Related papers: Power-fractional distributions and branching proce…
The simple Galton--Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of…
Branching processes pervade many models in statistical physics. We investigate the survival probability of a Galton-Watson branching process after a finite number of generations. We reveal the finite-size scaling law of the survival…
We investigate fractional moments and expectations of power means of complex-valued random variables by using fractional calculus. We deal with both negative and positive orders of the fractional derivatives. The one-dimensional…
Multiplicative random processes in (not necessaryly equilibrium or steady state) stochastic systems with many degrees of freedom lead to Boltzmann distributions when the dynamics is expressed in terms of the logarithm of the normalized…
Galaxies and clusters distributions show two major properties: (i) the positions of galaxies and clusters are characterized by a power law distribution indicating properties with respect to their positions. (ii) The distribution of masses…
Understanding the properties of response time distributions is a long-standing problem in cognitive science. We provide a tutorial overview of several contemporary models that assume power law scaling is a plausible description of the…
The statistics of natural catastrophes contains very counter-intuitive results. Using earthquakes as a working example, we show that the energy radiated by such events follows a power-law or Pareto distribution. This means, in theory, that…
Reinforced Galton--Watson processes describe the dynamics of a population where reproduction events are reinforced, in the sense that offspring numbers of forebears can be repeated randomly by descendants. More specifically, the evolution…
We review briefly the concepts underlying complex systems and probability distributions. The later are often taken as the first quantitative characteristics of complex systems, allowing one to detect the possible occurrence of regularities…
It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the…
Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment…
Power-law distributions with various exponents are studied. We first introduce a simple and generic model that reproduces Zipf's law. We can regard this model both as the time evolution of the population of cities and that of the asset…
Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the ``rich get…
We introduce a non-growth model that generates the power-law distribution with the Zipf exponent. There are N elements, each of which is characterized by a quantity, and at each time step these quantities are redistributed through binary…
The Zipf distribution also known as scale-free distribution or discrete Pareto distribution, is the particular case of Power Law distribution with support the strictly positive integers. It is a one-parameter distribution with a linear…
We propose a new mechanism for generating power laws. Starting from a random walk, we first outline a simple derivation of the Fokker-Planck equation. By analogy, starting from a certain Markov chain, we derive a master equation for power…
In this article we investigate the energy spectrum statistics of fractals at the quantum level. We show that the energy-level distribution of a fractal follows a power-law behaviour, if its energy spectrum is a limit set of piece-wise…
The Lindley distribution and its numerous generalizations are widely used in statistical and engineering practice. Recently, a power transformation of Lindley distribution, called the power Lindley distribution, has been introduced by M. E.…
Using a model based on generalised Lotka Volterra dynamics together with some recent results for the solution of generalised Langevin equations, we show that the equilibrium solution for the probability distribution of wealth has two…
We study multi-type Bienaym\'e-Galton-Watson processes with linear-fractional reproduction laws using various analytical tools like contour process, spinal representation, Perron-Frobenius theorem for countable matrices, renewal theory. For…