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Structural and topological phase transitions on the German Stock Exchange

机译：德国证券交易所的结构和拓扑相变

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摘要

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 the FSE were found. The 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-Kertész for the 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.
机译：我们发现，在全球金融崩溃的暂时范围内，（德国）法兰克福证券交易所（FSE）上存在动态，结构和拓扑相变的数值和经验证据。使用最小生成树（MST）技术（一种特别有用的图论规范工具），发现了代表FSE的复杂网络拓扑的两个过渡。第一个过渡是从代表最近一次全球金融崩溃之前的股票市场的无等级无等级MST到由代表市场崩溃状态的市场状态的无等级树状装饰的超级明星般的MST。随后，在全球金融崩溃之后，观察到了从崩溃的这种瞬态（亚）稳定状态过渡到由几棵星形树装饰的分层无标度MST。观察到的相变类似于我们早先在华沙证券交易所获得的相变，并且比Onnela-Chakraborti-Kaski-Kertész在黑色星期一（1987年10月19日）附近发现的标普500指数更明显，并且在1998年1月1日附近。我们的结果为未来金融市场的动态，结构和拓扑相变理论提供了经验基础。

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