Structural and topological phase transitions on the German Stock Exchange

TL;DR

This paper provides empirical evidence of dynamical, structural, and topological phase transitions in the German Stock Exchange around financial crashes, using network analysis with Minimal Spanning Trees to identify distinct topological states.

Contribution

It introduces a novel application of MST analysis to detect phase transitions in stock market networks during financial crises, extending previous studies to the German market.

Findings

Identification of a transition from hierarchical to star-like MSTs before the crash.

Observation of a return to hierarchical MSTs after the crash.

Transitions are more pronounced than those observed in other markets.

Abstract

We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two transitions of the topology of a complex network representing FSE were found. First transition is from a hierarchical scale-free MST representing the stock market before the recent worldwide financial crash, to a superstar-like MST decorated by a scale-free hierarchy of trees representing the market's state for the period containing the crash. Subsequently, a transition is observed from this transient, (meta)stable state of the crash, to a hierarchical scale-free MST decorated by several star-like trees after the worldwide financial crash. The phase transitions observed are analogous to the ones we obtained earlier for the Warsaw Stock Exchange and more pronounced than those found by Onnela-Chakraborti-Kaski-Kert\'esz for S&P 500 index in the vicinity of Black Monday (October 19, 1987) and also in the vicinity of January 1, 1998. Our results provide an empirical foundation for the future theory of dynamical, structural and topological phase transitions on financial markets.