Temporal condensation and dynamic λ-transition within the complex network: an application to real-life market evolution

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.