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