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Why Stock Markets CrashCritical Events in Complex Financial Systems$
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Didier Sornette

Print publication date: 2017

Print ISBN-13: 9780691175959

Published to Princeton Scholarship Online: May 2018

DOI: 10.23943/princeton/9780691175959.001.0001

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Hierarchies, Complex Fractal Dimensions, and Log-Periodicity

Hierarchies, Complex Fractal Dimensions, and Log-Periodicity

Chapter:
(p.171) Chapter 6 Hierarchies, Complex Fractal Dimensions, and Log-Periodicity
Source:
Why Stock Markets Crash
Author(s):

Didier Sornette

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691175959.003.0006

This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the information contained in the precursory patterns before large stock market crashes. The chapter first considers how models of cooperative behaviors resulting from imitation between agents organized within a hierarchical structure exhibit the announced critical phenomena decorated with “log-periodicity.” It then examines the underlying hierarchical structure of social networks, critical behavior in hierarchical networks, a hierarchical model of financial bubbles, and discrete scale invariance. It also discusses a technique, called the “renormalization group,” and a simple model exhibiting a finite-time singularity due to a positive feedback induced by trend following investment strategies. Finally, it looks at scenarios leading to discrete scale invariance and log-periodicity.

Keywords:   fractals, self-similarity, imitation, log-periodicity, social network, bubble, discrete scale invariance, renormalization group, finite-time singularity, positive feedback

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