What is a disaster distribution?

A disaster distribution, also known as a discrete Pareto distribution or a Yule-Simon distribution, is a discrete probability distribution that models the frequency of events or objects following a power-law behavior. It is often observed in situations where the "rich get richer," or preferential attachment occurs.

Key characteristics of a disaster distribution include:

• Power-Law Behavior: The probability of observing a certain value decreases rapidly as the value increases. This is a hallmark of power laws.

• Heavy Tail: The distribution has a "heavy tail," meaning that extreme values are more likely than in distributions like the normal distribution. This implies that there is a non-negligible probability of observing very large values.

• Discrete Nature: The distribution is defined over non-negative integers, representing counts or frequencies.

• Parameters: The disaster distribution typically has one or two parameters that control the shape and scale of the distribution. These parameters are often related to the initial conditions and the rate of preferential attachment. The most common parameter is related to the exponent of the power law.

Applications of the disaster distribution can be found in various fields:

• Linguistics: Modeling the frequency of words in a language (Zipf's law) is a good example of Zipf's%20law.

• Network Science: Describing the degree distribution of nodes in networks, where nodes with more connections are more likely to gain new connections.

• Economics: Modeling the distribution of wealth, where individuals with more wealth are more likely to accumulate even more wealth.

• City Sizes: Modeling the relative population size of cities.

The probability mass function (PMF) and cumulative distribution function (CDF) of the disaster distribution have specific mathematical forms, often involving hypergeometric functions or similar special functions. The exact form depends on the parameterization used. It's important to note that in some contexts, the disaster distribution is defined slightly differently, so careful attention should be paid to the specific definition being used.