"In contrast to gravitation, interatomic forces are typically modeled as inhomogeneous power laws with at least two different exponents. Such laws (and exponential laws, too) are not scale-free; they necessarily introduce a characteristic length, related to the size of the atoms. Power laws also govern the power spectra of all kinds of noises, most intriguing among them the ubiquitous (but sometimes difficult to explain)." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"[…] power laws, with integer or fractional exponents, are one of the most fertile fields and abundant sources of self-similarity." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"Scaling invariance results from the fact that homogeneous power laws lack natural scales; they do not harbor a characteristic unit (such as a unit length, a unit time, or a unit mass). Such laws are therefore also said to be scale-free or, somewhat paradoxically, "true on all scales." Of course, this is strictly true only for our mathematical models. A real spring will not expand linearly on all scales; it will eventually break, at some characteristic dilation length. And even Newton's law of gravitation, once properly quantized, will no doubt sprout a characteristic length." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"Nature normally hates power laws. In ordinary systems all quantities follow bell curves, and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition. Then power laws emerge-nature's unmistakable sign that chaos is departing in favor of order. The theory of phase transitions told us loud and clear that the road from disorder to order is maintained by the powerful forces of self-organization and is paved by power laws. It told us that power laws are not just another way of characterizing a system's behavior. They are the patent signatures of self-organization in complex systems." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"From a purely mathematical perspective, a power law signifies nothing in particular - it's just one of many possible kinds of algebraic relationship. But when a physicist sees a power law, his eyes light up. For power laws hint that a system may be organizing itself. They arise at phase transitions, when a system is poised at the brink, teetering between order and chaos. They arise in fractals, when an arbitrarily small piece of a complex shape is a microcosm of the whole. They arise in the statistics of natural hazards - avalanches and earthquakes, floods and forest fires - whose sizes fluctuate so erratically from one event to the next that the average cannot adequately stand in for the distribution as a whole."
"A power law pattern is also observed when one investigates the strength of the various genetic regulatory interactions provided by microarray datasets. Assigning each pair of genes a correlation coefficient which captures the degree to which they are coexpressed, one finds that the distribution of these pair-wise correlation coefficients follows a power law. ' That is, while the majority of gene pairs have only weak correlations, a few gene pairs display a signifi-cant correlation coefficient. These highly correlated pairs likely correspond to direct regulatory and protein interactions. This hypothesis is supported by the finding that the correlations are larger along the links of the protein interaction network and between proteins occurring in the same complex than for pairs of proteins that are not known to interact directly." (Eivind Almaas & Albert-Laszlo Barabasi, "Power Laws in Biological Networks", 2004)
"An event occurring at one node will cause a cascade of events: often this cascade or avalanche propagates to affect only one or two further elements, occasionally it affects more, and more rarely it affects many. The mathematical theory of this - which is very much part of complexity theory - shows that propagations of events causing further events show characteristic properties such as power laws (caused by many and frequent small propagations, few and infrequent large ones), heavy tailed probability distributions (lengthy propagations though rare appear more frequently than normal distributions would predict), and long correlations (events can and do propagate for long distances and times)." (W Brian Arthur, "Complexity and the Economy", 2015)
"The power law distributions transcend networks as such. The quantities that are so distributed include the number of genes in a family, the number of pseudogenes per gene, the number of people per city, the number of published papers per scientist, the number of citations per paper, and much, much more. In fact, the first distributions where power laws have been noticed are the distribution of people in a society by wealth (the Pareto law^^) and the distributions of words in a text by frequency (Zipf law)." (Eugene V Koonin et al, "The Genomic Revolution, Systems Biology, Power Laws, and Scale-Free Networks", 2006)
"But note that any heavy tailed process, even a power law, can be described in sample (that is finite number of observations necessarily discretized) by a simple Gaussian process with changing variance, a regime switching process, or a combination of Gaussian plus a series of variable jumps (though not one where jumps are of equal size […])." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)
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