The parabolic fractal distribution is a discrete probability distribution with probability mass function $$f(n;b,c)\propto n^{-b}\exp (-c(\log n)^{2}).$$ The function is given as a function of the rank $n$, where $b$ and $c$ are parameters.
I have tried to research about this distribution with little success. There are almost no papers on the topic, zero examples or notes, and no analysis or derivation.
Does anyone have any information on this probability distribution? And, since it is not a popular topic, is trying to research about this a total waste of time?
Any input would be extremely appreciated!
Like you said, this distribution is not often researched, but shows up predominantly in nature comparative to other things.
First off, it exhibits self-similarity across different scales, characterized by a central clustering of points with diminishing density towards the periphery, mirroring the geometric structure of a parabola. It typically arises in complex systems with nonlinear dynamics, where the underlying mechanisms generate patterns that are scale-invariant and exhibit fractal geometry. Analyzing such distributions often involves techniques from fractal analysis and nonlinear dynamics to elucidate underlying principles governing the system's behavior across multiple scales. This is how I understand it to be. Like previously mentioned, there is not much out and especially not much out recently, but speculation is it has application in other areas like finance and modeling and simulation.
In terms of derivation, you would first have to assume self-similarity. Next is deriving the power-law decay which is $n^{−b}$ in this case with $b$ controlling the decay rate. Then you find the exponential decay term as you move away from the peak and account for widening as we get towards higher ranks, which is given by $e^{−c(\log{n})^2}$ with $c$ being the control of the width of the parabolic peak. Now combing these two, we get our PMF while also remembering to normalize.