Mars General science Exponential growth and random observation (or killing) Progress through exponential growth and exponential diffusion of innovations Highly optimized tolerance Proposed form of experience curve effects Pink noise The law of stream numbers, and the law of stream lengths (Horton's laws describing river systems) Populations of cities (Gibrat's law) Bibliograms, and frequencies of words in a text (Zipf's law) 90–9–1 principle on wikis (also referred to as the 1% rule) Richardson's Law for the severity of violent conflicts (wars and terrorism) The relationship between a CPU's cache size and the number of cache misses follows the power law of cache misses. The spectral density of the weight matrices of deep neural networks Mathematics Fractals Pareto distribution and the Pareto principle also called the "80–20 rule" Zipf's law in corpus analysis and population distributions amongst others, where frequency of an item or event is inversely proportional to its frequency rank (i.e. the second most frequent item/event occurs half as often as the most frequent item, the third most frequent item/event occurs one third as often as the most frequent item, and so on). Zeta distribution (discrete) Yule–Simon distribution (discrete) Student's t-distribution (continuous), of which the Cauchy distribution is a special case Lotka's law The scale-free network model Economics Population sizes of cities in a region or urban network, Zipf's law. Distribution of artists by the average price of their artworks. Income distribution in a market economy. Distribution of degrees in banking networks. Finance The mean absolute change of the logarithmic mid-prices Number of tick counts over time Size of the maximum price move Average waiting time of a directional change Average waiting time of an overshoot Variants Broken power law A broken power law is a piecewise function, consisting of two or more power laws, combined with a threshold. For example, with two power laws: for . Power law with exponential cutoff A power law with an exponential cutoff is simply a power law multiplied by an exponential function: Curved power law Power-law probability distributions In a looser sense, a power-law probability distribution is a distribution whose density function (or mass function in the discrete case) has the form, for large values of , where , and is a slowly varying function, which is any function that satisfies for any positive factor . This property of follows directly from the requirement that be asymptotically scale invariant; thus, the form