Kara Sears
Power laws describe a wide variety of physical, biological, and human-made phenomena, including the sizes of craters on the moon, the foraging patterns of species, and the frequencies of family names. Power laws are also observed in gene networks, where only a few genes act as connectivity hubs, and in the study of macromolecules, where they are used to investigate the properties of proteins, peptides, and polymers. In statistics, a power law describes a phenomenon where a small number of items are clustered at the top or bottom of a distribution, taking up most of the resources. For instance, there are very few billionaires, while the majority of the population has modest wealth. The power-law behavior of natural processes can be modelled as an asymmetric Laplace (AL) distribution, which can handle incomplete data and has applications in computer vision and seismicity.
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What You'll Learn
Power-law behaviour in natural processes
Power laws are observed in a wide variety of physical, biological, and human-made phenomena. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities. These power laws can be used to study the frequency of extremely rare events like stock market crashes and large natural disasters.
In natural processes, power laws are observed in the sizes of craters on the moon, solar flares, cloud sizes, the foraging patterns of various species, the sizes of neuronal population activity patterns, and many other quantities. Power laws are also observed in the growth of cities, with some studies showing that Chinese cities exhibited faster growth in small and medium-sized cities compared to larger ones before the year 2000.
The study of power laws in natural processes is important because it can indicate specific kinds of mechanisms that underlie the observed phenomena and reveal deep connections with other seemingly unrelated systems. For example, the behavior of water and CO2 at their boiling points falls into the same universality class because they share identical critical exponents. This concept of universality highlights how different systems can share the same fundamental dynamics.
Furthermore, power laws play a fundamental role in mathematical convergence, similar to the normal distribution's role in the central limit theorem. This convergence effect explains the prevalence of the variance-to-mean power law in natural processes, such as Taylor's law in ecology and fluctuation scaling in physics. By understanding these power laws, we can gain insights into the underlying mechanisms driving these natural processes.
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Power-law in statistical analyses
Power laws describe a wide variety of physical, biological, and human-made phenomena. These include the sizes of craters on the moon, the sizes of clouds, the foraging patterns of species, and the frequencies of word usage in most languages. In the context of statistical analyses, power laws are particularly relevant in understanding the distributions of various quantities.
The power law, also known as the scaling law, states that a relative change in one quantity will result in a proportional relative change in another. This can be represented by the formula Y = k Xα, where a change in the value of X leads to a corresponding change in the value of Y. For example, if you double the length of a square (changing the value of X), the area (value of Y) will quadruple. Power laws can also describe inverse relationships, such as Y = X^-1, where a change in one quantity results in a negative change in the other.
Power laws are commonly observed in natural processes, but the data used in statistical analyses is often limited to a specific range where the observations are reliable. In some cases, the power-law behavior may only be evident above a certain threshold, and information below this threshold may be discarded due to detection limitations. This can result in incomplete data that can still be described by a power law, and the full range of data can be modelled using an asymmetric Laplace (AL) distribution.
The study of power laws in statistical distributions is closely related to the theory of large deviations or extreme value theory, which examines the frequency of rare events such as stock market crashes and natural disasters. Power laws can also be applied to various fields, including economics, biology, and medicine. For example, the Pareto distribution, also known as the "80-20 rule," describes how wealth is distributed in a population, with a small fraction of the population holding a disproportionately large share of the wealth. In biology, power laws can describe the behavior of bacteria and the formation of microcolonies.
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Power-law and asymmetric Laplace distribution
In probability theory and statistics, the asymmetric Laplace distribution (ALD) is a continuous probability distribution that is a generalization of the Laplace distribution. The Laplace distribution, named after Pierre-Simon Laplace, is a composite or double distribution, consisting of two equal-scale exponential distributions back-to-back about x = m. The ALD, on the other hand, consists of two exponential distributions of unequal scale back-to-back about x = m, adjusted for continuity and normalization.
The power-law behaviour of many natural processes is often observed only above a certain threshold, below which information is discarded due to detection limitations. This incomplete data can be described by a power law, and the distribution over the full quantity range can be reformulated as an asymmetric Laplace (AL) distribution. The ALD has applications in finance and neuroscience. In finance, for example, S.G. Kou developed a model for financial instrument prices that incorporated an ALD to address issues of skewness, kurtosis, and the volatility smile that can arise when using a normal distribution for pricing.
The ALD has been used to model various natural phenomena. For instance, using seismicity as an example, an asymmetric Laplace mixture model (ALMM) can be employed to account for ambiguous overlapping components observed in nature, based on a semi-supervised hard Expectation-Maximization algorithm. The ALMM has been shown to fit reasonably well with incomplete data, and the number of AL components can be related to the seismic network density.
In summary, the power-law and the asymmetric Laplace distribution are both important concepts in understanding and modelling complex systems and natural phenomena. The power-law describes the distributions of a wide range of magnitudes, from physical to human-made phenomena, while the ALD provides a flexible generalization of the Laplace distribution, allowing for the modelling of skewed data and addressing limitations in the normal distribution.
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Power-law and information asymmetry
Power laws describe a wide variety of physical, biological, and human-made phenomena, including the sizes of craters on the moon, the frequencies of words in most languages, and the distribution of car exhaust among cars. Incomplete data that exhibits power-law behavior can be described by an asymmetric Laplace (AL) distribution.
The concept of information asymmetry, on the other hand, is prevalent in economics, finance, sociology, psychology, political science, and media and communications. Information asymmetry occurs when one party in a transaction has more or better information than the other. This can lead to detrimental outcomes, as seen in the 2008 financial crisis. Information asymmetry can be mitigated through increased transparency, such as mandatory wage disclosure laws, and regulation, particularly in the banking and financial sectors.
The combination of power-law and information asymmetry can be observed in various contexts. For example, in the study of probability distributions, power laws are often used to describe the upper tail of a distribution, which represents large, rare events such as stock market crashes and natural disasters. In this case, the incomplete data exhibiting power-law behavior can be described by an asymmetric Laplace distribution.
Additionally, organizations may deliberately create information asymmetries to gain and maintain power. This can be achieved through various techniques, including media and propaganda, knowledge production, educational systems, legal and organizational structures, and exclusive information networks. For instance, governments may engage in secrecy and surveillance, resulting in an imbalance of knowledge between those who possess the data and the subjects of the data.
In summary, power-law and information asymmetry are distinct but interconnected concepts. Power laws describe the distribution of various phenomena, and information asymmetry refers to unequal access to information, which can lead to power imbalances and detrimental outcomes. The combination of these concepts can be observed in various fields, including economics and finance and probability distributions.
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Power-law in gene networks
Power laws are observed in a wide variety of physical, biological, and human-made phenomena. In the context of gene networks, power laws have been found to govern gene expression, indicating a universal mathematical dynamic at play. The study of gene expression has shifted from examining individual genes to understanding expression at the network or systems level, and power laws have been observed across different organisms, organs, and developmental times within the same organism.
The work of Hiroki Ueda and colleagues at the Center for Developmental Biology is notable in this area. They utilized data from public databases of whole-genome sequences and their own microarray analyses to uncover the mathematical principles underlying gene expression levels. Their findings revealed that highly expressed genes exhibit greater changes, while infrequently expressed genes exhibit smaller changes. This proportional relationship, known as "rich-travel-more," results in a heterogeneous distribution of gene expression levels governed by the power law of minus 2 exponent.
Ueda's findings also highlighted the surprising similarity between Escherichia coli and humans in terms of the underlying gene expression mechanisms. This discovery underscores the universality of the power law in gene networks. However, it is important to note that the power law is not universally applicable to all types of networks, and careful analyses are required to confirm its presence.
While the power law has been a subject of interest in understanding gene networks, there have been challenges and controversies along the way. Early literature on biological network topology often opposed the power law to Poisson distribution, which would be expected from random graphs. However, subsequent analyses revealed that many of the alleged power laws were based solely on visual inspections of degree distribution plots without rigorous statistical testing. This led to the realization that the power law's ubiquity was partly an illusion resulting from representation issues and that the data often did not fit the expected theoretical models.
Despite these challenges, the power law continues to be a topic of investigation in gene networks. The rapidly developing theory of complex networks suggests that real networks are not random but exhibit a highly robust large-scale architecture governed by strict organizational principles. The interplay between topology and reaction fluxes in metabolic networks, for example, demonstrates the inhomogeneous nature of cellular utilization, with "hot-spots" representing connected high-flux pathways.
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Frequently asked questions
A power law is used to describe a phenomenon where a small number of items are clustered at the top of a distribution, taking up most of the resources. For example, the distribution of income, where there are very few billionaires, and the bulk of the population holds very modest amounts of money.
The inverse-square laws of Newtonian gravity and electrostatics are examples of power laws.
Yes, power laws can be asymmetric. Incomplete data that follows a power law can be described by an asymmetric Laplace (AL) distribution.
An example of an asymmetric power law is the asymmetric Laplace mixture model (ALMM), which is used in the study of seismicity.
Power laws have applications in various fields, including physics, biology, and economics. For example, power laws can be used to model the behaviour of water and CO2 at their boiling points, the distribution of car exhaust among cars, and the diffusion coefficient in relation to molecular weight.
