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Abstract
Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are
typical ubiquitous general empirical observations across the individual sciences which
cannot be understood within the set of references developed within the specific scientific
domains. All these observations are associated with scaling laws and have caused a
broad research interest in the scientific circle. However, the inherent relationships
between these scaling phenomena are still pending questions remaining to be researched.
In this paper, theoretical derivation and mathematical experiments are employed to
reveal the analogy between fractal patterns, 1/f noise, and the Zipf distribution.
First, the multifractal process follows the generalized Zipf's law empirically. Second,
a 1/f spectrum is identical in mathematical form to Zipf's law. Third, both 1/f spectra
and Zipf's law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f
spectra, Zipf's law, and the occurrence of large catastrophic events can be described
with similar exponential laws and power laws. The self-similar hierarchy is a more
general framework or structure which can be used to encompass or unify different scaling
phenomena and rules in both physical and social systems such as cities, rivers, earthquakes,
fractals, 1/f noise, and rank-size distributions. The mathematical laws on the hierarchical
structure can provide us with a holistic perspective of looking at complexity such
as self-organized criticality (SOC).