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Temporal condensation and dynamic λtransition within the complex network: an application to reallife market evolution 
Mateusz J. Wiliński ^{1}, Bartłomiej Szewczak ^{2}, Tomasz Gubiec ^{1}, Ryszard Kutner ^{1}, Zbigniew R. Struzik ^{3} 
1. Uniwersytet Warszawski, Wydział Fizyki, ul. Pasteura 5, Warszawa 02093, Poland 
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 reallife 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 nonequilibrium 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 λpeak 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 λpeak logarithmically relaxes over three quarters of the year, resulting in a few loosely connected subgraphs. This λpeak (comparable to that of the specific heat vs. temperature forming during the equilibrium continuous phase transition from the normal fluid I 4 He to the superfluid II 4 He) is considered as a prominent result of a nonequilibrium superstarlike superhub or a dragonking’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 dragonking 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 λpeak allows us to derive a generic nonlinear constitutive equation of the dragonking dynamics describing the complexity of the MST network by the corresponding inherent nonlinearity of the underlying dynamical processes. The European Journal of Physics B (2015) 88:34, 115. 
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Presentation: Article at Econophysics group of Ryszard Kutner, by Ryszard Kutner Submitted: 20150813 16:26 Revised: 20150813 16:50 