Abstract
We fill a void in merging empirical and phenomenological characterisation of the dynamical phase transitions in complex networks by identifying and thoroughly characterising a triple sequence of such transitions on a real-life financial market. We extract and interpret the empirical, numerical, and analytical evidences for the existence of these dynamical phase transitions, by considering the medium size Frankfurt stock exchange (FSE), as a typical example of a financial market. By using the canonical object for the graph theory, i.e. the minimal spanning tree (MST) network, we observe: (i) the (initial) dynamical phase transition from equilibrium to non-equilibrium nucleation phase of the MST network, occurring at some critical time. Coalescence of edges on the FSE’s transient leader (defined by its largest degree) is observed within the nucleation phase; (ii) subsequent acceleration of the process of nucleation and the emergence of the condensation phase (the second dynamical phase transition), forming a logarithmically diverging temporal λ-peak of the leader’s degree at the second critical time; (iii) the third dynamical fragmentation phase transition (after passing the second critical time), where the λ-peak logarithmically relaxes over three quarters of the year, resulting in a few loosely connected sub-graphs. This λ-peak (comparable to that of the specific heat vs. temperature forming during the equilibrium continuous phase transition from the normal fluid I 4He to the superfluid II 4He) is considered as a prominent result of a non-equilibrium superstar-like superhub or a dragon-king’s abrupt evolution over about two and a half year of market evolution. We capture and meticulously characterise a remarkable phenomenon in which a peripheral company becomes progressively promoted to become the dragon-king strongly dominating the complex network over an exceptionally long period of time containing the crash. Detailed analysis of the complete trio of the dynamical phase transitions constituting the λ-peak allows us to derive a generic nonlinear constitutive equation of the dragon-king dynamics describing the complexity of the MST network by the corresponding inherent nonlinearity of the underlying dynamical processes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Albert, L.A. Barabási, Phys. Rev. Lett. 85, 5234 (2000)
R. Albert, L.A. Barabási, Rev. Mod. Phys. 74, 47 (2002)
G. Bonanno, G. Caldarelli, F. Lillo, R.N. Mantegna, Phys. Rev. E 68, 046130 (2003)
I. Derényi, I. Farkas, G. Palla, T. Vicsek, Physica A 334, 583 (2004)
S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Rev. Mod. Phys. 80, 1275 (2008)
J. Lorenz, S. Battiston, F. Schweitzer, Eur. Phys. J. B 71, 441 (2009)
T. Di Matteo, F. Pozi, T. Aste, Eur. Phys. J. B 73, 3 (2010)
J. Kwapień, S. Drożdż, Phys. Rep. 515, 115 (2012)
K.J. Mizigier, S.M. Wagner, J.A. Hołyst, Int. J. Prod. Econ. 135, 14 (2012)
D. Helbing, Nature 497, 51 (2013)
Th. Bury, J. Stat. Mech. 2013, P11004 (2013)
R.N. Mantegna, J. Kértesz, New J. Phys. 13, 025011 (2011)
S. Drożdż, J. Kwapień, P. Oświȩcimka, R. Rak, New J. Phys. 12, 105003 (2010)
P. Sieczka, J.A. Hołyst, Eur. Phys. J. B 71, 461 (2009)
T. Jia, M. Pósfai, Sci. Rep. 4, 1 (2014)
S.N. Dorogovtsev, Lectures on Complex Networks (Clarendon Press, Oxford, 2010)
B. Bollobás, Modern Graph Theory (Springer, Berlin, 1998)
R.N. Mantegna, Eur. Phys. J. B 11, 193 (1999)
G. Bonanno, G. Calderelli, F. Lillo, S. Micciche, N. Vandewalle, R.N. Mantegna, Eur. Phys. J. B 38, 363 (1999)
R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics. Correlations and Complexity in Finance (Cambridge University Press, Cambridge, 2000)
G. Bonanno, F. Lillo, R.N. Mantegna, Quant. Financ. 1, 96 (2001)
N. Vandewalle, F. Brisbois, X. Tordoir, Quant. Financ. 1, 372 (2001)
L. Kullmann, J. Kertész, K. Kaski, Phys. Rev. E 66, 026125 (2002)
G. Bonanno, G. Caldarelli, F. Lillo, R.N. Mantegna, Phys. Rev. E 68, 046130 (2003)
M. Tumminello, T. Di. Matteo, T. Aste, R.N. Mantegna, Eur. Phys. J. B 55, 209 (2007)
M. Tumminello, C. Coronello, F. Lillo, S. Micciche, R.N. Mantegna, Int. J. Bifurc. Chaos 17, 2319 (2007)
T. Ibuki, S. Suzuki, J. Inoue, in Econophysics of Systemic Risk and Network Dynamics (New Economic Windows), edited by F. Abergel, B.K. Chakrabarti, A. Chakraborti, A. Ghosh (Springer-Verlag, Milan, 2012), Chap. 15, p. 239
T. Ibuki, S. Higano, S. Suzuki, J. Inoue, A. Chakraborti, Statistical inference of co-movements of stocks during a finacial crisis, in Proceed. Int. Meeting on Inference, Computation, and Spin Glasses, Sapporo, 2013
A. Nobi, S.E. Maeng, G.G. Ha, J.W. Lee, arXiv:1307. 6974 [q-fin.GN] (2013)
A. Sienkiewicz, T. Gubiec, R. Kutner, Z. Struzik, Acta Physica Polonica A 123, 615 (2013)
G. Sugihara, R. May, H. Ye, C.-H. Hsieh, E. Deyle, M. Fogarty, S. Munch, Science 338, 496 (2012)
Z. Burda, J.D. Correia, A. Krzywicki, Phys. Rev. E 64, 046118 (2001)
N.G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd edn. (Elsevier, Amsterdam, 2007)
M. Wiliński, B. Szewczak, T. Gubiec, R. Kutner, Z.R. Struzik, arXiv:1311.5753 [q-fin.ST] (2013)
M. Wiliński, A. Sienkiewicz, T. Gubiec, R. Kutner, Z. Struzik, Physica A 392, 5963 (2013)
D. Sornette, Why Stock Markets Crash (Princeton University Press, Princeton, Oxford, 2003)
W. Weidlich, G. Haag, Concepts and Models of a Quantitative Sociology. The Dynamics of Interacting Populations (Springer-Verlag, Berlin, 1983)
P. Bak, How Nature Works: the Science of Self-organized Criticality (Copernicus, New York, 1996)
J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertész, Eur. Phys. J. B 30, 285 (2002)
J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertész, Physica A 324, 247 (2003)
J.G. Brida, W.A. Risso, Exp. Syst. Appl. 37, 3846 (2010)
B.M. Tabak, T.R. Serra, D.O. Cajueiro, Eur. Phys. J. B 74, 243 (2010)
J. Rehmeyer, Nature News (2013), DOI:10.1038/nature.2013.12447
G. Nicolis, I. Prigogine, Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations (J. Wiley & Sons, New York, 1977)
P. Kondratiuk, G. Siudem, J. Hołyst, Phys. Rev. E 85, 066126 (2012)
G. Kondrat, K. Sznajd-Weron, Phys. Rev. E 77, 021127 (2008)
G. Kondrat, K. Sznajd-Weron, Phys. Rev. E 79, 011119 (2009)
M. Henkel, M. Pleimling, in Non-Equilibrium Phase Transitions, Aging and Dynamical Scaling Far from Equilibrium (Springer-Verlag, Berlin, 2010), Vol. 2
T. Di Matteo, F. Pozzi, T. Aste, Eur. Phys. J. B 73, 3 (2010)
F. Pozzi, T. Di Matteo, T. Aste, Adv. Complex Syst. 11, 927 (2008)
P. Bonacich, Am. J. Sociol. 92, 1170 (1987)
R.N. Silver, in Superfluid Helium and Neutron Scattering a New Chapter in the Condensate Saga (Los Alamos Science, Summer, 1990), p. 159
K. Huang, Statistical Mechanics (J. Wiley & Sons, New York, 1963)
N. Proukakis, S. Gardiner, M. Davis, M. Szymańska, in Quantum Gases. Finite Temperature and Non-Equilibrium Dynamics in Cold Atoms (Imperial College Press, London, 2013), Vol. 1
D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer Series in Synergetics, 2nd edn. (Springer-Verlag, Heidelberg, 2004)
N.J. Cowan, E.J. Chastain, D.A. Vilhena, J.S. Freudenberg, C.T. Bergstrom, Plos One 7, e38398 (2012)
D. Beysens, Y. Garrabos, C. Chabot, AIP Conf. Proc. 469, 222 (1998)
I.S. Gutzow, J.W.P. Schmelzer, The Vitreous State. Thermodynamics, Structure, Rheology, and Crystallization, 2nd edn. (Springer-Verlag, Heidelberg, 2013)
M. Toda, R. Kubo, N. Saitô, Statistical Physics I. Equilibrium Statistical Mechanics (Springer-Verlag, Berlin, 1983)
D. Sornette, Int. J. Terraspace Eng. 2, 1 (2009)
A. Majdandzic, B. Podobnik, S.V. Buldyrev, D.Y. Kenett, S. Havlin, Nat. Phys. 10, 34 (2014)
M. Tumminello, T. Aste, T. Di Matteo, R.N. Mantegna, Proc. Natl. Acad. Sci. 102, 10421 (2005)
S.N. Dorogovtsev, J.F.F. Mendes, A.M. Povolotsky, A.N. Samukhin, Phys. Rev. Lett. 95, 195701 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Wiliński, M., Szewczak, B., Gubiec, T. et al. Temporal condensation and dynamic λ-transition within the complex network: an application to real-life market evolution. Eur. Phys. J. B 88, 1 (2015). https://doi.org/10.1140/epjb/e2014-50167-4
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2014-50167-4