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We fill a void in merging empirical and phenomenological characterisation ofthe dynamical phase transitions in complex systems by identifying three of themon real-life financial markets. We extract and interpret the empirical,numerical, and semi-analytical evidences for the existence of these phasetransitions, by considering the Frankfurt Stock Exchange (FSE), as a typicalexample of a financial market of a medium size. Using the canonical object forthe graph theory, i.e. the Minimal Spanning Tree (MST) network, we observe: (i)The initial phase transition from the equilibrium to non-equilibrium MSTnetwork in its nucleation phase, occurring at some critical time. Coalescenceof edges on the FSE's transient leader is observed within the nucleation and isapproximately characterized by the Lifsthiz-Slyozov growth exponent; (ii) Thenucleation accelerates and transforms to the condensation process, in thesecond phase transition, forming a logarithmically diverging lambda-peak ofshort-range order parameters at the subsequent critical time - an analogon ofsuch a transition in superfluidity; (iii) In the third phase transition, thepeak logarithmically decreases over three quarters of the year, resulting in afew loosely connected sub-graphs. This peak is reminiscent of a non-equilibriumsuperstar-like superhub or a `dragon king' effect, abruptly accelerating theevolution of the leader company. All these phase transitions are caused by thefew richest vertices, which drift towards the leader and provide the most ofthe edges increasing the leader's degree. Thus, we capture an amazingphenomenon, likely of a more universal character, where a peripheral vertexbecomes the one which is over dominating the complex network during anexceptionally long period of time.

Nucleation, condensation and lambda-transition on a real-life stock market