Power law - Knowledge

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plot. This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in the plot tend to "converge" to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published. A disadvantage of these plots is that, in order for them to provide reliable results, they require huge amounts of data. In addition, they are appropriate only for discrete (or grouped) data.

1751: 6349: 3879:(RQFs), also called residual percentile functions, which provide a full characterization of the tail behavior of many well-known probability distributions, including power-law distributions, distributions with other types of heavy tails, and even non-heavy-tailed distributions. Bundle plots do not have the disadvantages of Pareto Q–Q plots, mean residual life plots and log–log plots mentioned above (they are robust to outliers, allow visually identifying power laws with small values of 3857: 3805:. Another, more robust graphical method uses bundles of residual quantile functions. (Please keep in mind that power-law distributions are also called Pareto-type distributions.) It is assumed here that a random sample is obtained from a probability distribution, and that we want to know if the tail of the distribution follows a power law (in other words, we want to know if the distribution has a "Pareto tail"). Here, the random sample is called "the data". 3853:, where n is the size of the random sample. If the resultant scatterplot suggests that the plotted points tend to "stabilize" about a horizontal straight line, then a power-law distribution should be suspected. Since the mean residual life plot is very sensitive to outliers (it is not robust), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots 43: 6003:

power law. As a solution to this problem, Diaz proposed a graphical methodology based on random samples that allow visually discerning between different types of tail behavior. This methodology uses bundles of residual quantile functions, also called percentile residual life functions, which characterize many different types of distribution tails, including both heavy and non-heavy tails. However,

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without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a survival function representation is favored over that of the pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended.

5995:, yielding a "bowed" shape in a log–log plot, if the quadratic term is small relative to the linear term then the result can appear almost linear, and the lognormal behavior is only visible when the quadratic term dominates, which may require significantly more data. Therefore, a log–log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law. 4199: 2200: 644:). On the other hand, this also allows for cost-efficient interventions. For example, given that car exhaust is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially. 5296: 1493:

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

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are an alternative way of graphically examining the tail of a distribution using a random sample. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for a power law relationship, as many non power-law distributions will appear as straight lines on a log–log

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One method to validate a power-law relation tests many orthogonal predictions of a particular generative mechanism against data. Simply fitting a power-law relation to a particular kind of data is not considered a rational approach. As such, the validation of power-law claims remains a very active

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proposed plotting the empirical cumulative distribution function in the log-log domain and claimed that a candidate power-law should cover at least two orders of magnitude. Also, researchers usually have to face the problem of deciding whether or not a real-world probability distribution follows a

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are often mistaken for power-law distributions: a data set drawn from a lognormal distribution will be approximately linear for large values (corresponding to the upper tail of the lognormal being close to a power law), but for small values the lognormal will drop off significantly (bowing down),

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This criterion can be applied for the estimation of power-law exponent in the case of scale-free distributions and provides a more convergent estimate than the maximum likelihood method. It has been applied to study probability distributions of fracture apertures. In some contexts the probability

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estimators (like the mean and the variance) for diverging moments will never converge – as more data is accumulated, they continue to grow. These power-law probability distributions are also called Pareto-type distributions, distributions with Pareto tails, or distributions with regularly varying

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of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws.

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Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them. The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can

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Although it can be convenient to log-bin the data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided. The survival function, on the other hand, is more robust to (but not

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of the log-transformed data to the corresponding quantiles of an exponential distribution with mean 1 (or to the quantiles of a standard Pareto distribution) by plotting the former versus the latter. If the resultant scatterplot suggests that the plotted points " asymptotically converge" to a

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Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow a power-law relation requires more than simply fitting a particular model to the data. This is important for understanding the mechanism that gives rise to the distribution:

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at their boiling points fall in the same universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by a small set of universality classes. Similar observations have been made, though not as comprehensively, for various

3448: 4682: 4335:. Alternative methods are often based on making a linear regression on either the log–log probability, the log–log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent. 3980: 1975: 1958: 415:. Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both 7463:

Müller, Ulrich A.; Dacorogna, Michel M.; Olsen, Richard B.; Pictet, Olivier V.; Schwarz, Matthias; Morgenegg, Claude (1990-12-01). "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis".

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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. The rate of change exhibited in these relationships is said to be multiplicative.

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On the other hand, in its version for identifying power-law probability distributions, the mean residual life plot consists of first log-transforming the data, and then plotting the average of those log-transformed data that are higher than the

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under additive and reproductive convolution as well as under scale transformation. Consequently, these models all express a power-law relationship between the variance and the mean. These models have a fundamental role as foci of mathematical

115:, volcanic eruptions, human judgments of stimulus intensity and many other quantities. Empirical distributions can only fit a power law for a limited range of values, because a pure power law would allow for arbitrarily large or small values. 3246: 4440: 5703: 7351: 3796:

Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or Pareto

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More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income). Among them are:

5808: 5036: 4194:{\displaystyle P(x)=\Pr(X>x)=C\int _{x}^{\infty }p(X)\,\mathrm {d} X={\frac {\alpha -1}{x_{\min }^{-\alpha +1}}}\int _{x}^{\infty }X^{-\alpha }\,\mathrm {d} X=\left({\frac {x}{x_{\min }}}\right)^{-(\alpha -1)}.} 3307: 3662: 3722:. This distribution does not scale and is thus not asymptotically as a power law; however, it does approximately scale over a finite region before the cutoff. The pure form above is a subset of this family, with 891:, which can represent uncertainty in the observed values (perhaps measurement or sampling errors) or provide a simple way for observations to deviate from the power-law function (perhaps for stochastic reasons): 5958:. Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope). 4557: 816:: The distributions of a wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of 5982:

about proportional growth processes produce distributions that are lognormal, although their log–log plots look linear over a limited range. An explanation of this is that although the logarithm of the

2934: 336: 2265: 2195:{\displaystyle \ln \left({\frac {y}{y_{0}}}+a\right)=c_{0}\ln \left({\frac {x}{x_{0}}}\right)+\sum _{i=1}^{n}{\frac {c_{i}-c_{i-1}}{f_{i}}}\ln \left(1+\left({\frac {x}{x_{i}}}\right)^{f_{i}}\right)} 2631: 87:

The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the

5349: 2800: 5913:, respectively. As this method does not assume iid data, it provides an alternative way to determine the power-law exponent for data sets in which the temporal correlation can not be ignored. 6234:

Humphries NE, Queiroz N, Dyer JR, Pade NG, Musyl MK, Schaefer KM, Fuller DW, Brunnschweiler JM, Doyle TK, Houghton JD, Hays GC, Jones CS, Noble LR, Wearmouth VJ, Southall EJ, Sims DW (2010).

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Yu, Frank H.; Willson, Timothy; Frye, Stephen; Edwards, Aled; Bader, Gary D.; Isserlin, Ruth (2011-02-02). "The human genome and drug discovery after a decade. Roads (still) not taken".

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is a scaling factor to ensure that the total area is 1, as required by a probability distribution. More often one uses an asymptotic power law – one that is only true in the limit; see

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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

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The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example,

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Martin, Charles H.; Mahoney, Michael W. (2018-10-02). "Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning".

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The cdf is also a power-law function, but with a smaller scaling exponent. For data, an equivalent form of the cdf is the rank-frequency approach, in which we first sort the

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Another graphical method for the identification of power-law probability distributions using random samples has been proposed. This methodology consists of plotting a

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claimed the need for both a statistical and a theoretical background in order to support a power-law in the underlying mechanism driving the data generating process.

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behavior. This can be seen in the following thought experiment: imagine a room with your friends and estimate the average monthly income in the room. Now imagine the

3575:, the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the 677: 611: 581: 513: 413: 5637: 4313: 2556: 2524: 2295: 5586: 5441: 3070: 3041: 3012: 2983: 2858: 442: 2374: 6674:

Andriani, P.; McKelvey, B. (2007). "Beyond Gaussian averages: redirecting international business and management research toward extreme events and power laws".

5629: 4876: 4222: 3720: 3145: 3090: 2954: 2722: 1070: 462: 383: 226: 206: 4328: 5291:{\displaystyle {\frac {\zeta '({\hat {\alpha }},x_{\min })}{\zeta ({\hat {\alpha }},x_{\min })}}=-{\frac {1}{n}}\sum _{i=1}^{n}\ln {\frac {x_{i}}{x_{\min }}}} 3860:

A straight line on a log–log plot is necessary but insufficient evidence for power-laws, the slope of the straight line corresponds to the power law exponent.

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superficially similar distributions may arise for significantly different reasons, and different models yield different predictions, such as extrapolation.

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in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the

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straight line, then a power-law distribution should be suspected. A limitation of Pareto Q–Q plots is that they behave poorly when the tail index

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Jóhannesson, Gudlaugur; Björnsson, Gunnlaugur; Gudmundsson, Einar H. (2006). "Afterglow Light Curves and Broken Power Laws: A Statistical Study".

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More about these methods, and the conditions under which they can be used, can be found in . Further, this comprehensive review article provides

3443:{\displaystyle \mathbb {E} \left(X^{m}\right)=\int _{x_{\min }}^{\infty }x^{m}p(x)\,\mathrm {d} x={\frac {\alpha -1}{\alpha -1-m}}x_{\min }^{m}} 7779: 4964: 613:; most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of 5975:

corresponding to the lower tail of the lognormal being small (there are very few small values, rather than many small values in a power law).

4677:{\displaystyle {\mathcal {L}}(\alpha )=\log \prod _{i=1}^{n}{\frac {\alpha -1}{x_{\min }}}\left({\frac {x_{i}}{x_{\min }}}\right)^{-\alpha }} 8888: 3589: 9002: 7109: 8675:

Laherrère, J.; Sornette, D. (1998). "Stretched exponential distributions in nature and economy: "fat tails" with characteristic scales".

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Glattfelder, J. B.; Dupuis, A.; Olsen, R. B. (2011-04-01). "Patterns in high-frequency FX data: discovery of 12 empirical scaling laws".

3833:(also called Pareto index) is close to 0, because Pareto Q–Q plots are not designed to identify distributions with slowly varying tails. 3776:

in ecology and with fluctuation scaling in physics. It can also be shown that this variance-to-mean power law, when demonstrated by the

3748:. This distribution is a common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. 711:

of the system. Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach

95:, cloud sizes, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of 3899:, and do not demand the collection of much data). In addition, other types of tail behavior can be identified using bundle plots. 5604: 4344: 8933: 2867: 234: 1648: 5984: 137: 8822: 8769: 7923:

Kendal, WS; Jørgensen, B (2011). "Taylor's power law and fluctuation scaling explained by a central-limit-like convergence".

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Schmittlein, D. C.; Morrison, D. G. (1981). "The median residual lifetime: A characterization theorem and an application".

5355:. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for 2572: 8207:

Arnold, B. C.; Brockett, P. L. (1983). "When does the βth percentile residual life function determine the distribution?".

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Jeong, H.; Tombor, B. Albert; Oltvai, Z.N.; Barabasi, A.-L. (2000). "The large-scale organization of metabolic networks".

3772:. This convergence effect explains why the variance-to-mean power law manifests so widely in natural processes, as with 6625: 6185:

DeWitt, Thomas D.; Garrett, Timothy J.; Rees, Karlie N.; Bois, Corey; Krueger, Steven K.; Ferlay, Nicolas (2024-01-05).

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There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield

2730: 1332: 8939: 8902: 1584: 8618: 8599: 6712:; Terzuolo, Carlo; Viviani, Paolo (1983). "The law relating the kinematic and figural aspects of drawing movements". 2644: 1457: 4713: 4481: 3254: 5923: 5608: 2704:

is a distribution whose density function (or mass function in the discrete case) has the form, for large values of

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Mackay, D. M. (1963). "Psychophysics of perceived intensity:A theoretical basis for Fechner's and Stevens' laws".

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Online Citizen Science and the Widening of Academia: Distributed Engagement with Research and Knowledge Production

1953:{\displaystyle f(x)\propto x_{\text{th}}^{\alpha _{1}-\alpha _{2}}x^{\alpha _{2}}{\text{ for }}x>x_{\text{th}}} 625:

US$ . What happens to the average income in the room? Income is distributed according to a power-law known as the

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Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated

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Lux, Thomas A.; Alfarano, Simone (2016). "Financial power laws: Empirical evidence, models, and mechanisms".

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The pieces of a broken power law can be smoothly spliced together to construct a smoothly broken power law.

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Critical Phenomena in Natural Sciences: Chaos, Fractals, Self-organization and Disorder: Concepts and Tools

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Caballero, Ethan; Gupta, Kshitij; Rish, Irina; Krueger, David (2023-04-24). "Broken Neural Scaling Laws".

7019:"From gene families and genera to incomes and internet file sizes: Why power laws are so common in nature" 7255: 5093: 4881: 1821: 1728: 1509: 1503: 998: 629:(for example, the net worth of Americans is distributed according to a power law with an exponent of 2). 8345:

Morrison, D. G.; Schmittlein, D. C. (1980). "Jobs, strikes, and wars: Probability models for duration".

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Bolmatov, D.; Brazhkin, V. V.; Trachenko, K. (2013). "Thermodynamic behaviour of supercritical matter".

3911:, which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) 3095: 950: 8275: 7352:"On the distribution of links in the interbank network: evidence from the e-MID overnight money market" 5384:

are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa.

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for relationships between biological variables are among the best known power-law functions in nature.

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In general, many alternative functional forms can appear to follow a power-law form for some extent.

5502: 5473: 5358: 1567: 522: 9007: 6026: 4448: 3488: 1766:, consisting of two or more power laws, combined with a threshold. For example, with two power laws: 1384: 813: 618: 8399:

Bauke, H. (2007). "Parameter estimation for power-law distributions by maximum likelihood methods".

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A power law with an exponential cutoff is simply a power law multiplied by an exponential function:

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Clauset, A.; Shalizi, C. R.; Newman, M. E. J. (2009). "Power-Law Distributions in Empirical Data".

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N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation. Cambridge University Press, 1989

7251: 3520: 2701: 1553: 1366: 1321: 3241:{\displaystyle p(x)={\frac {\alpha -1}{x_{\min }}}\left({\frac {x}{x_{\min }}}\right)^{-\alpha },} 2808: 874: 143: 8885: 7018: 6021: 5971: 4435:{\displaystyle p(x)={\frac {\alpha -1}{x_{\min }}}\left({\frac {x}{x_{\min }}}\right)^{-\alpha }} 2861: 1611: 1607: 1559: 8723: 5946:

is a variable real number. As an example, the cumulative distribution of the fracture aperture,

3725: 7439: 6071: 5535: 5446: 5390: 5060: 4921: 4828: 4527: 1724: 1637: 1407: 832:. It is primarily in the study of statistical distributions that the name "power law" is used. 362: 67: 28: 8260:

Joe, H. (1985), "Characterizations of life distributions from percentile residual lifetimes",

5698:{\displaystyle {\hat {\alpha }}={\underset {\alpha }{\operatorname {arg\,min} }}\,D_{\alpha }} 3583:

A modification, which does not satisfy the general form above, with an exponential cutoff, is

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There are different possible ways to splice together power laws. One example is the following:

838: 344: 8992: 6187:"Climatologically invariant scale invariance seen in distributions of cloud horizontal sizes" 6046: 5896: 5595:(Matlab, Python, R and C++) for estimation and testing routines for power-law distributions. 4690: 3882: 3816: 3777: 3769: 3756: 3752: 1755: 1249: 1179: 1173: 1031: 716: 7146:

Li, W. (November 1999). "Random texts exhibit Zipf's-law-like word frequency distribution".

4707:, setting the result equal to zero. Upon rearrangement, this yields the estimator equation: 656: 590: 560: 488: 388: 8948: 8843: 8694: 8647: 8502: 8418: 8165: 7986: 7932: 7834: 7700: 7657: 7606: 7508: 7306: 7083: 7033: 6991: 6948: 6844: 6789: 6709: 6381: 6310: 6250: 6198: 6148: 6129: 5047: 5043: 4521: 4227: 3298: 3294: 2529: 2502: 2274: 1669: 1546: 1193: 821: 817: 762: 754: 632:

On the one hand, this makes it incorrect to apply traditional statistics that are based on

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theory, to share the same fundamental dynamics. For instance, the behavior of water and CO

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Gerchak, Y (1984). "Decreasing failure rates and related issues in the social sciences".

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Diaz, F. J. (1999). "Identifying Tail Behavior by Means of Residual Quantile Functions".

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Guerriero, Vincenzo; Vitale, Stefano; Ciarcia, Sabatino; Mazzoli, Stefano (2011-05-09).

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Moret, M.; Zebende, G. (2007). "Amino acid hydrophobicity and accessible surface area".

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used above) is greater than 1 (otherwise the tail has infinite area), the minimum value

736:, and systems with precisely the same critical exponents are said to belong to the same 46:

An example power-law graph that demonstrates ranking of popularity. To the right is the

35:"Scaling law" redirects here. For statistical laws of scaling deep learning models, see 8867: 8798: 8748: 8710: 8684: 8663: 8637: 8471: 8467: 8434: 8408: 8189: 8155: 8070: 7869: 7850: 7824: 7716: 7630: 7596: 7569: 7543: 7382: 7332: 7277: 6964: 6938: 6873: 6832: 6813: 6779: 6737: 6691: 6606: 6593: 6558: 6410: 6369: 6333: 6298: 6274: 6164: 6138: 5954:

elements is defined as 'the number of fractures per meter having aperture greater than

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The relationship between a CPU's cache size and the number of cache misses follows the

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Albert, J. S.; Bart, H. J.; Reis, R. E. "Species richness & cladal diversity". In

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The maximum of this likelihood is found by differentiating with respect to parameter

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relating the number of species found in an area as a function of the size of the area

708: 614: 7854: 7712: 6968: 6098: 5934:, defined as the number of elements per meter (or area unit, second etc.) for which 8916: 8851: 8790: 8752: 8743: 8738: 8702: 8667: 8655: 8510: 8463: 8426: 8381: 8354: 8327: 8298: 8290: 8243: 8216: 8193: 8173: 8101: 8062: 7994: 7940: 7842: 7758: 7750: 7708: 7665: 7614: 7553: 7516: 7473: 7412: 7366: 7322: 7314: 7155: 7091: 7041: 6999: 6956: 6909: 6868: 6852: 6817: 6797: 6741: 6721: 6683: 6588: 6570: 6485: 6477: 6442: 6405: 6389: 6328: 6318: 6299:"Statistical Analyses Support Power Law Distributions Found in Neuronal Avalanches" 6278: 6258: 6206: 6186: 6156: 5603:

Another method for the estimation of the power-law exponent, which does not assume

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Corral, A, Osso, A, Llebot, JE (2010). "Scaling of tropical cyclone dissipation".

6708: 6695: 6687: 6538:"9na CEPAL Charlas Sobre Sistemas Complejos Sociales (CCSSCS): Leyes de potencias" 8972: 8892: 8810: 8452:

Hall, P. (1982). "On Some Simple Estimates of an Exponent of Regular Variation".

8430: 7557: 7003: 6323: 6056: 5979: 5130:, the maximum likelihood exponent is the solution to the transcendental equation 3773: 3297:. We can now consider several properties of this distribution. For instance, its 1665: 1444: 1350: 1228: 758: 119:

follows frequency power-laws within wide frequency bands for many complex media.

8764:. Lecture Notes in Economics and Mathematical Systems. Vol. 632. Springer. 8724:"A Brief History of Generative Models for Power Law and Lognormal Distributions" 8514: 7520: 7318: 3072:

is the constant function, then we have a power law that holds for all values of

7998: 7944: 7754: 7669: 7071: 7045: 6856: 6481: 5631:, between the cumulative distribution functions of the data and the power law: 5039: 3576: 1603: 1519: 1214: 944: 774: 75: 7370: 6537: 6235: 6160: 6036: 3043:

only controls the shape and finite extent of the lower tail. For instance, if

8986: 8953: 8522: 8106: 8089: 7565: 7485: 7424: 7378: 7167: 6898:"Structural characteristics and radial properties of tropical cloud clusters" 6864: 6629: 6584: 6401: 6220: 1711: 1673: 1661:

relating to maximum simultaneous current and voltage in power semiconductors.

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indicate a deep connection with other, seemingly unrelated systems; see also

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Newman, M. E. J. (2005). "Power laws, Pareto distributions and Zipf's law".

5592: 761:, power laws are often thought to be signatures of hierarchy or of specific 8962:

which bins data to illustrate power-law distributions (if any) in the data.

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Saravia, Leonardo A.; Doyle, Santiago R.; Bond-Lamberty, Ben (2018-12-10).

6809: 6602: 6454: 6419: 6342: 6270: 5803:{\displaystyle D_{\alpha }=\max _{x}|P_{\mathrm {emp} }(x)-P_{\alpha }(x)|} 3798: 1563: 1380:

Richardson's Law for the severity of violent conflicts (wars and terrorism)

1255: 1221: 1198: 812:

However, much of the recent interest in power laws comes from the study of

794: 112: 8976: 8706: 8385: 8331: 8247: 8220: 6733: 5559:

depends strongly on the particular form of the lower tail, represented by

2271:

When the function is plotted as a log-log plot with horizontal axis being

1453:

Large price changes, volatility, and transaction volume on stock exchanges

1231:

relating mean population size and variance of populations sizes in ecology

8832:

Stumpf, M.P.H.; Porter, M.A. (2012). "Critical Truths about Power Laws".

8689: 8303: 8160: 7829: 6982:

Lorenz RD (2009). "Power Law of Dust Devil Diameters on Earth and Mars".

6557:

Taleb, Nassim Nicholas; Bar-Yam, Yaneer; Cirillo, Pasquale (2020-10-20).

6143: 6061: 5031:{\displaystyle \sigma ={\frac {{\hat {\alpha }}-1}{\sqrt {n}}}+O(n^{-1})} 1694: 1297: 798: 786: 100: 92: 21: 8915:: power-law distributions in homelessness and other social problems; by 7327: 6262: 4331:. Some of the most reliable techniques are often based on the method of 8817:. Springer Series in Synergetics (2nd ed.). Heidelberg: Springer. 8802: 8781: 8475: 8074: 7618: 7095: 3856: 1571: 1356: 1337: 1186: 790: 753:

above. The ubiquity of power-law relations in physics is partly due to

59: 8659: 7763: 7400: 7159: 6960: 6490: 6393: 3657:{\displaystyle p(x)\propto L(x)x^{-\alpha }\mathrm {e} ^{-\lambda x}.} 8177: 7968:"Tweedie convergence: a mathematical basis for Taylor's power law, 1/ 7416: 7072:"Scale-free power-laws as interaction between progress and diffusion" 6446: 6041: 4224:

observed values in ascending order, and plot them against the vector

3788:

noise can arise as a consequence of this Tweedie convergence effect.

3702:

eventually overwhelms the power-law behavior at very large values of

1690:

The damping force over speed relation in antiseismic dampers calculus

1677: 1625: 806: 802: 729: 228:

causes only a proportionate scaling of the function itself. That is,

47: 8794: 8234:

Joe, H.; Proschan, F. (1984). "Percentile residual life functions".

8121: 8066: 6801: 6784: 4961: > 100. Further, the standard error of the estimate is 7874: 7846: 7282: 6517: 6515: 6513: 6511: 6509: 6507: 6505: 6503: 6501: 6370:"On a Possible Unified Scaling Law for Volcanic Eruption Durations" 4918:. This estimator exhibits a small finite sample-size bias of order 3809: 1374: 1275: 1220:

The two-thirds power law, relating speed to curvature in the human

770: 633: 584: 8975:

visualizes the distribution of the degrees of the webgraph on the

8779:

Simon, H. A. (1955). "On a Class of Skew Distribution Functions".

8760:

Saichev, Alexander; Malevergne, Yannick; Sornette, Didier (2009).

8642: 8413: 7601: 7548: 6943: 6468:

Staddon, J. E. R. (1978). "Theory of behavioral power functions".

385:

simply multiplies the original power-law relation by the constant

6542: 1475: 778: 6498: 7814: 3127:

from which the law holds. Combining these two cases, and where

1588: 1421:

Distribution of artists by the average price of their artworks.

1245: 824:), which considers the frequency of extremely rare events like 477:

models is an active area of research in statistics; see below.

108: 50:, and to the left are the few that dominate (also known as the 8122:"So You Think You Have a Power Law — Well Isn't That Special?" 7780:"The Wickelgren Power Law and the Ebbinghaus Savings Function" 4322: 3543:, the average and all higher-order moments are infinite; when 8541: 8488: 1111:

below for details. Typically the exponent falls in the range

1049: 8949:

So You Think You Have a Power Law – Well Isn't That Special?

7777: 5893:

denote the cdfs of the data and the power law with exponent

3147:

is a continuous variable, the power law has the form of the

42: 8558: 8556: 6915:

10.1175/1520-0493(1993)121<3234:scarpo>2.0.co;2

5499:, while choosing it too large increases the uncertainty in 1370: 516: 96: 88: 7462: 3092:. In many cases, it is convenient to assume a lower bound 2929:{\displaystyle \lim _{x\rightarrow \infty }L(r\,x)/L(x)=1} 1342:

The law of stream numbers, and the law of stream lengths (

1099:

is needed otherwise the distribution has infinite area as

331:{\displaystyle f(cx)=a(cx)^{-k}=c^{-k}f(x)\propto f(x),\!} 8759: 8145: 8034:

An introduction to statistical modeling of extreme values

7867: 7586: 7365:(4). Springer Science and Business Media LLC: 1463–1495. 6833:"Power laws and critical fragmentation in global forests" 6830: 1287:

Sizes of cloud areas and perimeters, as viewed from space

8956:, Professor of Statistics at Carnegie-Mellon University. 8582:

Historical Biogeography of Neotropical Freshwater Fishes

8553: 8529: 6233: 5387:

Further, both of these estimators require the choice of

1108: 732:. Formally, this sharing of dynamics is referred to as 8965: 3791: 2260:{\displaystyle 0<x_{0}<x_{1}<\cdots <x_{n}} 835:

In empirical contexts, an approximation to a power-law

7533: 2695: 2526:

determines the sharpness of splicing between segments

1450:

The mean absolute change of the logarithmic mid-prices

769:

of income distribution, structural self-similarity of

728:

systems, where the critical point of the system is an

7299:

Physica A: Statistical Mechanics and Its Applications

7297:

Etro, F.; Stepanova, E. (2018). "Power-laws in art".

6184: 5899: 5863: 5819: 5717: 5640: 5617: 5565: 5538: 5505: 5476: 5449: 5420: 5393: 5361: 5307: 5139: 5096: 5063: 4967: 4924: 4884: 4864: 4831: 4716: 4693: 4560: 4530: 4484: 4451: 4356: 4230: 4210: 3983: 3925: 3885: 3819: 3728: 3708: 3673: 3592: 3549: 3523: 3491: 3459: 3310: 3257: 3159: 3133: 3098: 3078: 3049: 3020: 3014:

be asymptotically scale invariant; thus, the form of

2991: 2962: 2942: 2870: 2837: 2811: 2733: 2710: 2647: 2626:{\displaystyle f(x)\propto x^{-\alpha }e^{-\beta x}.} 2575: 2532: 2505: 2447: 2382: 2356: 2303: 2277: 2208: 1978: 1861: 1824: 1775: 1117: 1078: 1058: 1034: 1001: 953: 900: 877: 841: 659: 593: 563: 525: 491: 480: 450: 421: 391: 371: 347: 237: 214: 194: 146: 8627: 8276:"Confidence bands for percentile residual lifetimes" 7234: 6769: 6643: 6559:"On single point forecasts for fat-tailed variables" 6521: 6433:

Stevens, S. S. (1957). "On the psychophysical law".

3755:

are a family of statistical models characterized by

2561: 695:

is the minimum value for which the power law holds.

621:

entering the room, with a monthly income of about 1

8344: 8317: 6556: 6011:field of research in many areas of modern science. 3907:In general, power-law distributions are plotted on 3902: 8905:by Benoit Mandelbrot & Nassim Nicholas Taleb. 8455:Journal of the Royal Statistical Society, Series B 8019:Beirlant, J., Teugels, J. L., Vynckier, P. (1996) 7065: 7063: 6297:Klaus A, Yu S, Plenz D (2011). Zochowski M (ed.). 5905: 5885: 5849: 5802: 5697: 5623: 5580: 5551: 5520: 5491: 5462: 5435: 5406: 5376: 5344:{\displaystyle \zeta (\alpha ,x_{\mathrm {min} })} 5343: 5290: 5122: 5082: 5030: 4949: 4910: 4870: 4850: 4814: 4699: 4676: 4543: 4512: 4470: 4434: 4347:data, we fit a power-law distribution of the form 4307: 4216: 4193: 3966: 3891: 3825: 3740: 3714: 3694: 3656: 3567: 3535: 3509: 3477: 3442: 3285: 3240: 3139: 3119: 3084: 3064: 3035: 3006: 2977: 2948: 2928: 2852: 2823: 2795:{\displaystyle P(X>x)\sim L(x)x^{-(\alpha -1)}} 2794: 2716: 2684: 2625: 2550: 2518: 2491: 2433: 2368: 2342: 2289: 2259: 2194: 1952: 1846: 1810: 1758:use a broken power law; here Kroupa (2001) in red. 1241:(number of species) in clades of freshwater fishes 1135: 1091: 1064: 1040: 1020: 987: 932: 883: 863: 671: 605: 575: 549: 507: 456: 436: 407: 377: 353: 330: 220: 200: 180: 8886:Zipf, Power-laws, and Pareto – a ranking tutorial 8674: 8547: 8055:Journal of Computational and Graphical Statistics 6655: 6368:Cannavò, Flavio; Nunnari, Giuseppe (2016-03-01). 3667:In this distribution, the exponential decay term 1693:Folded solvent-exposed surface areas of centered 1687:in dielectric responses to step DC voltage input. 929: 327: 8984: 7401:"The Granular Origins of Aggregate Fluctuations" 7211:Media/Society: Industries, Images, and Audiences 6673: 6623: 6126: 5732: 5598: 5544: 5455: 5399: 5281: 5210: 5174: 5115: 4903: 4791: 4654: 4625: 4536: 4503: 4463: 4412: 4390: 4156: 4080: 3999: 3430: 3347: 3276: 3215: 3193: 2872: 647:The median does exist, however: for a power law 7965: 7922: 7060: 6895: 4520:is included to ensure that the distribution is 2685:{\displaystyle f(x)\propto x^{\alpha +\beta x}} 1731:with demonstrations that it may be logarithmic) 1244:The Harlow Knapp effect, where a subset of the 8206: 7208:Croteau, David; Hoynes, William (2013-11-06). 6754: 6367: 5038:. This estimator is equivalent to the popular 4815:{\displaystyle {\hat {\alpha }}=1+n\left^{-1}} 4513:{\displaystyle {\frac {\alpha -1}{x_{\min }}}} 3286:{\displaystyle {\frac {\alpha -1}{x_{\min }}}} 1248:found in the human body compose a majority of 16:Functional relationship between two quantities 8934:Critical Mass: How one thing leads to another 8347:Organizational Behavior and Human Performance 8283:Journal of Statistical Planning and Inference 8273: 7296: 7275: 7207: 5532:of our model. In general, the best choice of 1649:Behaviour near second-order phase transitions 8831: 8721: 8562: 8535: 7647: 6928: 6296: 6004: 5999: 5077: 5064: 4845: 4832: 1964: 1430:Distribution of degrees in banking networks. 8585:. Berkeley: University of California Press. 8233: 7959: 7916: 7498: 5916: 4323:Estimating the exponent from empirical data 2985:follows directly from the requirement that 1811:{\displaystyle f(x)\propto x^{\alpha _{1}}} 1558:The input-voltage–output-current curves of 765:. A few notable examples of power laws are 8903:"How the Finance Gurus Get Risk All Wrong" 8578: 8090:"Heavy Tail Modeling and Teletraffic Data" 7350:Fricke, Daniel; Lux, Thomas (2015-02-13). 7016: 7010: 6756: 6702: 6532: 6530: 6355: 6096: 3967:{\displaystyle P(x)=\mathrm {Pr} (X>x)} 1155: 474: 8742: 8688: 8641: 8579:Albert, J. S.; Reis, R. E., eds. (2011). 8412: 8398: 8302: 8159: 8105: 7873: 7828: 7762: 7600: 7547: 7349: 7326: 7281: 6942: 6913: 6872: 6783: 6676:Journal of International Business Studies 6592: 6574: 6489: 6409: 6332: 6322: 6210: 6142: 5684: 5668: 5607:(iid) data, uses the minimization of the 5470:too small produces a significant bias in 4129: 4050: 3695:{\displaystyle \mathrm {e} ^{-\lambda x}} 3381: 3312: 2896: 2499:, smoothly spliced together. The size of 1052:, not to be confused with scaling factor 8926:Benoit Mandelbrot & Richard Hudson: 8809: 8608: 8451: 6649: 6227: 6180: 6178: 6122: 6120: 6118: 6116: 5961: 3855: 1749: 1213:relating animal metabolism to size, and 933:{\displaystyle y=ax^{k}+\varepsilon .\!} 41: 8371: 8087: 8048: 8046: 8044: 8042: 7733: 7437: 7249: 7148:IEEE Transactions on Information Theory 7069: 6889: 6527: 6467: 6432: 6292: 6290: 6288: 6101:. New England Complex Systems Institute 5605:independent and identically distributed 4551:, the log likelihood function becomes: 4345:independent and identically distributed 2864:, which is any function that satisfies 8985: 7736:"Theory of behavioral power functions" 7690: 7180: 7132:Sutton, J. (1997), "Gibrat's Legacy", 6981: 6975: 6922: 6624:Malcolm Gladwell (February 13, 2006). 5922:distribution is described, not by the 1359:, and frequencies of words in a text ( 743: 8778: 6661: 6175: 6113: 5850:{\displaystyle P_{\mathrm {emp} }(x)} 4338: 3873:bundle for the log-transformed sample 2434:{\displaystyle c_{0},c_{1},...,c_{n}} 1672:, such as supercritical exponents of 1645:relating light intensity with voltage 136:One attribute of power laws is their 82: 8898:Stream Morphometry and Horton's Laws 8445: 8052: 8039: 8021:Practical Analysis of Extreme Values 7778:John T. Wixted; Shana K. Carpenter. 6563:International Journal of Forecasting 6285: 3792:Graphical methods for identification 1705: 1583:A 3/2-power law can be found in the 1185:The differential energy spectrum of 9003:Theory of probability distributions 8952:from Three-Toed Sloth, the blog of 8589: 7017:Reed, W. J.; Hughes, B. D. (2002). 5414:. For functions with a non-trivial 5123:{\displaystyle x_{i}\geq x_{\min }} 4911:{\displaystyle x_{i}\geq x_{\min }} 2696:Power-law probability distributions 2636: 1847:{\displaystyle x<x_{\text{th}},} 1745: 1494:the most frequent item, and so on). 1313:and random observation (or killing) 1109:power-law probability distributions 1021:{\displaystyle x>x_{\text{min}}} 131: 13: 8928:The Misbehaviour of Markets (2004) 8468:10.1111/j.2517-6161.1982.tb01183.x 7966:Kendal, WS; Jørgensen, BR (2011). 7145: 6522:Clauset, Shalizi & Newman 2009 5832: 5829: 5826: 5758: 5755: 5752: 5675: 5672: 5669: 5665: 5662: 5659: 5332: 5329: 5326: 4563: 4131: 4111: 4052: 4033: 3945: 3942: 3676: 3635: 3383: 3354: 3120:{\displaystyle x_{\mathrm {min} }} 3111: 3108: 3105: 2882: 1304: 1281: 988:{\displaystyle p(x)=Cx^{-\alpha }} 775:scaling laws in biological systems 541: 481:Lack of well-defined average value 99:in most languages, frequencies of 14: 9019: 8879: 8023:, Leuven: Leuven University Press 7237:The Statistics of Deadly Quarrels 6191:Atmospheric Chemistry and Physics 3801:), mean residual life plots and 3568:{\displaystyle 2<\alpha <3} 2562:Power law with exponential cutoff 2492:{\displaystyle x=x_{1},...,x_{n}} 1580:(ratio of surface area to volume) 1346:'s laws describing river systems) 1136:{\displaystyle 2<\alpha <3} 750: 365:. That is, scaling by a constant 8274:Csorgo, S.; Viharos, L. (1992). 7466:Journal of Banking & Finance 7250:Berreby, David (July 31, 2014). 6576:10.1016/j.ijforecast.2020.08.008 5924:cumulative distribution function 5521:{\displaystyle {\hat {\alpha }}} 5492:{\displaystyle {\hat {\alpha }}} 5377:{\displaystyle {\hat {\alpha }}} 3903:Plotting power-law distributions 871:often includes a deviation term 550:{\displaystyle x\in [1,\infty )} 70:between two quantities, where a 8762:Theory of Zipf's law and beyond 8482: 8392: 8365: 8338: 8311: 8267: 8254: 8227: 8200: 8139: 8114: 8081: 8026: 8013: 7907: 7882: 7861: 7808: 7771: 7727: 7713:10.1126/science.139.3560.1213-a 7684: 7641: 7580: 7527: 7492: 7456: 7431: 7393: 7343: 7290: 7269: 7243: 7228: 7201: 7174: 7139: 7126: 7102: 6824: 6763: 6748: 6667: 6617: 6550: 4471:{\displaystyle x\geq x_{\min }} 4329:unbiased and consistent answers 3849: = 1, ..., 3841:-th order statistic versus the 3510:{\displaystyle m\geq \alpha -1} 3453:which is only well defined for 2700:In a looser sense, a power-law 1103:approaches 0, and the constant 698: 464:, and the straight-line on the 8744:10.1080/15427951.2004.10129088 7501:Chaos, Solitons & Fractals 7134:Journal of Economic Literature 6896:Machado L, Rossow, WB (1993). 6461: 6426: 6361: 6090: 5886:{\displaystyle P_{\alpha }(x)} 5880: 5874: 5844: 5838: 5796: 5792: 5786: 5770: 5764: 5742: 5647: 5575: 5569: 5512: 5483: 5430: 5424: 5368: 5338: 5311: 5215: 5196: 5187: 5179: 5160: 5151: 5025: 5009: 4983: 4944: 4928: 4723: 4574: 4568: 4366: 4360: 4183: 4171: 4047: 4041: 4014: 4002: 3993: 3987: 3961: 3949: 3935: 3929: 3764:similar to the role that the 3617: 3611: 3602: 3596: 3478:{\displaystyle m<\alpha -1} 3378: 3372: 3169: 3163: 3059: 3053: 3030: 3024: 3001: 2995: 2972: 2966: 2917: 2911: 2900: 2890: 2879: 2847: 2841: 2787: 2775: 2764: 2758: 2749: 2737: 2657: 2651: 2585: 2579: 2343:{\displaystyle \ln(y/y_{0}+a)} 2337: 2310: 1871: 1865: 1785: 1779: 1469: 1293:Energy dissipation in cyclones 1092:{\displaystyle x_{\text{min}}} 963: 957: 858: 845: 544: 532: 431: 425: 321: 315: 306: 300: 269: 259: 250: 241: 156: 150: 1: 8613:. Weidenfeld & Nicolson. 8548:Laherrère & Sornette 1998 7438:Neumann, Jerry (2015-06-25). 7252:"Cloudy With a Chance of War" 7235:Lewis Fry Richardson (1950). 7181:Curtis, Vickie (2018-04-20). 6688:10.1057/palgrave.jibs.8400324 6078: 5599:Kolmogorov–Smirnov estimation 3808:Pareto Q–Q plots compare the 3536:{\displaystyle \alpha \leq 2} 1718: 1290:The size of rain-shower cells 1258:globally follows a power law 473:Thus, accurately fitting and 126: 8359:10.1016/0030-5073(80)90065-3 8295:10.1016/0378-3758(92)90159-p 7558:10.1080/14697688.2010.481632 7478:10.1016/0378-4266(90)90009-Q 7004:10.1016/j.icarus.2009.06.029 6726:10.1016/0001-6918(83)90027-6 6324:10.1371/journal.pone.0019779 5609:Kolmogorov–Smirnov statistic 3780:, implies the presence of 1/ 2824:{\displaystyle \alpha >1} 2376:linear segments with slopes 1545:The frequency-dependency of 1487:also called the "80–20 rule" 1401: 1269: 1167: 884:{\displaystyle \varepsilon } 181:{\displaystyle f(x)=ax^{-k}} 7: 8677:European Physical Journal B 8594:. Oxford University Press. 8515:10.1016/j.tecto.2011.01.003 8401:European Physical Journal B 7521:10.1016/j.chaos.2016.01.020 7319:10.1016/j.physa.2018.04.057 6014: 5057:integer-valued data points 1740: 1735:The power law of forgetting 1585:plate characteristic curves 1570:relationship, a factor in " 1512:(continuous), of which the 1462:Average waiting time of an 1146: 111:of organisms, the sizes of 10: 9024: 8944:from The Econophysics Blog 8431:10.1140/epjb/e2007-00219-y 8036:. Springer-Verlag, London. 7999:10.1103/physreve.84.066120 7972:noise and multifractality" 7945:10.1103/physreve.83.066115 7755:10.1037/0033-295x.85.4.305 7734:Staddon, J. E. R. (1978). 7670:10.1103/PhysRevE.75.011920 7046:10.1103/physreve.66.067103 6857:10.1038/s41598-018-36120-w 6482:10.1037/0033-295x.85.4.305 6027:Heavy-tailed distributions 6005:Stumpf & Porter (2012) 6000:Stumpf & Porter (2012) 5985:lognormal density function 3741:{\displaystyle \lambda =0} 2350:, the plot is composed of 1618:Self-organized criticality 1532: 1456:Average waiting time of a 1437: 1408:Population sizes of cities 1394:of the weight matrices of 1327:Highly optimized tolerance 1204: 818:theory of large deviations 34: 18: 8966:The Erdős Webgraph Server 8722:Mitzenmacher, M. (2004). 8262:Ann. Inst. Statist. Math. 7817:The Astrophysical Journal 7371:10.1007/s00181-015-0919-x 7110:"Horton's Laws – Example" 6161:10.1080/00107510500052444 5552:{\displaystyle x_{\min }} 5463:{\displaystyle x_{\min }} 5407:{\displaystyle x_{\min }} 5083:{\displaystyle \{x_{i}\}} 4950:{\displaystyle O(n^{-1})} 4851:{\displaystyle \{x_{i}\}} 4544:{\displaystyle x_{\min }} 3845:-th order statistic, for 1965:Smoothly broken power law 1385:power law of cache misses 1373:(also referred to as the 1263:species–area relationship 814:probability distributions 8941:Tyranny of the Power Law 8563:Stumpf & Porter 2012 8094:The Annals of Statistics 7078:(Submitted manuscript). 7070:Hilbert, Martin (2013). 6757:Albert & Reis (2011) 5972:log-normal distributions 5917:Two-point fitting method 5353:incomplete zeta function 4478:, where the coefficient 3778:method of expanding bins 3251:where the pre-factor to 2936:for any positive factor 2702:probability distribution 2297:and vertical axis being 1762:A broken power law is a 1685:Curie–von Schweidler law 1560:field-effect transistors 1510:Student's t-distribution 1433:Firm-size distributions. 1333:experience curve effects 1322:diffusion of innovations 864:{\displaystyle o(x^{k})} 354:{\displaystyle \propto } 20:Not to be confused with 8913:"Million-dollar Murray" 8856:10.1126/science.1216142 8609:Buchanan, Mark (2000). 8088:Resnick, S. I. (1997). 7440:"Power Laws in Venture" 6626:"Million-Dollar Murray" 6212:10.5194/acp-24-109-2024 6022:Fat-tailed distribution 5906:{\displaystyle \alpha } 4700:{\displaystyle \alpha } 3913:cumulative distribution 3909:doubly logarithmic axes 3892:{\displaystyle \alpha } 3826:{\displaystyle \alpha } 3485:. That is, all moments 2862:slowly varying function 1636:Force and potential in 1612:Electrostatic potential 1608:gravitational potential 1504:Yule–Simon distribution 1349:Populations of cities ( 1156:Artificial Intelligence 1041:{\displaystyle \alpha } 755:dimensional constraints 726:self-organized critical 188:, scaling the argument 121:Allometric scaling laws 68:functional relationship 8107:10.1214/aos/1069362376 7114:www.engr.colostate.edu 6902:Monthly Weather Review 6356:Albert & Reis 2011 5907: 5887: 5851: 5804: 5699: 5625: 5582: 5553: 5522: 5493: 5464: 5437: 5408: 5378: 5345: 5292: 5257: 5124: 5084: 5032: 4957:, which is small when 4951: 4912: 4872: 4852: 4816: 4767: 4701: 4678: 4606: 4545: 4524:. Given a choice for 4514: 4472: 4436: 4309: 4218: 4195: 3968: 3893: 3861: 3827: 3768:has as a focus in the 3742: 3716: 3696: 3658: 3569: 3537: 3511: 3479: 3444: 3287: 3242: 3141: 3121: 3086: 3066: 3037: 3008: 2979: 2950: 2930: 2854: 2825: 2796: 2718: 2686: 2627: 2552: 2520: 2493: 2435: 2370: 2344: 2291: 2261: 2196: 2085: 1954: 1848: 1812: 1759: 1638:simple harmonic motion 1606:, as evidenced by the 1443:Returns for high-risk 1137: 1093: 1066: 1042: 1022: 989: 934: 885: 865: 681:, it takes the value 2 673: 672:{\displaystyle k>1} 619:world's richest person 607: 606:{\displaystyle k>3} 583:, and it has a finite 577: 576:{\displaystyle k>2} 551: 509: 508:{\displaystyle x^{-k}} 458: 438: 409: 408:{\displaystyle c^{-k}} 379: 363:direct proportionality 355: 332: 222: 202: 182: 55: 29:Power (disambiguation) 8707:10.1007/s100510050276 8386:10.1287/opre.32.3.537 8332:10.1287/opre.29.2.392 8248:10.1287/opre.32.3.668 8221:10.1287/opre.31.2.391 7787:Psychological Science 7589:Nature Communications 7214:. SAGE Publications. 6710:Lacquaniti, Francesco 6099:"Concepts: Power Law" 6047:Low-degree saturation 5962:Validating power laws 5908: 5888: 5852: 5805: 5700: 5626: 5583: 5554: 5523: 5494: 5465: 5438: 5409: 5379: 5346: 5293: 5237: 5125: 5085: 5033: 4952: 4913: 4873: 4853: 4817: 4747: 4702: 4679: 4586: 4546: 4515: 4473: 4437: 4310: 4308:{\displaystyle \left} 4219: 4196: 3969: 3894: 3859: 3828: 3770:central limit theorem 3753:Tweedie distributions 3743: 3717: 3697: 3659: 3570: 3538: 3512: 3480: 3445: 3288: 3243: 3142: 3122: 3087: 3067: 3038: 3009: 2980: 2951: 2931: 2855: 2826: 2797: 2719: 2687: 2628: 2553: 2551:{\displaystyle i-1,i} 2521: 2519:{\displaystyle f_{i}} 2494: 2436: 2371: 2345: 2292: 2290:{\displaystyle \ln x} 2262: 2197: 2065: 1955: 1849: 1813: 1756:initial mass function 1753: 1180:initial mass function 1143:, though not always. 1138: 1094: 1067: 1043: 1023: 990: 935: 886: 866: 717:renormalization group 674: 608: 578: 552: 510: 459: 439: 410: 380: 356: 333: 223: 208:by a constant factor 203: 183: 45: 8960:Simple MATLAB script 8923:, February 13, 2006. 8731:Internet Mathematics 8264:37, Part A, 165–172. 7743:Psychological Review 7536:Quantitative Finance 7411:(3): 733–772. 2011. 6470:Psychological Review 6435:Psychological Review 6130:Contemporary Physics 5928:cumulative frequency 5897: 5861: 5817: 5715: 5638: 5615: 5581:{\displaystyle L(x)} 5563: 5536: 5503: 5474: 5447: 5436:{\displaystyle L(x)} 5418: 5391: 5359: 5305: 5137: 5094: 5061: 5048:extreme value theory 5044:quantitative finance 4965: 4922: 4882: 4862: 4829: 4714: 4691: 4558: 4528: 4482: 4449: 4354: 4228: 4208: 3981: 3923: 3915:(ccdf) that is, the 3883: 3817: 3726: 3706: 3671: 3590: 3547: 3521: 3489: 3457: 3308: 3295:normalizing constant 3255: 3157: 3131: 3096: 3076: 3065:{\displaystyle L(x)} 3047: 3036:{\displaystyle L(x)} 3018: 3007:{\displaystyle p(x)} 2989: 2978:{\displaystyle L(x)} 2960: 2940: 2868: 2853:{\displaystyle L(x)} 2835: 2809: 2731: 2708: 2645: 2573: 2530: 2503: 2445: 2380: 2354: 2301: 2275: 2206: 1976: 1859: 1822: 1773: 1670:supercritical fluids 1554:Stefan–Boltzmann law 1547:acoustic attenuation 1427:in a market economy. 1396:deep neural networks 1115: 1076: 1056: 1032: 999: 951: 898: 875: 839: 826:stock market crashes 822:extreme value theory 763:stochastic processes 657: 591: 561: 523: 489: 475:validating power-law 468:is often called the 448: 437:{\displaystyle f(x)} 419: 389: 369: 345: 235: 212: 192: 144: 117:Acoustic attenuation 27:For other uses, see 8848:2012Sci...335..665S 8699:1998EPJB....2..525L 8652:2009SIAMR..51..661C 8507:2011Tectp.504...14G 8423:2007EPJB...58..167B 8374:Operations Research 8320:Operations Research 8236:Operations Research 8209:Operations Research 8170:2000Natur.407..651J 7991:2011PhRvE..84f6120K 7937:2011PhRvE..83f6115K 7839:2006ApJ...640L...5J 7705:1963Sci...139.1213M 7699:(3560): 1213–1216. 7662:2007PhRvE..75a1920M 7611:2013NatCo...4.2331B 7513:2016CSF....88....3L 7359:Empirical Economics 7311:2018PhyA..506..217E 7088:2014Cmplx..19d..56H 7038:2002PhRvE..66f7103R 6996:2009Icar..203..683L 6953:2010NatPh...6..693C 6849:2018NatSR...817766S 6794:2011Natur.470..163E 6546:. 31 December 2013. 6386:2016NatSR...622289C 6315:2011PLoSO...619779K 6263:10.1038/nature09116 6255:2010Natur.465.1066H 6249:(7301): 1066–1069. 6203:2024ACP....24..109D 6153:2005ConPh..46..323N 6072:Stevens's power law 6067:Stable distribution 6052:Pareto distribution 5443:function, choosing 5090:, again where each 4115: 4098: 4037: 3766:normal distribution 3439: 3358: 3149:Pareto distribution 2956:. This property of 2369:{\displaystyle n+1} 1911: 1754:Some models of the 1725:Stevens's power law 1659:safe operating area 1632:van der Waals force 1596:inverse-square laws 1514:Cauchy distribution 1481:Pareto distribution 1425:Income distribution 1234:Neuronal avalanches 1028:where the exponent 744:Power-law functions 715:—can be shown, via 642:regression analysis 627:Pareto distribution 515:has a well-defined 140:. Given a relation 8971:2021-03-01 at the 8891:2007-10-26 at the 7890:"Curved-power law" 7656:(1 Pt 1). 011920. 7619:10.1038/ncomms3331 7096:10.1002/cplx.21485 6837:Scientific Reports 6759:, pp. 89–104. 6374:Scientific Reports 5950:, for a sample of 5903: 5883: 5847: 5800: 5740: 5695: 5682: 5621: 5578: 5549: 5528:, and reduces the 5518: 5489: 5460: 5433: 5404: 5374: 5341: 5288: 5120: 5080: 5028: 4947: 4908: 4868: 4848: 4812: 4697: 4674: 4541: 4510: 4468: 4432: 4339:Maximum likelihood 4333:maximum likelihood 4305: 4214: 4191: 4101: 4075: 4023: 3964: 3889: 3877:quantile functions 3862: 3823: 3738: 3712: 3692: 3654: 3565: 3533: 3507: 3475: 3440: 3425: 3337: 3283: 3238: 3137: 3117: 3082: 3062: 3033: 3004: 2975: 2946: 2926: 2886: 2850: 2821: 2792: 2714: 2682: 2623: 2548: 2516: 2489: 2431: 2366: 2340: 2287: 2257: 2192: 1950: 1877: 1844: 1808: 1764:piecewise function 1760: 1727:of psychophysics ( 1653:critical exponents 1526:scale-free network 1458:directional change 1318:exponential growth 1311:Exponential growth 1300:on Earth and Mars 1250:published research 1174:Kepler's third law 1162:Neural scaling law 1133: 1089: 1062: 1038: 1018: 985: 930: 881: 861: 738:universality class 709:critical exponents 669: 638:standard deviation 603: 573: 547: 505: 454: 434: 405: 375: 351: 328: 218: 198: 178: 83:Empirical examples 56: 37:Neural scaling law 8842:(6069): 665–666. 8824:978-3-540-30882-9 8771:978-3-642-02945-5 8660:10.1137/070710111 8590:Bak, Per (1997). 8536:Mitzenmacher 2004 8154:(6804): 651–654. 8032:Coles, S. (2001) 7650:Physical Review E 7256:Nautilus Magazine 7221:978-1-4833-2355-8 7194:978-3-319-77664-4 7160:10.1109/18.165464 6961:10.1038/nphys1725 6908:(12): 3234–3260. 6778:(7333): 163–165. 6714:Acta Psychologica 6394:10.1038/srep22289 6032:Hyperbolic growth 5731: 5657: 5650: 5624:{\displaystyle D} 5530:statistical power 5515: 5486: 5371: 5286: 5235: 5219: 5199: 5163: 5001: 5000: 4986: 4871:{\displaystyle n} 4796: 4726: 4659: 4630: 4508: 4417: 4395: 4343:For real-valued, 4298: 4279: 4258: 4217:{\displaystyle n} 4161: 4099: 3917:survival function 3784:noise and that 1/ 3715:{\displaystyle x} 3423: 3281: 3220: 3198: 3140:{\displaystyle x} 3085:{\displaystyle x} 2949:{\displaystyle r} 2871: 2717:{\displaystyle x} 2168: 2129: 2056: 2005: 1947: 1932: 1884: 1838: 1714:of assembly sizes 1706:Political Science 1699:protein structure 1600:Newtonian gravity 1542:in aerosol optics 1540:Angstrom exponent 1516:is a special case 1498:Zeta distribution 1331:Proposed form of 1316:Progress through 1086: 1065:{\displaystyle a} 1015: 830:natural disasters 705:phase transitions 457:{\displaystyle x} 378:{\displaystyle c} 221:{\displaystyle c} 201:{\displaystyle x} 9015: 9008:Statistical laws 8917:Malcolm Gladwell 8909:, July 11, 2005. 8875: 8828: 8811:Sornette, Didier 8806: 8789:(3/4): 425–440. 8775: 8756: 8746: 8728: 8718: 8692: 8690:cond-mat/9801293 8671: 8645: 8624: 8605: 8592:How nature works 8586: 8566: 8560: 8551: 8545: 8539: 8533: 8527: 8526: 8486: 8480: 8479: 8449: 8443: 8442: 8416: 8396: 8390: 8389: 8369: 8363: 8362: 8342: 8336: 8335: 8315: 8309: 8308: 8306: 8280: 8271: 8265: 8258: 8252: 8251: 8231: 8225: 8224: 8204: 8198: 8197: 8178:10.1038/35036627 8163: 8161:cond-mat/0010278 8143: 8137: 8136: 8134: 8132: 8118: 8112: 8111: 8109: 8100:(5): 1805–1869. 8085: 8079: 8078: 8050: 8037: 8030: 8024: 8017: 8011: 8010: 7976: 7963: 7957: 7956: 7920: 7914: 7911: 7905: 7904: 7902: 7901: 7892:. Archived from 7886: 7880: 7879: 7877: 7865: 7859: 7858: 7832: 7830:astro-ph/0602219 7812: 7806: 7805: 7803: 7801: 7796:on April 8, 2016 7795: 7789:. Archived from 7784: 7775: 7769: 7768: 7766: 7740: 7731: 7725: 7724: 7688: 7682: 7681: 7645: 7639: 7638: 7604: 7584: 7578: 7577: 7551: 7531: 7525: 7524: 7496: 7490: 7489: 7472:(6): 1189–1208. 7460: 7454: 7453: 7451: 7450: 7435: 7429: 7428: 7417:10.3982/ecta8769 7397: 7391: 7390: 7356: 7347: 7341: 7340: 7330: 7294: 7288: 7287: 7285: 7273: 7267: 7266: 7264: 7262: 7247: 7241: 7240: 7232: 7226: 7225: 7205: 7199: 7198: 7178: 7172: 7171: 7154:(6): 1842–1845. 7143: 7137: 7130: 7124: 7123: 7121: 7120: 7106: 7100: 7099: 7067: 7058: 7057: 7023: 7014: 7008: 7007: 6979: 6973: 6972: 6946: 6926: 6920: 6919: 6917: 6893: 6887: 6886: 6876: 6828: 6822: 6821: 6787: 6767: 6761: 6760: 6752: 6746: 6745: 6720:(1–3): 115–130. 6706: 6700: 6699: 6682:(7): 1212–1230. 6671: 6665: 6659: 6653: 6647: 6641: 6640: 6638: 6637: 6628:. Archived from 6621: 6615: 6614: 6596: 6578: 6554: 6548: 6547: 6534: 6525: 6519: 6496: 6495: 6493: 6465: 6459: 6458: 6447:10.1037/h0046162 6430: 6424: 6423: 6413: 6365: 6359: 6353: 6347: 6346: 6336: 6326: 6294: 6283: 6282: 6240: 6231: 6225: 6224: 6214: 6182: 6173: 6172: 6146: 6144:cond-mat/0412004 6124: 6111: 6110: 6108: 6106: 6097:Yaneer Bar-Yam. 6094: 5994: 5987:is quadratic in 5938: > 5912: 5910: 5909: 5904: 5892: 5890: 5889: 5884: 5873: 5872: 5856: 5854: 5853: 5848: 5837: 5836: 5835: 5809: 5807: 5806: 5801: 5799: 5785: 5784: 5763: 5762: 5761: 5745: 5739: 5727: 5726: 5704: 5702: 5701: 5696: 5694: 5693: 5683: 5678: 5652: 5651: 5643: 5630: 5628: 5627: 5622: 5587: 5585: 5584: 5579: 5558: 5556: 5555: 5550: 5548: 5547: 5527: 5525: 5524: 5519: 5517: 5516: 5508: 5498: 5496: 5495: 5490: 5488: 5487: 5479: 5469: 5467: 5466: 5461: 5459: 5458: 5442: 5440: 5439: 5434: 5413: 5411: 5410: 5405: 5403: 5402: 5383: 5381: 5380: 5375: 5373: 5372: 5364: 5350: 5348: 5347: 5342: 5337: 5336: 5335: 5297: 5295: 5294: 5289: 5287: 5285: 5284: 5275: 5274: 5265: 5256: 5251: 5236: 5228: 5220: 5218: 5214: 5213: 5201: 5200: 5192: 5182: 5178: 5177: 5165: 5164: 5156: 5150: 5141: 5129: 5127: 5126: 5121: 5119: 5118: 5106: 5105: 5089: 5087: 5086: 5081: 5076: 5075: 5037: 5035: 5034: 5029: 5024: 5023: 5002: 4996: 4995: 4988: 4987: 4979: 4975: 4956: 4954: 4953: 4948: 4943: 4942: 4917: 4915: 4914: 4909: 4907: 4906: 4894: 4893: 4877: 4875: 4874: 4869: 4857: 4855: 4854: 4849: 4844: 4843: 4821: 4819: 4818: 4813: 4811: 4810: 4802: 4798: 4797: 4795: 4794: 4785: 4784: 4775: 4766: 4761: 4728: 4727: 4719: 4706: 4704: 4703: 4698: 4683: 4681: 4680: 4675: 4673: 4672: 4664: 4660: 4658: 4657: 4648: 4647: 4638: 4631: 4629: 4628: 4619: 4608: 4605: 4600: 4567: 4566: 4550: 4548: 4547: 4542: 4540: 4539: 4519: 4517: 4516: 4511: 4509: 4507: 4506: 4497: 4486: 4477: 4475: 4474: 4469: 4467: 4466: 4441: 4439: 4438: 4433: 4431: 4430: 4422: 4418: 4416: 4415: 4403: 4396: 4394: 4393: 4384: 4373: 4314: 4312: 4311: 4306: 4304: 4300: 4299: 4291: 4280: 4275: 4264: 4259: 4254: 4243: 4223: 4221: 4220: 4215: 4200: 4198: 4197: 4192: 4187: 4186: 4166: 4162: 4160: 4159: 4147: 4134: 4128: 4127: 4114: 4109: 4100: 4097: 4083: 4074: 4063: 4055: 4036: 4031: 3973: 3971: 3970: 3965: 3948: 3898: 3896: 3895: 3890: 3832: 3830: 3829: 3824: 3747: 3745: 3744: 3739: 3721: 3719: 3718: 3713: 3701: 3699: 3698: 3693: 3691: 3690: 3679: 3663: 3661: 3660: 3655: 3650: 3649: 3638: 3632: 3631: 3574: 3572: 3571: 3566: 3542: 3540: 3539: 3534: 3516: 3514: 3513: 3508: 3484: 3482: 3481: 3476: 3449: 3447: 3446: 3441: 3438: 3433: 3424: 3422: 3405: 3394: 3386: 3368: 3367: 3357: 3352: 3351: 3350: 3333: 3329: 3328: 3315: 3292: 3290: 3289: 3284: 3282: 3280: 3279: 3270: 3259: 3247: 3245: 3244: 3239: 3234: 3233: 3225: 3221: 3219: 3218: 3206: 3199: 3197: 3196: 3187: 3176: 3146: 3144: 3143: 3138: 3126: 3124: 3123: 3118: 3116: 3115: 3114: 3091: 3089: 3088: 3083: 3071: 3069: 3068: 3063: 3042: 3040: 3039: 3034: 3013: 3011: 3010: 3005: 2984: 2982: 2981: 2976: 2955: 2953: 2952: 2947: 2935: 2933: 2932: 2927: 2907: 2885: 2859: 2857: 2856: 2851: 2830: 2828: 2827: 2822: 2801: 2799: 2798: 2793: 2791: 2790: 2723: 2721: 2720: 2715: 2691: 2689: 2688: 2683: 2681: 2680: 2637:Curved power law 2632: 2630: 2629: 2624: 2619: 2618: 2603: 2602: 2557: 2555: 2554: 2549: 2525: 2523: 2522: 2517: 2515: 2514: 2498: 2496: 2495: 2490: 2488: 2487: 2463: 2462: 2440: 2438: 2437: 2432: 2430: 2429: 2405: 2404: 2392: 2391: 2375: 2373: 2372: 2367: 2349: 2347: 2346: 2341: 2330: 2329: 2320: 2296: 2294: 2293: 2288: 2266: 2264: 2263: 2258: 2256: 2255: 2237: 2236: 2224: 2223: 2201: 2199: 2198: 2193: 2191: 2187: 2186: 2185: 2184: 2183: 2173: 2169: 2167: 2166: 2154: 2130: 2128: 2127: 2118: 2117: 2116: 2098: 2097: 2087: 2084: 2079: 2061: 2057: 2055: 2054: 2042: 2030: 2029: 2017: 2013: 2006: 2004: 2003: 1991: 1959: 1957: 1956: 1951: 1949: 1948: 1945: 1933: 1930: 1928: 1927: 1926: 1925: 1910: 1909: 1908: 1896: 1895: 1885: 1882: 1853: 1851: 1850: 1845: 1840: 1839: 1836: 1817: 1815: 1814: 1809: 1807: 1806: 1805: 1804: 1746:Broken power law 1643:Gamma correction 1549:in complex media 1485:Pareto principle 1392:spectral density 1367:90–9–1 principle 1320:and exponential 1239:species richness 1194:M–sigma relation 1142: 1140: 1139: 1134: 1098: 1096: 1095: 1090: 1088: 1087: 1084: 1071: 1069: 1068: 1063: 1047: 1045: 1044: 1039: 1027: 1025: 1024: 1019: 1017: 1016: 1013: 994: 992: 991: 986: 984: 983: 939: 937: 936: 931: 919: 918: 890: 888: 887: 882: 870: 868: 867: 862: 857: 856: 783:computer science 680: 678: 676: 675: 670: 651:, with exponent 612: 610: 609: 604: 582: 580: 579: 574: 556: 554: 553: 548: 514: 512: 511: 506: 504: 503: 463: 461: 460: 455: 443: 441: 440: 435: 414: 412: 411: 406: 404: 403: 384: 382: 381: 376: 360: 358: 357: 352: 337: 335: 334: 329: 296: 295: 280: 279: 227: 225: 224: 219: 207: 205: 204: 199: 187: 185: 184: 179: 177: 176: 138:scale invariance 132:Scale invariance 105:species richness 32: 25: 9023: 9022: 9018: 9017: 9016: 9014: 9013: 9012: 8983: 8982: 8973:Wayback Machine 8893:Wayback Machine 8882: 8825: 8795:10.2307/2333389 8772: 8726: 8621: 8602: 8570: 8569: 8561: 8554: 8546: 8542: 8534: 8530: 8487: 8483: 8450: 8446: 8397: 8393: 8370: 8366: 8343: 8339: 8316: 8312: 8278: 8272: 8268: 8259: 8255: 8232: 8228: 8205: 8201: 8144: 8140: 8130: 8128: 8120: 8119: 8115: 8086: 8082: 8067:10.2307/1390871 8051: 8040: 8031: 8027: 8018: 8014: 7974: 7964: 7960: 7921: 7917: 7912: 7908: 7899: 7897: 7888: 7887: 7883: 7866: 7862: 7813: 7809: 7799: 7797: 7793: 7782: 7776: 7772: 7738: 7732: 7728: 7689: 7685: 7646: 7642: 7585: 7581: 7532: 7528: 7497: 7493: 7461: 7457: 7448: 7446: 7436: 7432: 7399: 7398: 7394: 7354: 7348: 7344: 7295: 7291: 7274: 7270: 7260: 7258: 7248: 7244: 7233: 7229: 7222: 7206: 7202: 7195: 7179: 7175: 7144: 7140: 7131: 7127: 7118: 7116: 7108: 7107: 7103: 7068: 7061: 7021: 7015: 7011: 6980: 6976: 6927: 6923: 6894: 6890: 6829: 6825: 6802:10.1038/470163a 6768: 6764: 6753: 6749: 6707: 6703: 6672: 6668: 6660: 6656: 6648: 6644: 6635: 6633: 6622: 6618: 6555: 6551: 6536: 6535: 6528: 6520: 6499: 6466: 6462: 6431: 6427: 6366: 6362: 6354: 6350: 6295: 6286: 6238: 6232: 6228: 6183: 6176: 6125: 6114: 6104: 6102: 6095: 6091: 6081: 6076: 6057:Power-law fluid 6017: 5992: 5988: 5964: 5942:applies, where 5919: 5898: 5895: 5894: 5868: 5864: 5862: 5859: 5858: 5825: 5824: 5820: 5818: 5815: 5814: 5795: 5780: 5776: 5751: 5750: 5746: 5741: 5735: 5722: 5718: 5716: 5713: 5712: 5689: 5685: 5658: 5656: 5642: 5641: 5639: 5636: 5635: 5616: 5613: 5612: 5601: 5564: 5561: 5560: 5543: 5539: 5537: 5534: 5533: 5507: 5506: 5504: 5501: 5500: 5478: 5477: 5475: 5472: 5471: 5454: 5450: 5448: 5445: 5444: 5419: 5416: 5415: 5398: 5394: 5392: 5389: 5388: 5363: 5362: 5360: 5357: 5356: 5325: 5324: 5320: 5306: 5303: 5302: 5280: 5276: 5270: 5266: 5264: 5252: 5241: 5227: 5209: 5205: 5191: 5190: 5183: 5173: 5169: 5155: 5154: 5143: 5142: 5140: 5138: 5135: 5134: 5114: 5110: 5101: 5097: 5095: 5092: 5091: 5071: 5067: 5062: 5059: 5058: 5016: 5012: 4978: 4977: 4976: 4974: 4966: 4963: 4962: 4935: 4931: 4923: 4920: 4919: 4902: 4898: 4889: 4885: 4883: 4880: 4879: 4863: 4860: 4859: 4839: 4835: 4830: 4827: 4826: 4803: 4790: 4786: 4780: 4776: 4774: 4762: 4751: 4746: 4742: 4741: 4718: 4717: 4715: 4712: 4711: 4692: 4689: 4688: 4665: 4653: 4649: 4643: 4639: 4637: 4633: 4632: 4624: 4620: 4609: 4607: 4601: 4590: 4562: 4561: 4559: 4556: 4555: 4535: 4531: 4529: 4526: 4525: 4502: 4498: 4487: 4485: 4483: 4480: 4479: 4462: 4458: 4450: 4447: 4446: 4423: 4411: 4407: 4402: 4398: 4397: 4389: 4385: 4374: 4372: 4355: 4352: 4351: 4341: 4325: 4290: 4265: 4263: 4244: 4242: 4235: 4231: 4229: 4226: 4225: 4209: 4206: 4205: 4167: 4155: 4151: 4146: 4142: 4141: 4130: 4120: 4116: 4110: 4105: 4084: 4079: 4064: 4062: 4051: 4032: 4027: 3982: 3979: 3978: 3941: 3924: 3921: 3920: 3905: 3884: 3881: 3880: 3818: 3815: 3814: 3794: 3727: 3724: 3723: 3707: 3704: 3703: 3680: 3675: 3674: 3672: 3669: 3668: 3639: 3634: 3633: 3624: 3620: 3591: 3588: 3587: 3548: 3545: 3544: 3522: 3519: 3518: 3490: 3487: 3486: 3458: 3455: 3454: 3434: 3429: 3406: 3395: 3393: 3382: 3363: 3359: 3353: 3346: 3342: 3341: 3324: 3320: 3316: 3311: 3309: 3306: 3305: 3275: 3271: 3260: 3258: 3256: 3253: 3252: 3226: 3214: 3210: 3205: 3201: 3200: 3192: 3188: 3177: 3175: 3158: 3155: 3154: 3132: 3129: 3128: 3104: 3103: 3099: 3097: 3094: 3093: 3077: 3074: 3073: 3048: 3045: 3044: 3019: 3016: 3015: 2990: 2987: 2986: 2961: 2958: 2957: 2941: 2938: 2937: 2903: 2875: 2869: 2866: 2865: 2836: 2833: 2832: 2810: 2807: 2806: 2771: 2767: 2732: 2729: 2728: 2709: 2706: 2705: 2698: 2667: 2663: 2646: 2643: 2642: 2639: 2608: 2604: 2595: 2591: 2574: 2571: 2570: 2564: 2531: 2528: 2527: 2510: 2506: 2504: 2501: 2500: 2483: 2479: 2458: 2454: 2446: 2443: 2442: 2441:, separated at 2425: 2421: 2400: 2396: 2387: 2383: 2381: 2378: 2377: 2355: 2352: 2351: 2325: 2321: 2316: 2302: 2299: 2298: 2276: 2273: 2272: 2270: 2251: 2247: 2232: 2228: 2219: 2215: 2207: 2204: 2203: 2179: 2175: 2174: 2162: 2158: 2153: 2149: 2148: 2141: 2137: 2123: 2119: 2106: 2102: 2093: 2089: 2088: 2086: 2080: 2069: 2050: 2046: 2041: 2037: 2025: 2021: 1999: 1995: 1990: 1989: 1985: 1977: 1974: 1973: 1967: 1944: 1940: 1931: for 1929: 1921: 1917: 1916: 1912: 1904: 1900: 1891: 1887: 1886: 1881: 1860: 1857: 1856: 1835: 1831: 1823: 1820: 1819: 1800: 1796: 1795: 1791: 1774: 1771: 1770: 1748: 1743: 1721: 1708: 1666:state of matter 1614:, respectively. 1578:Square–cube law 1535: 1472: 1445:venture capital 1440: 1410:in a region or 1404: 1307: 1305:General science 1284: 1282:Climate science 1272: 1215:allometric laws 1207: 1170: 1158: 1149: 1116: 1113: 1112: 1083: 1079: 1077: 1074: 1073: 1057: 1054: 1053: 1033: 1030: 1029: 1012: 1008: 1000: 997: 996: 976: 972: 952: 949: 948: 914: 910: 899: 896: 895: 876: 873: 872: 852: 848: 840: 837: 836: 759:complex systems 746: 722: 701: 694: 687: 658: 655: 654: 652: 592: 589: 588: 562: 559: 558: 524: 521: 520: 496: 492: 490: 487: 486: 483: 449: 446: 445: 420: 417: 416: 396: 392: 390: 387: 386: 370: 367: 366: 346: 343: 342: 288: 284: 272: 268: 236: 233: 232: 213: 210: 209: 193: 190: 189: 169: 165: 145: 142: 141: 134: 129: 85: 72:relative change 40: 33: 26: 19: 17: 12: 11: 5: 9021: 9011: 9010: 9005: 9000: 8995: 8981: 8980: 8963: 8957: 8945: 8937: 8930: 8924: 8921:The New Yorker 8910: 8900: 8895: 8881: 8880:External links 8878: 8877: 8876: 8829: 8823: 8807: 8776: 8770: 8757: 8737:(2): 226–251. 8719: 8683:(4): 525–539. 8672: 8636:(4): 661–703. 8625: 8619: 8606: 8600: 8587: 8568: 8567: 8552: 8540: 8528: 8495:Tectonophysics 8481: 8444: 8407:(2): 167–173. 8391: 8380:(3): 537–546. 8364: 8353:(2): 224–251. 8337: 8326:(2): 392–399. 8310: 8289:(3): 327–337. 8266: 8253: 8242:(3): 668–678. 8226: 8215:(2): 391–396. 8199: 8138: 8113: 8080: 8061:(3): 493–509. 8038: 8025: 8012: 7958: 7915: 7906: 7881: 7860: 7847:10.1086/503294 7807: 7770: 7749:(4): 305–320. 7726: 7683: 7640: 7579: 7542:(4): 599–614. 7526: 7491: 7455: 7430: 7392: 7342: 7289: 7268: 7242: 7227: 7220: 7200: 7193: 7173: 7138: 7125: 7101: 7059: 7009: 6990:(2): 683–684. 6974: 6937:(9): 693–696. 6931:Nature Physics 6921: 6888: 6823: 6762: 6747: 6701: 6666: 6654: 6642: 6616: 6569:(2): 413–422. 6549: 6526: 6497: 6476:(4): 305–320. 6460: 6441:(3): 153–181. 6425: 6360: 6348: 6284: 6226: 6197:(1): 109–122. 6174: 6137:(5): 323–351. 6112: 6088: 6087: 6080: 6077: 6075: 6074: 6069: 6064: 6059: 6054: 6049: 6044: 6039: 6034: 6029: 6024: 6018: 6016: 6013: 5990: 5963: 5960: 5930:of a property 5918: 5915: 5902: 5882: 5879: 5876: 5871: 5867: 5846: 5843: 5840: 5834: 5831: 5828: 5823: 5811: 5810: 5798: 5794: 5791: 5788: 5783: 5779: 5775: 5772: 5769: 5766: 5760: 5757: 5754: 5749: 5744: 5738: 5734: 5730: 5725: 5721: 5706: 5705: 5692: 5688: 5681: 5677: 5674: 5671: 5667: 5664: 5661: 5655: 5649: 5646: 5620: 5600: 5597: 5577: 5574: 5571: 5568: 5546: 5542: 5514: 5511: 5485: 5482: 5457: 5453: 5432: 5429: 5426: 5423: 5401: 5397: 5370: 5367: 5340: 5334: 5331: 5328: 5323: 5319: 5316: 5313: 5310: 5299: 5298: 5283: 5279: 5273: 5269: 5263: 5260: 5255: 5250: 5247: 5244: 5240: 5234: 5231: 5226: 5223: 5217: 5212: 5208: 5204: 5198: 5195: 5189: 5186: 5181: 5176: 5172: 5168: 5162: 5159: 5153: 5149: 5146: 5117: 5113: 5109: 5104: 5100: 5079: 5074: 5070: 5066: 5040:Hill estimator 5027: 5022: 5019: 5015: 5011: 5008: 5005: 4999: 4994: 4991: 4985: 4982: 4973: 4970: 4946: 4941: 4938: 4934: 4930: 4927: 4905: 4901: 4897: 4892: 4888: 4867: 4847: 4842: 4838: 4834: 4823: 4822: 4809: 4806: 4801: 4793: 4789: 4783: 4779: 4773: 4770: 4765: 4760: 4757: 4754: 4750: 4745: 4740: 4737: 4734: 4731: 4725: 4722: 4696: 4685: 4684: 4671: 4668: 4663: 4656: 4652: 4646: 4642: 4636: 4627: 4623: 4618: 4615: 4612: 4604: 4599: 4596: 4593: 4589: 4585: 4582: 4579: 4576: 4573: 4570: 4565: 4538: 4534: 4505: 4501: 4496: 4493: 4490: 4465: 4461: 4457: 4454: 4443: 4442: 4429: 4426: 4421: 4414: 4410: 4406: 4401: 4392: 4388: 4383: 4380: 4377: 4371: 4368: 4365: 4362: 4359: 4340: 4337: 4324: 4321: 4303: 4297: 4294: 4289: 4286: 4283: 4278: 4274: 4271: 4268: 4262: 4257: 4253: 4250: 4247: 4241: 4238: 4234: 4213: 4202: 4201: 4190: 4185: 4182: 4179: 4176: 4173: 4170: 4165: 4158: 4154: 4150: 4145: 4140: 4137: 4133: 4126: 4123: 4119: 4113: 4108: 4104: 4096: 4093: 4090: 4087: 4082: 4078: 4073: 4070: 4067: 4061: 4058: 4054: 4049: 4046: 4043: 4040: 4035: 4030: 4026: 4022: 4019: 4016: 4013: 4010: 4007: 4004: 4001: 3998: 3995: 3992: 3989: 3986: 3963: 3960: 3957: 3954: 3951: 3947: 3944: 3940: 3937: 3934: 3931: 3928: 3904: 3901: 3888: 3822: 3793: 3790: 3737: 3734: 3731: 3711: 3689: 3686: 3683: 3678: 3665: 3664: 3653: 3648: 3645: 3642: 3637: 3630: 3627: 3623: 3619: 3616: 3613: 3610: 3607: 3604: 3601: 3598: 3595: 3577:central moment 3564: 3561: 3558: 3555: 3552: 3532: 3529: 3526: 3517:diverge: when 3506: 3503: 3500: 3497: 3494: 3474: 3471: 3468: 3465: 3462: 3451: 3450: 3437: 3432: 3428: 3421: 3418: 3415: 3412: 3409: 3404: 3401: 3398: 3392: 3389: 3385: 3380: 3377: 3374: 3371: 3366: 3362: 3356: 3349: 3345: 3340: 3336: 3332: 3327: 3323: 3319: 3314: 3278: 3274: 3269: 3266: 3263: 3249: 3248: 3237: 3232: 3229: 3224: 3217: 3213: 3209: 3204: 3195: 3191: 3186: 3183: 3180: 3174: 3171: 3168: 3165: 3162: 3136: 3113: 3110: 3107: 3102: 3081: 3061: 3058: 3055: 3052: 3032: 3029: 3026: 3023: 3003: 3000: 2997: 2994: 2974: 2971: 2968: 2965: 2945: 2925: 2922: 2919: 2916: 2913: 2910: 2906: 2902: 2899: 2895: 2892: 2889: 2884: 2881: 2878: 2874: 2849: 2846: 2843: 2840: 2820: 2817: 2814: 2803: 2802: 2789: 2786: 2783: 2780: 2777: 2774: 2770: 2766: 2763: 2760: 2757: 2754: 2751: 2748: 2745: 2742: 2739: 2736: 2713: 2697: 2694: 2693: 2692: 2679: 2676: 2673: 2670: 2666: 2662: 2659: 2656: 2653: 2650: 2638: 2635: 2634: 2633: 2622: 2617: 2614: 2611: 2607: 2601: 2598: 2594: 2590: 2587: 2584: 2581: 2578: 2563: 2560: 2547: 2544: 2541: 2538: 2535: 2513: 2509: 2486: 2482: 2478: 2475: 2472: 2469: 2466: 2461: 2457: 2453: 2450: 2428: 2424: 2420: 2417: 2414: 2411: 2408: 2403: 2399: 2395: 2390: 2386: 2365: 2362: 2359: 2339: 2336: 2333: 2328: 2324: 2319: 2315: 2312: 2309: 2306: 2286: 2283: 2280: 2254: 2250: 2246: 2243: 2240: 2235: 2231: 2227: 2222: 2218: 2214: 2211: 2190: 2182: 2178: 2172: 2165: 2161: 2157: 2152: 2147: 2144: 2140: 2136: 2133: 2126: 2122: 2115: 2112: 2109: 2105: 2101: 2096: 2092: 2083: 2078: 2075: 2072: 2068: 2064: 2060: 2053: 2049: 2045: 2040: 2036: 2033: 2028: 2024: 2020: 2016: 2012: 2009: 2002: 1998: 1994: 1988: 1984: 1981: 1966: 1963: 1962: 1961: 1943: 1939: 1936: 1924: 1920: 1915: 1907: 1903: 1899: 1894: 1890: 1880: 1876: 1873: 1870: 1867: 1864: 1854: 1843: 1834: 1830: 1827: 1803: 1799: 1794: 1790: 1787: 1784: 1781: 1778: 1747: 1744: 1742: 1739: 1738: 1737: 1732: 1720: 1717: 1716: 1715: 1707: 1704: 1703: 1702: 1691: 1688: 1681: 1664:Supercritical 1662: 1655: 1646: 1640: 1634: 1628: 1622:critical point 1615: 1604:electrostatics 1592: 1581: 1575: 1566:approximate a 1556: 1550: 1543: 1534: 1531: 1530: 1529: 1522: 1517: 1507: 1501: 1495: 1488: 1478: 1471: 1468: 1467: 1466: 1460: 1454: 1451: 1448: 1439: 1436: 1435: 1434: 1431: 1428: 1422: 1419: 1403: 1400: 1399: 1398: 1388: 1381: 1378: 1364: 1354: 1347: 1340: 1335: 1329: 1324: 1314: 1306: 1303: 1302: 1301: 1294: 1291: 1288: 1283: 1280: 1279: 1278: 1271: 1268: 1267: 1266: 1259: 1256:forest patches 1252: 1242: 1235: 1232: 1225: 1218: 1206: 1203: 1202: 1201: 1196: 1190: 1183: 1176: 1169: 1166: 1165: 1164: 1157: 1154: 1148: 1145: 1132: 1129: 1126: 1123: 1120: 1082: 1061: 1048:(Greek letter 1037: 1011: 1007: 1004: 982: 979: 975: 971: 968: 965: 962: 959: 956: 945:power function 941: 940: 928: 925: 922: 917: 913: 909: 906: 903: 880: 860: 855: 851: 847: 844: 745: 742: 720: 700: 697: 692: 685: 668: 665: 662: 602: 599: 596: 572: 569: 566: 546: 543: 540: 537: 534: 531: 528: 502: 499: 495: 482: 479: 453: 433: 430: 427: 424: 402: 399: 395: 374: 350: 339: 338: 326: 323: 320: 317: 314: 311: 308: 305: 302: 299: 294: 291: 287: 283: 278: 275: 271: 267: 264: 261: 258: 255: 252: 249: 246: 243: 240: 217: 197: 175: 172: 168: 164: 161: 158: 155: 152: 149: 133: 130: 128: 125: 84: 81: 15: 9: 6: 4: 3: 2: 9020: 9009: 9006: 9004: 9001: 8999: 8996: 8994: 8991: 8990: 8988: 8978: 8977:download page 8974: 8970: 8967: 8964: 8961: 8958: 8955: 8954:Cosma Shalizi 8951: 8950: 8946: 8943: 8942: 8938: 8935: 8932:Philip Ball: 8931: 8929: 8925: 8922: 8918: 8914: 8911: 8908: 8904: 8901: 8899: 8896: 8894: 8890: 8887: 8884: 8883: 8873: 8869: 8865: 8861: 8857: 8853: 8849: 8845: 8841: 8837: 8836: 8830: 8826: 8820: 8816: 8812: 8808: 8804: 8800: 8796: 8792: 8788: 8784: 8783: 8777: 8773: 8767: 8763: 8758: 8754: 8750: 8745: 8740: 8736: 8732: 8725: 8720: 8716: 8712: 8708: 8704: 8700: 8696: 8691: 8686: 8682: 8678: 8673: 8669: 8665: 8661: 8657: 8653: 8649: 8644: 8639: 8635: 8631: 8626: 8622: 8620:0-297-64376-2 8616: 8612: 8607: 8603: 8601:0-19-850164-1 8597: 8593: 8588: 8584: 8583: 8577: 8576: 8575: 8574: 8564: 8559: 8557: 8549: 8544: 8537: 8532: 8524: 8520: 8516: 8512: 8508: 8504: 8500: 8496: 8492: 8485: 8477: 8473: 8469: 8465: 8461: 8457: 8456: 8448: 8440: 8436: 8432: 8428: 8424: 8420: 8415: 8410: 8406: 8402: 8395: 8387: 8383: 8379: 8375: 8368: 8360: 8356: 8352: 8348: 8341: 8333: 8329: 8325: 8321: 8314: 8305: 8304:2027.42/30190 8300: 8296: 8292: 8288: 8284: 8277: 8270: 8263: 8257: 8249: 8245: 8241: 8237: 8230: 8222: 8218: 8214: 8210: 8203: 8195: 8191: 8187: 8183: 8179: 8175: 8171: 8167: 8162: 8157: 8153: 8149: 8142: 8127: 8123: 8117: 8108: 8103: 8099: 8095: 8091: 8084: 8076: 8072: 8068: 8064: 8060: 8056: 8049: 8047: 8045: 8043: 8035: 8029: 8022: 8016: 8008: 8004: 8000: 7996: 7992: 7988: 7985:(6): 066120. 7984: 7980: 7973: 7971: 7962: 7954: 7950: 7946: 7942: 7938: 7934: 7931:(6): 066115. 7930: 7926: 7919: 7910: 7896:on 2016-02-08 7895: 7891: 7885: 7876: 7871: 7864: 7856: 7852: 7848: 7844: 7840: 7836: 7831: 7826: 7822: 7818: 7811: 7792: 7788: 7781: 7774: 7765: 7760: 7756: 7752: 7748: 7744: 7737: 7730: 7722: 7718: 7714: 7710: 7706: 7702: 7698: 7694: 7687: 7679: 7675: 7671: 7667: 7663: 7659: 7655: 7651: 7644: 7636: 7632: 7628: 7624: 7620: 7616: 7612: 7608: 7603: 7598: 7594: 7590: 7583: 7575: 7571: 7567: 7563: 7559: 7555: 7550: 7545: 7541: 7537: 7530: 7522: 7518: 7514: 7510: 7506: 7502: 7495: 7487: 7483: 7479: 7475: 7471: 7467: 7459: 7445: 7444:ReactionWheel 7441: 7434: 7426: 7422: 7418: 7414: 7410: 7406: 7402: 7396: 7388: 7384: 7380: 7376: 7372: 7368: 7364: 7360: 7353: 7346: 7338: 7334: 7329: 7324: 7320: 7316: 7312: 7308: 7304: 7300: 7293: 7284: 7279: 7272: 7257: 7253: 7246: 7238: 7231: 7223: 7217: 7213: 7212: 7204: 7196: 7190: 7186: 7185: 7177: 7169: 7165: 7161: 7157: 7153: 7149: 7142: 7135: 7129: 7115: 7111: 7105: 7097: 7093: 7089: 7085: 7081: 7077: 7073: 7066: 7064: 7055: 7051: 7047: 7043: 7039: 7035: 7032:(6): 067103. 7031: 7027: 7020: 7013: 7005: 7001: 6997: 6993: 6989: 6985: 6978: 6970: 6966: 6962: 6958: 6954: 6950: 6945: 6940: 6936: 6932: 6925: 6916: 6911: 6907: 6903: 6899: 6892: 6884: 6880: 6875: 6870: 6866: 6862: 6858: 6854: 6850: 6846: 6842: 6838: 6834: 6827: 6819: 6815: 6811: 6807: 6803: 6799: 6795: 6791: 6786: 6781: 6777: 6773: 6766: 6758: 6751: 6743: 6739: 6735: 6731: 6727: 6723: 6719: 6715: 6711: 6705: 6697: 6693: 6689: 6685: 6681: 6677: 6670: 6663: 6658: 6651: 6650:Sornette 2006 6646: 6632:on 2015-03-18 6631: 6627: 6620: 6612: 6608: 6604: 6600: 6595: 6590: 6586: 6582: 6577: 6572: 6568: 6564: 6560: 6553: 6545: 6544: 6539: 6533: 6531: 6523: 6518: 6516: 6514: 6512: 6510: 6508: 6506: 6504: 6502: 6492: 6487: 6483: 6479: 6475: 6471: 6464: 6456: 6452: 6448: 6444: 6440: 6436: 6429: 6421: 6417: 6412: 6407: 6403: 6399: 6395: 6391: 6387: 6383: 6379: 6375: 6371: 6364: 6357: 6352: 6344: 6340: 6335: 6330: 6325: 6320: 6316: 6312: 6309:(5). e19779. 6308: 6304: 6300: 6293: 6291: 6289: 6280: 6276: 6272: 6268: 6264: 6260: 6256: 6252: 6248: 6244: 6237: 6230: 6222: 6218: 6213: 6208: 6204: 6200: 6196: 6192: 6188: 6181: 6179: 6170: 6166: 6162: 6158: 6154: 6150: 6145: 6140: 6136: 6132: 6131: 6123: 6121: 6119: 6117: 6100: 6093: 6089: 6086: 6085: 6073: 6070: 6068: 6065: 6063: 6060: 6058: 6055: 6053: 6050: 6048: 6045: 6043: 6040: 6038: 6035: 6033: 6030: 6028: 6025: 6023: 6020: 6019: 6012: 6008: 6006: 6001: 5996: 5986: 5981: 5978:For example, 5976: 5973: 5970:For example, 5968: 5959: 5957: 5953: 5949: 5945: 5941: 5937: 5933: 5929: 5925: 5914: 5900: 5877: 5869: 5865: 5841: 5821: 5789: 5781: 5777: 5773: 5767: 5747: 5736: 5728: 5723: 5719: 5711: 5710: 5709: 5690: 5686: 5679: 5653: 5644: 5634: 5633: 5632: 5618: 5610: 5606: 5596: 5594: 5589: 5572: 5566: 5540: 5531: 5509: 5480: 5451: 5427: 5421: 5395: 5385: 5365: 5354: 5321: 5317: 5314: 5308: 5277: 5271: 5267: 5261: 5258: 5253: 5248: 5245: 5242: 5238: 5232: 5229: 5224: 5221: 5206: 5202: 5193: 5184: 5170: 5166: 5157: 5147: 5144: 5133: 5132: 5131: 5111: 5107: 5102: 5098: 5072: 5068: 5056: 5053:For a set of 5051: 5049: 5045: 5041: 5020: 5017: 5013: 5006: 5003: 4997: 4992: 4989: 4980: 4971: 4968: 4960: 4939: 4936: 4932: 4925: 4899: 4895: 4890: 4886: 4865: 4840: 4836: 4807: 4804: 4799: 4787: 4781: 4777: 4771: 4768: 4763: 4758: 4755: 4752: 4748: 4743: 4738: 4735: 4732: 4729: 4720: 4710: 4709: 4708: 4694: 4669: 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