Abstract
We develop a topology data analysis-based method to detect early signs for critical transitions in financial data. From the time-series of multiple stock prices, we build time-dependent correlation networks, which exhibit topological structures. We compute the persistent homology associated to these structures in order to track the changes in topology when approaching a critical transition. As a case study, we investigate a portfolio of stocks during a period prior to the US financial crisis of 2007–2008, and show the presence of early signs of the critical transition.
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References
Adler, R.J., Bobrowski, O., Borman, M.S., et al.: Persistent homology for random fields and complexes. Inst. Math. Stat. Collect. 6, 124–143 (2010)
Berwald, J., Gidea, M.: Critical transitions in a model of a genetic regulatory system. Math. Biol. Eng. 11 (4), 723–740 (2014)
Berwald, J., Gidea, M., Vejdemo-Johansson, M.: Automatic recognition and tagging of topologically different regimes in dynamical systems. Discontinuity Nonlinearity Complex. 3 (4), 413–426 (2015)
Carstens, C.J., Horadam, K.J.: Persistent homology of collaboration networks. Math. Probl. Eng. 2013, 1–7 (2013)
Chazal, F., Cohen-Steiner, D., Guibas, L.J., et al.: Gromov-Hausdorff stable signatures for shapes using persistence. In: Computer Graphics Forum (Proc. SGP 2009), pp. 1393–1403 (2009)
Chazal, F., de Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedicata 173, 193–214 (2013). doi:10.1007/s10711-013-9937-z
Cohen-Steiner, D., Edelsbrunner, H., Harer, J., et al.: Lipschitz functions have L p -stable persistence. Found. Comput. Math. 10, 127–139 (2010). doi:10.1007/s10208-010-9060-6
Edelsbrunner, H., Harer, J.: Persistent homology — a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry. Twenty Years Later. Contemporary Mathematics, vol. 453, pp. 257–282. American Mathematical Society, Providence, RI (2008)
Edelsbrunner, H., Morozov, M.: Persistent homology: theory and practice. In: European Congress of Mathematics, Krakow, 2–7 July 2012, pp. 31–50. European Mathematical Society, Zürich (2012)
Fasy, B.T., Kim, J., Lecci, F., Maria, C.: Introduction to the R package TDA, arXiv:1411.1830 (2014)
Horak, D., Maletic, S., Rajkovic, M.: Persistent homology of complex networks. J. Stat. Mech. Theory Exp. 2009, 1–25 (2009)
Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, New York (2004)
Khasawneh, F., Munch, E.: Chatter detection in turning using persistent homology. Mech. Syst. Signal Process. 70–71, 527–541 (2016)
Kramár, M., Levanger, R., Tithof, J., et al.: Analysis of Kolmogorov flow and Rayleigh-Bénard convection using persistent homology. Phys. D Nonlinear Phenom. 334, 82–98 (2016). ISSN 0167-2789, http://dx.doi.org/10.1016/j.physd.2016.02.003
Münnix, M.C., Shimada, T., Schäfer, R., et al.: Identifying states of a financial market. Sci. Rep. 2 (644), 1–6 (2012)
Nicolau, M., Levine, A.J., Carlsson, G.: Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proc. Natl. Acad. Sci. 108 (17), 7265–7270 (2011)
Nobi, A., Sungmin, L., Kim, D.H., et al.: Correlation and network topologies in global and local stock indices. Phys. Lett. A 378 (34), 2482–2489 (2014)
Nobi, A., Maeng, S.E., Ha, G.G., et al.: Effects of global financial crisis on network structure in a local stock market. Phys. A Stat. Mech. Appl. 407, 135–143 (2014)
Scheffer, M., Bascompte, J., Brock, W.A., et al.: Early-warning signals for critical transitions. Nature 461 (3), 53–59 (2009)
Smerlak, M., Stoll, B., Gupta, A., Magdanz, J.S.: Mapping systemic risk: critical degree and failures distribution in financial networks. PLoS ONE 10 (7), e0130948 (2015). doi:10.1371/journal.pone.0130948
Acknowledgements
Research of Marian Gidea was partially supported by the Alfred P. Sloan Foundation grant G-2016-7320, and by the NSF grant DMS-0635607.
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Gidea, M. (2017). Topological Data Analysis of Critical Transitions in Financial Networks. In: Shmueli, E., Barzel, B., Puzis, R. (eds) 3rd International Winter School and Conference on Network Science . NetSci-X 2017. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-55471-6_5
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DOI: https://doi.org/10.1007/978-3-319-55471-6_5
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