mi}$ | mining | ${C}_{sp}$ | sport & lifestyle | ${F}_{in}$ | investment services |

${B}_{nr}$ | natural resources | ${C}_{te}$ | textile | ${F}_{re}$ | real estate |

${C}_{ho}$ | household | ${F}_{ti}$ | travel & accident insurance | ||

I | Industrial goods | S | Services | T | Technology |

${I}_{ae}$ | aerospace & defense | ${S}_{ai}$ | airlines | ${T}_{co}$ | communications |

${I}_{ag}$ | agriculture industry | ${S}_{en}$ | entertainment | ${T}_{di}$ | digital services |

${I}_{hi}$ | heavy industry | ${S}_{me}$ | media | ${T}_{ht}$ | high-tech industry |

${I}_{in}$ | infrastructure | ${S}_{mo}$ | movie production | ${T}_{it}$ | information technology |

${I}_{ma}$ | machinery | ${S}_{se}$ | security services | ${T}_{on}$ | online services |

${I}_{ra}$ | railway construction | ${S}_{sh}$ | shipping | ${T}_{op}$ | optical & nano technology |

${I}_{sh}$ | ship construction | ${S}_{tr}$ | transportation | ${T}_{so}$ | software |

${}_{Ive}$ | vehicle industry | ${S}_{tv}$ | television | ${T}_{wi}$ | wireless services |

H | Healthcare | U | Utilities | Country | |

${H}_{hb}$ | health & beauty | ${U}_{en}$ | energy | ${}^{cn}$ | China |

${H}_{me}$ | medical equipment | ${U}_{ga}$ | gas & oil | ${}^{hk}$ | Hongkong |

${H}_{ph}$ | pharmacy | ${U}_{re}$ | renewable energy | ${}^{ma}$ | Macau |

${}^{in}$ | international | ||||

V | Various and conglomerates | ${}^{ge}$ | major German companies |

## References

- Gabaix, X.; Gopikrishnan, P.; Plerou, V.; Stanley, H.E. A theory of power-law distributions in financial market fluctuations. Nature
**2003**, 423, 267. [Google Scholar] [CrossRef] - Jung, W.S.; Chae, S.; Yang, J.S.; Moon, H.T. Characteristics of the Korean stock market correlations. Physica A
**2006**, 361, 263–271. [Google Scholar] [CrossRef] - Lillo, F.; Mantegna, R.N. Power-law relaxation in a complex system: Omori law after a financial market crash. Phys. Rev. E
**2003**, 68, 016119. [Google Scholar] [CrossRef] [PubMed] - Aste, T.; Shaw, W.; Di Matteo, T. Correlation structure and dynamics in volatile markets. New J. Phys.
**2010**, 12, 085009. [Google Scholar] [CrossRef] - Longin, F.; Solnik, B. Extreme correlation of international equity markets. J. Financ.
**2001**, 56, 649–676. [Google Scholar] [CrossRef] - Jiang, X.F.; Zheng, B. Anti-correlation and subsector structure in financial systems. Europhys. Lett.
**2012**, 97, 48006. [Google Scholar] [CrossRef] - Ramchand, L.; Susmel, R. Volatility and Cross Correlation across Major Stock Markets. J. Empir. Financ.
**1998**, 5, 397–416. [Google Scholar] [CrossRef] - Mattia, K.; Ashkenazy, Y.; Stanley, H.E. Multifractal properties of price fluctuations of stocks and commodities. Europhys. Lett.
**2003**, 61, 422. [Google Scholar] [CrossRef] - Schmitt, F.; Schertzer, D.; Lovejoy, S. Multifractal fluctuations in finance. Int. J. Theor. Appl. Financ.
**2000**, 3, 361–364. [Google Scholar] [CrossRef] - Di Matteo, T. Multi-scaling in finance. Quant. Financ.
**2007**, 7, 21–36. [Google Scholar] [CrossRef] - Ouyang, F.Y.; Zheng, B.; Jiang, X.F. Intrinsic Multi-Scale Dynamic Behaviors of Complex Financial Systems. PLoS ONE
**2015**, 10, e0139420. [Google Scholar] [CrossRef] [PubMed] - Hilgers, A.; Beck, C. Turbulent Behavior of Stock Exchange Indices and Foreign Currency Exchange Rates. Int. J. Bifurc. Chaos
**1997**, 7, 1855–1859. [Google Scholar] [CrossRef] - Preis, T.; Golke, S.; Paul, W.; Schneider, J.J. Multi-agent–based Order Book Model of financial markets. Europhys. Lett.
**2006**, 75, 510. [Google Scholar] [CrossRef] - Raberto, M.; Cincotti, S.; Focardi, S.M.; Marchesi, M. Agent-based simulation of a financial market. Physica A
**2001**, 299, 319–327. [Google Scholar] [CrossRef] - Preis, T.; Moat, H.S.; Stanley, H.E. Quantifying trading behavior in financial markets using google trends. Sci. Rep.
**2013**, 3, 1801. [Google Scholar] [CrossRef] - Garlaschelli, D. The weighted random graph model. New J. Phys.
**2009**, 11, 073005. [Google Scholar] [CrossRef] - Squartini, T.; Garlaschelli, D. Analytical maximum-likelihood method to detect patterns in real networks. New J. Phys.
**2011**, 13, 083001. [Google Scholar] [CrossRef] - Granger, C.W.J.; Huang, B.N.; Yang, C.W. A bivariate causality between stock prices and exchange rates: evidence from recent Asian flu. Quat. Rev. Econ. Financ.
**2000**, 40, 337–354. [Google Scholar] [CrossRef] - Cheung, Y.W.; Lilian, K.N. A causality-in-variance test and its application to financial market prices. J. Econ.
**1996**, 33, 33–48. [Google Scholar] [CrossRef] - Schreiber, T. Measuring Information Transfer. Phys. Rev. Lett.
**2000**, 85, 461. [Google Scholar] [CrossRef] - Marschinski, R.; Kantz, H. Analysing the information flow between financial time series. Eur. Phys. J. B
**2002**, 30, 275–281. [Google Scholar] [CrossRef] - Kwon, O.; Yang, J.S. Information flow between stock indices. Europhys. Lett.
**2008**, 82, 68003. [Google Scholar] [CrossRef] - Paluš, M.; Vejmelka, M. Directionality of coupling from bivariate time series: How to avoid false causalities and missed connections. Phys. Rev. E
**2007**, 75, 056211. [Google Scholar] [CrossRef] [PubMed] - Hlaváčková-Schindler, K.; Paluš, M.; Vejmelka, M. Causality detection based on information-theoretic approaches in time series analysis. Phys. Rep.
**2007**, 441, 1–46. [Google Scholar] [CrossRef] - Lungarella, M.; Pitti, A.; Kuniyoshi, Y. Information transfer at multiple scales. Phys. Rev. E
**2007**, 76, 056117. [Google Scholar] [CrossRef] - Barnett, L.; Barrett, A.B.; Seth, A.K. Granger causality and transfer entropy are equivalent for Gaussian variables. Phys. Rev. Lett.
**2009**, 103, 238701. [Google Scholar] [CrossRef] - Jizba, P.; Kleinert, H.; Shefaat, M. Rényi’s information transfer between financial time series. Physica A
**2012**, 391, 2971–2989. [Google Scholar] [CrossRef] - Sarbu, S. Rényi information transfer: Partial Rényi transfer entropy and partial Rényi mutual information. In Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy, 4–9 May 2014; pp. 5666–5670. [Google Scholar]
- Deng, Z.; Wu, J.; Guo, W. Rényi information flow in the Ising model with single-spin dynamics. Phys. Rev. E
**2014**, 90, 063308. [Google Scholar] [CrossRef] - Choi, H. Localization and regularization of normalized transfer entropy. Neurocomputing
**2014**, 139, 408–414. [Google Scholar] [CrossRef] - Kang, J.; Shang, P. Information flow and cross-correlation of Chinese stock markets based on transfer entropy and DCCA. Dyn. Contin. Discr. Impulsive Syst. Ser. B
**2013**, 20, 577–589. [Google Scholar] - Teng, Y.; Shang, P. Transfer entropy coefficient: Quantifying level of information flow between financial time series. Physica A
**2017**, 469, 60–70. [Google Scholar] [CrossRef] - Newman, M.E.J.; Girvan, M. Finding and evaluating community structure in networks. Phys. Rev. E
**2004**, 69, 026113. [Google Scholar] [CrossRef] [PubMed][Green Version] - Newman, M.E.J. Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA
**2006**, 49, 8577–8582. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rosvall, M.; Bergstrom, C.T. Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. USA
**2008**, 105, 1118–1123. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rosvall, M.; Axelsson, D.; Bergstrom, C.T. The map equation. Eur. Phys. J. Spec. Top.
**2009**, 178, 13–23. [Google Scholar] [CrossRef] - Kim, D.H.; Jeong, H. Systematic analysis of group identification in stock markets. Phys. Rev. E
**2005**, 72, 046133. [Google Scholar] [CrossRef][Green Version] - Pan, R.K.; Sinha, S. Collective behavior of stock price movements in an emerging market. Phys. Rev. E
**2007**, 76, 046116. [Google Scholar] [CrossRef][Green Version] - Sengupta, A.M.; Mitra, P.P. Distributions of singular values for some random matrices. Phys. Rev. E
**1999**, 60, 3389. [Google Scholar] [CrossRef][Green Version] - Dyson, F.J. Distribution of eigenvalues for a class of real symmetric matrices. Revis. Mex. Fís.
**1971**, 20, 231. [Google Scholar] - Noh, J.D. Model for correlations in stock markets. Phys. Rev. E
**2000**, 61, 5981. [Google Scholar] [CrossRef][Green Version] - Kruskal, J.B. On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc.
**1956**, 7, 48–50. [Google Scholar] [CrossRef] - Tumminello, M.; Aste, T.; Di Matteo, T.; Mantegna, R.N. A tool for filtering information in complex systems. Proc. Natl. Acad. Sci. USA
**2005**, 102, 10421–10426. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jiang, X.F.; Chen, T.T.; Zheng, B. Structure of local interactions in complex financial dynamics. Sci. Rep.
**2014**, 4, 5321. [Google Scholar] [CrossRef] [PubMed][Green Version] - Campbell, L.L. A Coding Theorem and Renyi’s Entropy. Inf. Control
**1965**, 8, 423–429. [Google Scholar] [CrossRef][Green Version] - Jizba, P.; Arimitsu, T. The world according to Rényi: thermodynamics of multifractal systems. Ann. Phys.
**2004**, 312, 17–59. [Google Scholar] [CrossRef] - Bercher, J.F. Source Coding with Escort Distributions and Renyi Entropy Bounds. Phys. Lett. A
**2009**, 373, 3235–3238. [Google Scholar] [CrossRef][Green Version] - Hentschel, H.G.E.; Procaccia, I. The infinite number of generalized dimensions of fractals and strange attractors. Physica D
**1983**, 8, 435–444. [Google Scholar] [CrossRef] - Beck, C.; Schlögl, F. Thermodynamics of Chaotic Systems; CUP: Cambridge, UK, 1993. [Google Scholar]
- Dimpfl, T.; Peter, F.J. Using transfer entropy to measure information flows between financial markets. Stud. Nonlinear Dyn. Econ.
**2013**, 17, 85–102. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Comparison of correlation network and transfer entropy flows between communities of New York stock exchange (SE). The size of the circle denotes the number of stocks in the community, the strength of the line/arrow denotes the strength of the interaction.

**Figure 2.**Information flows between communities for London SE. The size of the circle denotes the number of stocks in the community, the strength of the line/arrow denotes the strength of the interaction.

**Figure 3.**Information flows between communities for Tokyo SE. The size of the circle denotes the number of stocks in the community, the strength of the line/arrow denotes the strength of the interaction.

**Figure 4.**Information flows between communities for Shanghai SE. The size of the circle denotes amount of stocks in the community, the strength of the line/arrow denotes strength of the interaction.

**Figure 5.**Information flows between communities for Hong Kong SE. The size of the circle denotes amount of stocks in the community, the strength of the line/arrow denotes strength of the interaction.

**Table 1.**Statistical properties of the investigated markets. For each market, the table contains number of investigated stocks, average length of trading period, equal-time cross-correlation, lagged cross-correlation, Shannon and Rényi transfer entropy and number of communities obtained from the InfoMap algorithm.

Market | New York | London | Tokyo | Shanghai | Hong Kong |
---|---|---|---|---|---|

Index | S&P 500 | FTSE AS | NKY 225 | SSE 300 | HSI Comp. |

stocks | 485 | 527 | 185 | 283 | 411 |

av. length (days) | 3905 | 2206 | 2205 | 2929 | 3677 |

$\overline{C}\left(0\right)$ | 0.2122 | 0.0390 | 0.2185 | 0.2601 | 0.1448 |

$\overline{C}\left(1\right)$ | −0.0093 | 0.0069 | 0.0002 | 0.0193 | 0.0229 |

$\overline{STE}$ | 0.0047 | 0.0008 | 0.0027 | 0.0030 | 0.0019 |

${\overline{RTE}}_{0.75}$ | −0.0260 | −0.0062 | 0.0027 | −0.0218 | −0.0162 |

communities | 22 | 22 | 10 | 16 | 25 |

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**MDPI and ACS Style**

Korbel, J.; Jiang, X.; Zheng, B.
Transfer Entropy between Communities in Complex Financial Networks. *Entropy* **2019**, *21*, 1124.
https://doi.org/10.3390/e21111124

**AMA Style**

Korbel J, Jiang X, Zheng B.
Transfer Entropy between Communities in Complex Financial Networks. *Entropy*. 2019; 21(11):1124.
https://doi.org/10.3390/e21111124

**Chicago/Turabian Style**

Korbel, Jan, Xiongfei Jiang, and Bo Zheng.
2019. "Transfer Entropy between Communities in Complex Financial Networks" *Entropy* 21, no. 11: 1124.
https://doi.org/10.3390/e21111124