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Publication# Power law size distribution of supercritical random trees

Abstract

The probability distribution P(k) of the sizes k of critical trees ( branching ratio m = 1) is well known to show a power law behavior k(-3/2). Such behavior corresponds to the mean-field approximation for many critical and self-organized critical phenomena. Here we show numerically and analytically that also supercritical trees (branching ration m > 1) are critical in that their size distribution obeys a power law k(-2). We mention some possible applications of these results.

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Related concepts (32)

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Related publications (38)

Poisson distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson ('pwɑːsɒn; pwasɔ̃). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.

Power law

In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to a power of the change, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.

Taylor's law

Taylor's power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship. It is named after the ecologist who first proposed it in 1961, Lionel Roy Taylor (1924–2007). Taylor's original name for this relationship was the law of the mean. The name Taylor's law was coined by Southwood in 1966. This law was originally defined for ecological systems, specifically to assess the spatial clustering of organisms.

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We explore statistical physics in both classical and open quantum systems. Additionally, we will cover probabilistic data analysis that is extremely useful in many applications.

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