The different between the inverse power function and the negative exponentialfunction is significant. The former suggests a complex distribution, while thelatter indicates a simple distribution. However, the association of thepower-law distribution with the exponential distribution has been seldomresearched. Using mathematical derivation and numerical experiments, I revealthat a power-law distribution can be created through averaging an exponentialdistribution. For the distributions defined in a 1-dimension space, the scalingexponent is 1; while for those defined in a 2-dimension space, the scalingexponent is 2. The findings of this study are as follows. First, theexponential distributions suggest a hidden scaling, but the scaling exponentssuggest a Euclidean dimension. Second, special power-law distributions can bederived from exponential distributions, but they differ from the typicalpower-law distribution. Third, it is the real power-law distribution that canbe related with fractal dimension. This study discloses the inherentrelationship between simplicity and complexity. In practice, maybe the resultpresented in this paper can be employed to distinguish the real power laws fromspurious power laws (e.g., the fake Zipf distribution).
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