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.

1762: 6360: 3890:(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 3868: 3816:. 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". 3864:, 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 54: 6014:

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.

6006:, 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. 4210: 2211: 655:). 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. 5307: 1504:

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

3459: 4693: 4346:. 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. 3991: 1986: 1969: 426:. 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 7474:

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

126:, 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. 3257: 4451: 5714: 7362: 3807:

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:

5819: 5047: 4205:{\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)}.} 3318: 3673: 3733:. 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 902:, 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): 5969:. 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). 4568: 827:: 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 5993:

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

2945: 347: 2276: 2206:{\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)} 2642: 98:

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

5360: 2811: 5924:, 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. 6245:

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

3526: 5902: 3494: 2359: 1108: 3552: 2840: 900: 197: 3886:. Originally proposed as a tool to explore the existence of moments and the moment generation function using random samples, the bundle methodology is based on residual 3757: 5568: 5479: 5423: 5099: 4966: 4867: 4560: 3167: 4364: 3882:

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

3586:, 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 688: 622: 592: 524: 424: 5648: 4324: 2567: 2535: 2306: 5597: 5452: 3081: 3052: 3023: 2994: 2869: 453: 2385: 6685:

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

5640: 4887: 4233: 3731: 3156: 3101: 2965: 2733: 1081: 473: 394: 237: 217: 4339: 5302:{\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 }}}} 3871:

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

3454:{\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}} 7790: 4975: 624:; 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 5986:

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).

4688:{\displaystyle {\mathcal {L}}(\alpha )=\log \prod _{i=1}^{n}{\frac {\alpha -1}{x_{\min }}}\left({\frac {x_{i}}{x_{\min }}}\right)^{-\alpha }} 8899: 3600: 9013: 7120: 8686:

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".

3844:(also called Pareto index) is close to 0, because Pareto Q–Q plots are not designed to identify distributions with slowly varying tails. 3787:

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

3759:. This distribution is a common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. 722:

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

106:, cloud sizes, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of 3910:, and do not demand the collection of much data). In addition, other types of tail behavior can be identified using bundle plots. 5615: 4355: 8944: 2878: 245: 1659: 5995: 148: 8833: 8780: 7934:

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".

5366:. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for 2583: 8218:

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".

3783:. This convergence effect explains why the variance-to-mean power law manifests so widely in natural processes, as with 6636: 6196:

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

2741: 1343: 8950: 8913: 1595: 8629: 8610: 6723:; Terzuolo, Carlo; Viviani, Paolo (1983). "The law relating the kinematic and figural aspects of drawing movements". 2655: 1468: 4724: 4492: 3265: 5934: 5619: 2715:

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

1964:{\displaystyle f(x)\propto x_{\text{th}}^{\alpha _{1}-\alpha _{2}}x^{\alpha _{2}}{\text{ for }}x>x_{\text{th}}} 636:

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".

7030:"From gene families and genera to incomes and internet file sizes: Why power laws are so common in nature" 7266: 5104: 4892: 1832: 1739: 1520: 1514: 1009: 640:(for example, the net worth of Americans is distributed according to a power law with an exponent of 2). 8356:

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".

3922:, which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) 3106: 961: 8286: 7363:"On the distribution of links in the interbank network: evidence from the e-MID overnight money market" 5395:

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.

5513: 5484: 5369: 1578: 533: 9018: 6037: 4459: 3499: 1777:, consisting of two or more power laws, combined with a threshold. For example, with two power laws: 1395: 824: 629: 17: 8410:

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

7262: 3531: 2712: 1564: 1377: 1332: 3252:{\displaystyle p(x)={\frac {\alpha -1}{x_{\min }}}\left({\frac {x}{x_{\min }}}\right)^{-\alpha },} 2819: 885: 154: 8896: 7029: 6032: 5982: 4446:{\displaystyle p(x)={\frac {\alpha -1}{x_{\min }}}\left({\frac {x}{x_{\min }}}\right)^{-\alpha }} 2872: 1622: 1618: 1570: 8734: 5957:

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

3736: 7450: 6082: 5546: 5457: 5401: 5071: 4932: 4839: 4538: 1735: 1648: 1418: 843:. It is primarily in the study of statistical distributions that the name "power law" is used. 373: 78: 39: 8271:

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

5709:{\displaystyle {\hat {\alpha }}={\underset {\alpha }{\operatorname {arg\,min} }}\,D_{\alpha }} 3594:

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:

849: 355: 9003: 6198:"Climatologically invariant scale invariance seen in distributions of cloud horizontal sizes" 6057: 5907: 5606:(Matlab, Python, R and C++) for estimation and testing routines for power-law distributions. 4701: 3893: 3827: 3788: 3780: 3767: 3763: 1766: 1260: 1190: 1184: 1042: 727: 7157:

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

4718:, setting the result equal to zero. Upon rearrangement, this yields the estimator equation: 667: 601: 571: 499: 399: 8959: 8854: 8705: 8658: 8513: 8429: 8176: 7997: 7943: 7845: 7711: 7668: 7617: 7519: 7317: 7094: 7044: 7002: 6959: 6855: 6800: 6720: 6392: 6321: 6261: 6209: 6159: 6140: 5058: 5054: 4532: 4238: 3309: 3305: 2540: 2513: 2285: 1680: 1557: 1204: 832: 828: 773: 765: 643:

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

747:, and systems with precisely the same critical exponents are said to belong to the same 57:

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

46:"Scaling law" redirects here. For statistical laws of scaling deep learning models, see 8878: 8809: 8759: 8721: 8695: 8674: 8648: 8482: 8478: 8445: 8419: 8200: 8166: 8081: 7880: 7861: 7835: 7727: 7641: 7607: 7580: 7554: 7393: 7343: 7288: 6975: 6949: 6884: 6843: 6824: 6790: 6748: 6702: 6617: 6604: 6569: 6421: 6380: 6344: 6309: 6285: 6175: 6149: 5965:

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

719: 625: 7865: 7723: 6979: 6109: 5945:, defined as the number of elements per meter (or area unit, second etc.) for which 8927: 8862: 8801: 8763: 8754: 8749: 8713: 8678: 8666: 8521: 8474: 8437: 8392: 8365: 8338: 8309: 8301: 8254: 8227: 8204: 8184: 8112: 8073: 8005: 7951: 7853: 7769: 7761: 7719: 7676: 7625: 7564: 7527: 7484: 7423: 7377: 7333: 7325: 7166: 7102: 7052: 7010: 6967: 6920: 6879: 6863: 6828: 6808: 6752: 6732: 6694: 6599: 6581: 6496: 6488: 6453: 6416: 6400: 6339: 6329: 6310:"Statistical Analyses Support Power Law Distributions Found in Neuronal Avalanches" 6289: 6269: 6217: 6197: 6167: 5614:

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".

6719: 6706: 6698: 6549:"9na CEPAL Charlas Sobre Sistemas Complejos Sociales (CCSSCS): Leyes de potencias" 8983: 8903: 8821: 8463:

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

8441: 7568: 7014: 6334: 6067: 5990: 5141:, the maximum likelihood exponent is the solution to the transcendental equation 3784: 3308:. We can now consider several properties of this distribution. For instance, its 1676: 1455: 1361: 1239: 769: 130:

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

8775:. Lecture Notes in Economics and Mathematical Systems. Vol. 632. Springer. 8735:"A Brief History of Generative Models for Power Law and Lognormal Distributions" 8525: 7531: 7329: 3083:

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

8009: 7955: 7765: 7680: 7082: 7056: 6867: 6492: 5642:, between the cumulative distribution functions of the data and the power law: 5050: 3587: 1614: 1530: 1225: 955: 785: 86: 7381: 6548: 6246: 6171: 6047: 3054:

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

8997: 8964: 8533: 8117: 8100: 7576: 7496: 7435: 7389: 7178: 6909:"Structural characteristics and radial properties of tropical cloud clusters" 6875: 6640: 6595: 6412: 6231: 1722: 1684: 1672:

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".

5603: 772:, power laws are often thought to be signatures of hierarchy or of specific 8973:

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).

6820: 6613: 6465: 6430: 6353: 6281: 5814:{\displaystyle D_{\alpha }=\max _{x}|P_{\mathrm {emp} }(x)-P_{\alpha }(x)|} 3809: 1574: 1391:

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

1266: 1232: 1209: 823:

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

805: 123: 8987: 8717: 8396: 8342: 8258: 8231: 6744: 5570:

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

2282:

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

1464:

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

1242:

relating mean population size and variance of populations sizes in ecology

8843:

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

8700: 8314: 8171: 7840: 6993:

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

6568:

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

6154: 6072: 5042:{\displaystyle \sigma ={\frac {{\hat {\alpha }}-1}{\sqrt {n}}}+O(n^{-1})} 1705: 1308: 809: 797: 111: 103: 32: 8926:: power-law distributions in homelessness and other social problems; by 7338: 6273: 4342:. Some of the most reliable techniques are often based on the method of 8828:. Springer Series in Synergetics (2nd ed.). Heidelberg: Springer. 8813: 8792: 8486: 8085: 7629: 7106: 3867: 1582: 1367: 1348: 1197: 801: 764:

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

70: 8670: 7774: 7411: 7170: 6971: 6501: 6404: 3668:{\displaystyle p(x)\propto L(x)x^{-\alpha }\mathrm {e} ^{-\lambda x}.} 8188: 7979:"Tweedie convergence: a mathematical basis for Taylor's power law, 1/ 7427: 7083:"Scale-free power-laws as interaction between progress and diffusion" 6457: 6052: 4235:

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

3799:

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

3713:

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

1701:

The damping force over speed relation in antiseismic dampers calculus

1688: 1636: 817: 813: 740: 239:

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

58: 8805: 8245:

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

8132: 8077: 6812: 6795: 4972: > 100. Further, the standard error of the estimate is 7885: 7857: 7293: 6528: 6526: 6524: 6522: 6520: 6518: 6516: 6514: 6512: 6381:"On a Possible Unified Scaling Law for Volcanic Eruption Durations" 4929:. This estimator exhibits a small finite sample-size bias of order 3820: 1385: 1286: 1231:

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

781: 644: 595: 8986:

visualizes the distribution of the degrees of the webgraph on the

8790:

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

8771:

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

8653: 8424: 7612: 7559: 6954: 6479:

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

396:

simply multiplies the original power-law relation by the constant

6553: 1486: 789: 6509: 7825: 3138:

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

1599: 1432:

Distribution of artists by the average price of their artworks.

1256: 835:), which considers the frequency of extremely rare events like 488:

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

119: 61:, and to the left are the few that dominate (also known as the 8133:"So You Think You Have a Power Law — Well Isn't That Special?" 7791:"The Wickelgren Power Law and the Ebbinghaus Savings Function" 4333: 3554:, the average and all higher-order moments are infinite; when 8552: 8499: 1122:

below for details. Typically the exponent falls in the range

1060: 8960:

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

7788: 5904:

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

3158:

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

53: 8569: 8567: 6926:

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

5510:, while choosing it too large increases the uncertainty in 1381: 527: 107: 99: 7473: 3103:. In many cases, it is convenient to assume a lower bound 2940:{\displaystyle \lim _{x\rightarrow \infty }L(r\,x)/L(x)=1} 1353:

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

1110:

is needed otherwise the distribution has infinite area as

342:{\displaystyle f(cx)=a(cx)^{-k}=c^{-k}f(x)\propto f(x),\!} 8770: 8156: 8045:

An introduction to statistical modeling of extreme values

7878: 7597: 7376:(4). Springer Science and Business Media LLC: 1463–1495. 6844:"Power laws and critical fragmentation in global forests" 6841: 1298:

Sizes of cloud areas and perimeters, as viewed from space

8967:, Professor of Statistics at Carnegie-Mellon University. 8593:

Historical Biogeography of Neotropical Freshwater Fishes

8564: 8540: 6244: 5398:

Further, both of these estimators require the choice of

1119: 743:. Formally, this sharing of dynamics is referred to as 8976: 3802: 2271:{\displaystyle 0<x_{0}<x_{1}<\cdots <x_{n}} 846:

In empirical contexts, an approximation to a power-law

7544: 2706: 2537:

determines the sharpness of splicing between segments

1461:

The mean absolute change of the logarithmic mid-prices

780:

of income distribution, structural self-similarity of

739:

systems, where the critical point of the system is an

7310:

Physica A: Statistical Mechanics and Its Applications

7308:

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

6195: 5910: 5874: 5830: 5728: 5651: 5628: 5576: 5549: 5516: 5487: 5460: 5431: 5404: 5372: 5318: 5150: 5107: 5074: 4978: 4935: 4895: 4875: 4842: 4727: 4704: 4571: 4541: 4495: 4462: 4367: 4241: 4221: 3994: 3936: 3896: 3830: 3739: 3719: 3684: 3603: 3560: 3534: 3502: 3470: 3321: 3268: 3170: 3144: 3109: 3089: 3060: 3031: 3025:

be asymptotically scale invariant; thus, the form of

3002: 2973: 2953: 2881: 2848: 2822: 2744: 2721: 2658: 2637:{\displaystyle f(x)\propto x^{-\alpha }e^{-\beta x}.} 2586: 2543: 2516: 2458: 2393: 2367: 2314: 2288: 2219: 1989: 1872: 1835: 1786: 1128: 1089: 1069: 1045: 1012: 964: 911: 888: 852: 670: 604: 574: 536: 502: 491: 461: 432: 402: 382: 358: 248: 225: 205: 157: 8638: 8287:"Confidence bands for percentile residual lifetimes" 7245: 6780: 6654: 6570:"On single point forecasts for fat-tailed variables" 6532: 6444:

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

3766:

are a family of statistical models characterized by

2572: 706:

is the minimum value for which the power law holds.

632:

entering the room, with a monthly income of about 1

8355: 8328: 6567: 6022:field of research in many areas of modern science. 3918:In general, power-law distributions are plotted on 3913: 8916:by Benoit Mandelbrot & Nassim Nicholas Taleb. 8466:Journal of the Royal Statistical Society, Series B 8030:Beirlant, J., Teugels, J. L., Vynckier, P. (1996) 7076: 7074: 6308:Klaus A, Yu S, Plenz D (2011). Zochowski M (ed.). 5916: 5896: 5860: 5813: 5708: 5634: 5591: 5562: 5531: 5502: 5473: 5446: 5417: 5387: 5355:{\displaystyle \zeta (\alpha ,x_{\mathrm {min} })} 5354: 5301: 5133: 5093: 5041: 4960: 4921: 4881: 4861: 4825: 4710: 4687: 4554: 4523: 4481: 4445: 4358:data, we fit a power-law distribution of the form 4318: 4227: 4204: 3977: 3902: 3836: 3751: 3725: 3705: 3667: 3578: 3546: 3520: 3488: 3453: 3296: 3251: 3150: 3130: 3095: 3075: 3046: 3017: 2988: 2959: 2939: 2863: 2834: 2806:{\displaystyle P(X>x)\sim L(x)x^{-(\alpha -1)}} 2805: 2727: 2695: 2636: 2561: 2529: 2502: 2444: 2379: 2353: 2300: 2270: 2205: 1963: 1857: 1821: 1769:use a broken power law; here Kroupa (2001) in red. 1252:(number of species) in clades of freshwater fishes 1146: 1102: 1075: 1051: 1031: 998: 943: 894: 874: 682: 616: 586: 560: 518: 467: 447: 418: 388: 364: 341: 231: 211: 191: 8897:Zipf, Power-laws, and Pareto – a ranking tutorial 8685: 8558: 8066:Journal of Computational and Graphical Statistics 6666: 6379:Cannavò, Flavio; Nunnari, Giuseppe (2016-03-01). 3678:In this distribution, the exponential decay term 1704:Folded solvent-exposed surface areas of centered 1698:in dielectric responses to step DC voltage input. 940: 338: 8995: 7412:"The Granular Origins of Aggregate Fluctuations" 7222:Media/Society: Industries, Images, and Audiences 6684: 6634: 6137: 5743: 5609: 5555: 5466: 5410: 5292: 5221: 5185: 5126: 4914: 4802: 4665: 4636: 4547: 4514: 4474: 4423: 4401: 4167: 4091: 4010: 3441: 3358: 3287: 3226: 3204: 2883: 658:The median does exist, however: for a power law 7976: 7933: 7071: 6906: 4531:is included to ensure that the distribution is 2696:{\displaystyle f(x)\propto x^{\alpha +\beta x}} 1742:with demonstrations that it may be logarithmic) 1255:The Harlow Knapp effect, where a subset of the 8217: 7219:Croteau, David; Hoynes, William (2013-11-06). 6765: 6378: 5049:. This estimator is equivalent to the popular 4826:{\displaystyle {\hat {\alpha }}=1+n\left^{-1}} 4524:{\displaystyle {\frac {\alpha -1}{x_{\min }}}} 3297:{\displaystyle {\frac {\alpha -1}{x_{\min }}}} 1259:found in the human body compose a majority of 27:Functional relationship between two quantities 8945:Critical Mass: How one thing leads to another 8358:Organizational Behavior and Human Performance 8294:Journal of Statistical Planning and Inference 8284: 7307: 7286: 7218: 5543:of our model. In general, the best choice of 1660:Behaviour near second-order phase transitions 8842: 8732: 8573: 8546: 7658: 6939: 6307: 6015: 6010: 5088: 5075: 4856: 4843: 1975: 1441:Distribution of degrees in banking networks. 8596:. Berkeley: University of California Press. 8244: 7970: 7927: 7509: 5927: 4334:Estimating the exponent from empirical data 2996:follows directly from the requirement that 1822:{\displaystyle f(x)\propto x^{\alpha _{1}}} 1569:The input-voltage–output-current curves of 776:. A few notable examples of power laws are 8914:"How the Finance Gurus Get Risk All Wrong" 8589: 8101:"Heavy Tail Modeling and Teletraffic Data" 7361:Fricke, Daniel; Lux, Thomas (2015-02-13). 7027: 7021: 6767: 6713: 6543: 6541: 6366: 6107: 3978:{\displaystyle P(x)=\mathrm {Pr} (X>x)} 1166: 485: 8753: 8699: 8652: 8590:Albert, J. S.; Reis, R. E., eds. (2011). 8423: 8409: 8313: 8170: 8116: 7884: 7839: 7773: 7611: 7558: 7360: 7337: 7292: 6953: 6924: 6883: 6794: 6687:Journal of International Business Studies 6603: 6585: 6500: 6420: 6343: 6333: 6221: 6153: 5695: 5679: 5618:(iid) data, uses the minimization of the 5481:too small produces a significant bias in 4140: 4061: 3706:{\displaystyle \mathrm {e} ^{-\lambda x}} 3392: 3323: 2907: 2510:, smoothly spliced together. The size of 1063:, not to be confused with scaling factor 8937:Benoit Mandelbrot & Richard Hudson: 8820: 8619: 8462: 6660: 6238: 6191: 6189: 6133: 6131: 6129: 6127: 5972: 3866: 1760: 1224:relating animal metabolism to size, and 944:{\displaystyle y=ax^{k}+\varepsilon .\!} 52: 8382: 8098: 8059: 8057: 8055: 8053: 7744: 7448: 7260: 7159:IEEE Transactions on Information Theory 7080: 6900: 6538: 6478: 6443: 6303: 6301: 6299: 6112:. New England Complex Systems Institute 5616:independent and identically distributed 4562:, the log likelihood function becomes: 4356:independent and identically distributed 2875:, which is any function that satisfies 14: 8996: 7747:"Theory of behavioral power functions" 7701: 7191: 7143:Sutton, J. (1997), "Gibrat's Legacy", 6992: 6986: 6933: 6635:Malcolm Gladwell (February 13, 2006). 5933:distribution is described, not by the 1370:, and frequencies of words in a text ( 754: 8789: 6672: 6186: 6124: 5861:{\displaystyle P_{\mathrm {emp} }(x)} 4349: 3884:bundle for the log-transformed sample 2445:{\displaystyle c_{0},c_{1},...,c_{n}} 1683:, such as supercritical exponents of 1656:relating light intensity with voltage 147:One attribute of power laws is their 93: 8909:Stream Morphometry and Horton's Laws 8456: 8063: 8050: 8032:Practical Analysis of Extreme Values 7789:John T. Wixted; Shana K. Carpenter. 6574:International Journal of Forecasting 6296: 3803:Graphical methods for identification 1716: 1594:A 3/2-power law can be found in the 1196:The differential energy spectrum of 9014:Theory of probability distributions 8963:from Three-Toed Sloth, the blog of 8600: 7028:Reed, W. J.; Hughes, B. D. (2002). 5425:. For functions with a non-trivial 5134:{\displaystyle x_{i}\geq x_{\min }} 4922:{\displaystyle x_{i}\geq x_{\min }} 2707:Power-law probability distributions 2647: 1858:{\displaystyle x<x_{\text{th}},} 1756: 1505:the most frequent item, and so on). 1324:and random observation (or killing) 1120:power-law probability distributions 1032:{\displaystyle x>x_{\text{min}}} 142: 24: 8939:The Misbehaviour of Markets (2004) 8479:10.1111/j.2517-6161.1982.tb01183.x 7977:Kendal, WS; Jørgensen, BR (2011). 7156: 6533:Clauset, Shalizi & Newman 2009 5843: 5840: 5837: 5769: 5766: 5763: 5686: 5683: 5680: 5676: 5673: 5670: 5343: 5340: 5337: 4574: 4142: 4122: 4063: 4044: 3956: 3953: 3687: 3646: 3394: 3365: 3131:{\displaystyle x_{\mathrm {min} }} 3122: 3119: 3116: 2893: 1315: 1292: 999:{\displaystyle p(x)=Cx^{-\alpha }} 786:scaling laws in biological systems 552: 492:Lack of well-defined average value 110:in most languages, frequencies of 25: 9030: 8890: 8034:, Leuven: Leuven University Press 7248:The Statistics of Deadly Quarrels 6202:Atmospheric Chemistry and Physics 3812:), mean residual life plots and 3579:{\displaystyle 2<\alpha <3} 2573:Power law with exponential cutoff 2503:{\displaystyle x=x_{1},...,x_{n}} 1591:(ratio of surface area to volume) 1357:'s laws describing river systems) 1147:{\displaystyle 2<\alpha <3} 761: 376:. That is, scaling by a constant 8285:Csorgo, S.; Viharos, L. (1992). 7477:Journal of Banking & Finance 7261:Berreby, David (July 31, 2014). 6587:10.1016/j.ijforecast.2020.08.008 5935:cumulative distribution function 5532:{\displaystyle {\hat {\alpha }}} 5503:{\displaystyle {\hat {\alpha }}} 5388:{\displaystyle {\hat {\alpha }}} 3914:Plotting power-law distributions 882:often includes a deviation term 561:{\displaystyle x\in [1,\infty )} 81:between two quantities, where a 8773:Theory of Zipf's law and beyond 8493: 8403: 8376: 8349: 8322: 8278: 8265: 8238: 8211: 8150: 8125: 8092: 8037: 8024: 7918: 7893: 7872: 7819: 7782: 7738: 7724:10.1126/science.139.3560.1213-a 7695: 7652: 7591: 7538: 7503: 7467: 7442: 7404: 7354: 7301: 7280: 7254: 7239: 7212: 7185: 7150: 7137: 7113: 6835: 6774: 6759: 6678: 6628: 6561: 4482:{\displaystyle x\geq x_{\min }} 4340:unbiased and consistent answers 3860: = 1, ..., 3852:-th order statistic versus the 3521:{\displaystyle m\geq \alpha -1} 3464:which is only well defined for 2711:In a looser sense, a power-law 1114:approaches 0, and the constant 709: 475:, and the straight-line on the 8755:10.1080/15427951.2004.10129088 7512:Chaos, Solitons & Fractals 7145:Journal of Economic Literature 6907:Machado L, Rossow, WB (1993). 6472: 6437: 6372: 6101: 5897:{\displaystyle P_{\alpha }(x)} 5891: 5885: 5855: 5849: 5807: 5803: 5797: 5781: 5775: 5753: 5658: 5586: 5580: 5523: 5494: 5441: 5435: 5379: 5349: 5322: 5226: 5207: 5198: 5190: 5171: 5162: 5036: 5020: 4994: 4955: 4939: 4734: 4585: 4579: 4377: 4371: 4194: 4182: 4058: 4052: 4025: 4013: 4004: 3998: 3972: 3960: 3946: 3940: 3775:similar to the role that the 3628: 3622: 3613: 3607: 3489:{\displaystyle m<\alpha -1} 3389: 3383: 3180: 3174: 3070: 3064: 3041: 3035: 3012: 3006: 2983: 2977: 2928: 2922: 2911: 2901: 2890: 2858: 2852: 2798: 2786: 2775: 2769: 2760: 2748: 2668: 2662: 2596: 2590: 2354:{\displaystyle \ln(y/y_{0}+a)} 2348: 2321: 1882: 1876: 1796: 1790: 1480: 1304:Energy dissipation in cyclones 1103:{\displaystyle x_{\text{min}}} 974: 968: 869: 856: 555: 543: 442: 436: 332: 326: 317: 311: 280: 270: 261: 252: 167: 161: 13: 1: 8624:. Weidenfeld & Nicolson. 8559:Laherrère & Sornette 1998 7449:Neumann, Jerry (2015-06-25). 7263:"Cloudy With a Chance of War" 7246:Lewis Fry Richardson (1950). 7192:Curtis, Vickie (2018-04-20). 6699:10.1057/palgrave.jibs.8400324 6089: 5610:Kolmogorov–Smirnov estimation 3819:Pareto Q–Q plots compare the 3547:{\displaystyle \alpha \leq 2} 1729: 1301:The size of rain-shower cells 1269:globally follows a power law 484:Thus, accurately fitting and 137: 8370:10.1016/0030-5073(80)90065-3 8306:10.1016/0378-3758(92)90159-p 7569:10.1080/14697688.2010.481632 7489:10.1016/0378-4266(90)90009-Q 7015:10.1016/j.icarus.2009.06.029 6737:10.1016/0001-6918(83)90027-6 6335:10.1371/journal.pone.0019779 5620:Kolmogorov–Smirnov statistic 3791:, implies the presence of 1/ 2835:{\displaystyle \alpha >1} 2387:linear segments with slopes 1556:The frequency-dependency of 1498:also called the "80–20 rule" 1412: 1280: 1178: 895:{\displaystyle \varepsilon } 192:{\displaystyle f(x)=ax^{-k}} 7: 8688:European Physical Journal B 8605:. Oxford University Press. 8526:10.1016/j.tecto.2011.01.003 8412:European Physical Journal B 7532:10.1016/j.chaos.2016.01.020 7330:10.1016/j.physa.2018.04.057 6025: 5068:integer-valued data points 1751: 1746:The power law of forgetting 1596:plate characteristic curves 1581:relationship, a factor in " 1523:(continuous), of which the 1473:Average waiting time of an 1157: 122:of organisms, the sizes of 10: 9035: 8955:from The Econophysics Blog 8442:10.1140/epjb/e2007-00219-y 8047:. Springer-Verlag, London. 8010:10.1103/physreve.84.066120 7983:noise and multifractality" 7956:10.1103/physreve.83.066115 7766:10.1037/0033-295x.85.4.305 7745:Staddon, J. E. R. (1978). 7681:10.1103/PhysRevE.75.011920 7057:10.1103/physreve.66.067103 6868:10.1038/s41598-018-36120-w 6493:10.1037/0033-295x.85.4.305 6038:Heavy-tailed distributions 6016:Stumpf & Porter (2012) 6011:Stumpf & Porter (2012) 5996:lognormal density function 3752:{\displaystyle \lambda =0} 2361:, the plot is composed of 1629:Self-organized criticality 1543: 1467:Average waiting time of a 1448: 1419:Population sizes of cities 1405:of the weight matrices of 1338:Highly optimized tolerance 1215: 829:theory of large deviations 45: 29: 8977:The Erdős Webgraph Server 8733:Mitzenmacher, M. (2004). 8273:Ann. Inst. Statist. Math. 7828:The Astrophysical Journal 7382:10.1007/s00181-015-0919-x 7121:"Horton's Laws – Example" 6172:10.1080/00107510500052444 5563:{\displaystyle x_{\min }} 5474:{\displaystyle x_{\min }} 5418:{\displaystyle x_{\min }} 5094:{\displaystyle \{x_{i}\}} 4961:{\displaystyle O(n^{-1})} 4862:{\displaystyle \{x_{i}\}} 4555:{\displaystyle x_{\min }} 3856:-th order statistic, for 1976:Smoothly broken power law 1396:power law of cache misses 1384:(also referred to as the 1274:species–area relationship 825:probability distributions 8952:Tyranny of the Power Law 8574:Stumpf & Porter 2012 8105:The Annals of Statistics 7089:(Submitted manuscript). 7081:Hilbert, Martin (2013). 6768:Albert & Reis (2011) 5983:log-normal distributions 5928:Two-point fitting method 5364:incomplete zeta function 4489:, where the coefficient 3789:method of expanding bins 3262:where the pre-factor to 2947:for any positive factor 2713:probability distribution 2308:and vertical axis being 1773:A broken power law is a 1696:Curie–von Schweidler law 1571:field-effect transistors 1521:Student's t-distribution 1444:Firm-size distributions. 1344:experience curve effects 1333:diffusion of innovations 875:{\displaystyle o(x^{k})} 365:{\displaystyle \propto } 31:Not to be confused with 8924:"Million-dollar Murray" 8867:10.1126/science.1216142 8620:Buchanan, Mark (2000). 8099:Resnick, S. I. (1997). 7451:"Power Laws in Venture" 6637:"Million-Dollar Murray" 6223:10.5194/acp-24-109-2024 6033:Fat-tailed distribution 5917:{\displaystyle \alpha } 4711:{\displaystyle \alpha } 3924:cumulative distribution 3920:doubly logarithmic axes 3903:{\displaystyle \alpha } 3837:{\displaystyle \alpha } 3496:. That is, all moments 2873:slowly varying function 1647:Force and potential in 1623:Electrostatic potential 1619:gravitational potential 1515:Yule–Simon distribution 1360:Populations of cities ( 1167:Artificial Intelligence 1052:{\displaystyle \alpha } 766:dimensional constraints 737:self-organized critical 199:, scaling the argument 132:Allometric scaling laws 79:functional relationship 8118:10.1214/aos/1069362376 7125:www.engr.colostate.edu 6913:Monthly Weather Review 6367:Albert & Reis 2011 5918: 5898: 5862: 5815: 5710: 5636: 5593: 5564: 5533: 5504: 5475: 5448: 5419: 5389: 5356: 5303: 5268: 5135: 5095: 5043: 4968:, which is small when 4962: 4923: 4883: 4863: 4827: 4778: 4712: 4689: 4617: 4556: 4535:. Given a choice for 4525: 4483: 4447: 4320: 4229: 4206: 3979: 3904: 3872: 3838: 3779:has as a focus in the 3753: 3727: 3707: 3669: 3580: 3548: 3522: 3490: 3455: 3298: 3253: 3152: 3132: 3097: 3077: 3048: 3019: 2990: 2961: 2941: 2865: 2836: 2807: 2729: 2697: 2638: 2563: 2531: 2504: 2446: 2381: 2355: 2302: 2272: 2207: 2096: 1965: 1859: 1823: 1770: 1649:simple harmonic motion 1617:, as evidenced by the 1454:Returns for high-risk 1148: 1104: 1077: 1053: 1033: 1000: 945: 896: 876: 692:, it takes the value 2 684: 683:{\displaystyle k>1} 630:world's richest person 618: 617:{\displaystyle k>3} 594:, and it has a finite 588: 587:{\displaystyle k>2} 562: 520: 519:{\displaystyle x^{-k}} 469: 449: 420: 419:{\displaystyle c^{-k}} 390: 374:direct proportionality 366: 343: 233: 213: 193: 66: 40:Power (disambiguation) 8718:10.1007/s100510050276 8397:10.1287/opre.32.3.537 8343:10.1287/opre.29.2.392 8259:10.1287/opre.32.3.668 8232:10.1287/opre.31.2.391 7798:Psychological Science 7600:Nature Communications 7225:. SAGE Publications. 6721:Lacquaniti, Francesco 6110:"Concepts: Power Law" 6058:Low-degree saturation 5973:Validating power laws 5919: 5899: 5863: 5816: 5711: 5637: 5594: 5565: 5534: 5505: 5476: 5449: 5420: 5390: 5357: 5304: 5248: 5136: 5096: 5044: 4963: 4924: 4884: 4864: 4828: 4758: 4713: 4690: 4597: 4557: 4526: 4484: 4448: 4321: 4319:{\displaystyle \left} 4230: 4207: 3980: 3905: 3870: 3839: 3781:central limit theorem 3764:Tweedie distributions 3754: 3728: 3708: 3670: 3581: 3549: 3523: 3491: 3456: 3299: 3254: 3153: 3133: 3098: 3078: 3049: 3020: 2991: 2962: 2942: 2866: 2837: 2808: 2730: 2698: 2639: 2564: 2562:{\displaystyle i-1,i} 2532: 2530:{\displaystyle f_{i}} 2505: 2447: 2382: 2356: 2303: 2301:{\displaystyle \ln x} 2273: 2208: 2076: 1966: 1860: 1824: 1767:initial mass function 1764: 1191:initial mass function 1154:, though not always. 1149: 1105: 1078: 1054: 1034: 1001: 946: 897: 877: 728:renormalization group 685: 619: 589: 563: 521: 470: 450: 421: 391: 367: 344: 234: 219:by a constant factor 214: 194: 56: 8971:Simple MATLAB script 8934:, February 13, 2006. 8742:Internet Mathematics 8275:37, Part A, 165–172. 7754:Psychological Review 7547:Quantitative Finance 7422:(3): 733–772. 2011. 6481:Psychological Review 6446:Psychological Review 6141:Contemporary Physics 5939:cumulative frequency 5908: 5872: 5828: 5726: 5649: 5626: 5592:{\displaystyle L(x)} 5574: 5547: 5514: 5485: 5458: 5447:{\displaystyle L(x)} 5429: 5402: 5370: 5316: 5148: 5105: 5072: 5059:extreme value theory 5055:quantitative finance 4976: 4933: 4893: 4873: 4840: 4725: 4702: 4569: 4539: 4493: 4460: 4365: 4239: 4219: 3992: 3934: 3926:(ccdf) that is, the 3894: 3828: 3737: 3717: 3682: 3601: 3558: 3532: 3500: 3468: 3319: 3306:normalizing constant 3266: 3168: 3142: 3107: 3087: 3076:{\displaystyle L(x)} 3058: 3047:{\displaystyle L(x)} 3029: 3018:{\displaystyle p(x)} 3000: 2989:{\displaystyle L(x)} 2971: 2951: 2879: 2864:{\displaystyle L(x)} 2846: 2820: 2742: 2719: 2656: 2584: 2541: 2514: 2456: 2391: 2365: 2312: 2286: 2217: 1987: 1870: 1833: 1784: 1681:supercritical fluids 1565:Stefan–Boltzmann law 1558:acoustic attenuation 1438:in a market economy. 1407:deep neural networks 1126: 1087: 1067: 1043: 1010: 962: 909: 886: 850: 837:stock market crashes 833:extreme value theory 774:stochastic processes 668: 602: 572: 534: 500: 486:validating power-law 479:is often called the 459: 448:{\displaystyle f(x)} 430: 400: 380: 356: 246: 223: 203: 155: 128:Acoustic attenuation 38:For other uses, see 8859:2012Sci...335..665S 8710:1998EPJB....2..525L 8663:2009SIAMR..51..661C 8518:2011Tectp.504...14G 8434:2007EPJB...58..167B 8385:Operations Research 8331:Operations Research 8247:Operations Research 8220:Operations Research 8181:2000Natur.407..651J 8002:2011PhRvE..84f6120K 7948:2011PhRvE..83f6115K 7850:2006ApJ...640L...5J 7716:1963Sci...139.1213M 7710:(3560): 1213–1216. 7673:2007PhRvE..75a1920M 7622:2013NatCo...4.2331B 7524:2016CSF....88....3L 7370:Empirical Economics 7322:2018PhyA..506..217E 7099:2014Cmplx..19d..56H 7049:2002PhRvE..66f7103R 7007:2009Icar..203..683L 6964:2010NatPh...6..693C 6860:2018NatSR...817766S 6805:2011Natur.470..163E 6557:. 31 December 2013. 6397:2016NatSR...622289C 6326:2011PLoSO...619779K 6274:10.1038/nature09116 6266:2010Natur.465.1066H 6260:(7301): 1066–1069. 6214:2024ACP....24..109D 6164:2005ConPh..46..323N 6083:Stevens's power law 6078:Stable distribution 6063:Pareto distribution 5454:function, choosing 5101:, again where each 4126: 4109: 4048: 3777:normal distribution 3450: 3369: 3160:Pareto distribution 2967:. This property of 2380:{\displaystyle n+1} 1922: 1765:Some models of the 1736:Stevens's power law 1670:safe operating area 1643:van der Waals force 1607:inverse-square laws 1525:Cauchy distribution 1492:Pareto distribution 1436:Income distribution 1245:Neuronal avalanches 1039:where the exponent 755:Power-law functions 726:—can be shown, via 653:regression analysis 638:Pareto distribution 526:has a well-defined 151:. Given a relation 8982:2021-03-01 at the 8902:2007-10-26 at the 7901:"Curved-power law" 7667:(1 Pt 1). 011920. 7630:10.1038/ncomms3331 7107:10.1002/cplx.21485 6848:Scientific Reports 6770:, pp. 89–104. 6385:Scientific Reports 5961:, for a sample of 5914: 5894: 5858: 5811: 5751: 5706: 5693: 5632: 5589: 5560: 5539:, and reduces the 5529: 5500: 5471: 5444: 5415: 5385: 5352: 5299: 5131: 5091: 5039: 4958: 4919: 4879: 4859: 4823: 4708: 4685: 4552: 4521: 4479: 4443: 4350:Maximum likelihood 4344:maximum likelihood 4316: 4225: 4202: 4112: 4086: 4034: 3975: 3900: 3888:quantile functions 3873: 3834: 3749: 3723: 3703: 3665: 3576: 3544: 3518: 3486: 3451: 3436: 3348: 3294: 3249: 3148: 3128: 3093: 3073: 3044: 3015: 2986: 2957: 2937: 2897: 2861: 2832: 2803: 2725: 2693: 2634: 2559: 2527: 2500: 2442: 2377: 2351: 2298: 2268: 2203: 1961: 1888: 1855: 1819: 1775:piecewise function 1771: 1738:of psychophysics ( 1664:critical exponents 1537:scale-free network 1469:directional change 1329:exponential growth 1322:Exponential growth 1311:on Earth and Mars 1261:published research 1185:Kepler's third law 1173:Neural scaling law 1144: 1100: 1073: 1049: 1029: 996: 941: 892: 872: 749:universality class 720:critical exponents 680: 649:standard deviation 614: 584: 558: 516: 465: 445: 416: 386: 362: 339: 229: 209: 189: 94:Empirical examples 67: 48:Neural scaling law 8853:(6069): 665–666. 8835:978-3-540-30882-9 8782:978-3-642-02945-5 8671:10.1137/070710111 8601:Bak, Per (1997). 8547:Mitzenmacher 2004 8165:(6804): 651–654. 8043:Coles, S. (2001) 7661:Physical Review E 7267:Nautilus Magazine 7232:978-1-4833-2355-8 7205:978-3-319-77664-4 7171:10.1109/18.165464 6972:10.1038/nphys1725 6919:(12): 3234–3260. 6789:(7333): 163–165. 6725:Acta Psychologica 6405:10.1038/srep22289 6043:Hyperbolic growth 5742: 5668: 5661: 5635:{\displaystyle D} 5541:statistical power 5526: 5497: 5382: 5297: 5246: 5230: 5210: 5174: 5012: 5011: 4997: 4882:{\displaystyle n} 4807: 4737: 4670: 4641: 4519: 4428: 4406: 4354:For real-valued, 4309: 4290: 4269: 4228:{\displaystyle n} 4172: 4110: 3928:survival function 3795:noise and that 1/ 3726:{\displaystyle x} 3434: 3292: 3231: 3209: 3151:{\displaystyle x} 3096:{\displaystyle x} 2960:{\displaystyle r} 2882: 2728:{\displaystyle x} 2179: 2140: 2067: 2016: 1958: 1943: 1895: 1849: 1725:of assembly sizes 1717:Political Science 1710:protein structure 1611:Newtonian gravity 1553:in aerosol optics 1551:Angstrom exponent 1527:is a special case 1509:Zeta distribution 1342:Proposed form of 1327:Progress through 1097: 1076:{\displaystyle a} 1026: 841:natural disasters 716:phase transitions 468:{\displaystyle x} 389:{\displaystyle c} 232:{\displaystyle c} 212:{\displaystyle x} 16:(Redirected from 9026: 9019:Statistical laws 8928:Malcolm Gladwell 8920:, July 11, 2005. 8886: 8839: 8822:Sornette, Didier 8817: 8800:(3/4): 425–440. 8786: 8767: 8757: 8739: 8729: 8703: 8701:cond-mat/9801293 8682: 8656: 8635: 8616: 8603:How nature works 8597: 8577: 8571: 8562: 8556: 8550: 8544: 8538: 8537: 8497: 8491: 8490: 8460: 8454: 8453: 8427: 8407: 8401: 8400: 8380: 8374: 8373: 8353: 8347: 8346: 8326: 8320: 8319: 8317: 8291: 8282: 8276: 8269: 8263: 8262: 8242: 8236: 8235: 8215: 8209: 8208: 8189:10.1038/35036627 8174: 8172:cond-mat/0010278 8154: 8148: 8147: 8145: 8143: 8129: 8123: 8122: 8120: 8111:(5): 1805–1869. 8096: 8090: 8089: 8061: 8048: 8041: 8035: 8028: 8022: 8021: 7987: 7974: 7968: 7967: 7931: 7925: 7922: 7916: 7915: 7913: 7912: 7903:. Archived from 7897: 7891: 7890: 7888: 7876: 7870: 7869: 7843: 7841:astro-ph/0602219 7823: 7817: 7816: 7814: 7812: 7807:on April 8, 2016 7806: 7800:. Archived from 7795: 7786: 7780: 7779: 7777: 7751: 7742: 7736: 7735: 7699: 7693: 7692: 7656: 7650: 7649: 7615: 7595: 7589: 7588: 7562: 7542: 7536: 7535: 7507: 7501: 7500: 7483:(6): 1189–1208. 7471: 7465: 7464: 7462: 7461: 7446: 7440: 7439: 7428:10.3982/ecta8769 7408: 7402: 7401: 7367: 7358: 7352: 7351: 7341: 7305: 7299: 7298: 7296: 7284: 7278: 7277: 7275: 7273: 7258: 7252: 7251: 7243: 7237: 7236: 7216: 7210: 7209: 7189: 7183: 7182: 7165:(6): 1842–1845. 7154: 7148: 7141: 7135: 7134: 7132: 7131: 7117: 7111: 7110: 7078: 7069: 7068: 7034: 7025: 7019: 7018: 6990: 6984: 6983: 6957: 6937: 6931: 6930: 6928: 6904: 6898: 6897: 6887: 6839: 6833: 6832: 6798: 6778: 6772: 6771: 6763: 6757: 6756: 6731:(1–3): 115–130. 6717: 6711: 6710: 6693:(7): 1212–1230. 6682: 6676: 6670: 6664: 6658: 6652: 6651: 6649: 6648: 6639:. Archived from 6632: 6626: 6625: 6607: 6589: 6565: 6559: 6558: 6545: 6536: 6530: 6507: 6506: 6504: 6476: 6470: 6469: 6458:10.1037/h0046162 6441: 6435: 6434: 6424: 6376: 6370: 6364: 6358: 6357: 6347: 6337: 6305: 6294: 6293: 6251: 6242: 6236: 6235: 6225: 6193: 6184: 6183: 6157: 6155:cond-mat/0412004 6135: 6122: 6121: 6119: 6117: 6108:Yaneer Bar-Yam. 6105: 6005: 5998:is quadratic in 5949: > 5923: 5921: 5920: 5915: 5903: 5901: 5900: 5895: 5884: 5883: 5867: 5865: 5864: 5859: 5848: 5847: 5846: 5820: 5818: 5817: 5812: 5810: 5796: 5795: 5774: 5773: 5772: 5756: 5750: 5738: 5737: 5715: 5713: 5712: 5707: 5705: 5704: 5694: 5689: 5663: 5662: 5654: 5641: 5639: 5638: 5633: 5598: 5596: 5595: 5590: 5569: 5567: 5566: 5561: 5559: 5558: 5538: 5536: 5535: 5530: 5528: 5527: 5519: 5509: 5507: 5506: 5501: 5499: 5498: 5490: 5480: 5478: 5477: 5472: 5470: 5469: 5453: 5451: 5450: 5445: 5424: 5422: 5421: 5416: 5414: 5413: 5394: 5392: 5391: 5386: 5384: 5383: 5375: 5361: 5359: 5358: 5353: 5348: 5347: 5346: 5308: 5306: 5305: 5300: 5298: 5296: 5295: 5286: 5285: 5276: 5267: 5262: 5247: 5239: 5231: 5229: 5225: 5224: 5212: 5211: 5203: 5193: 5189: 5188: 5176: 5175: 5167: 5161: 5152: 5140: 5138: 5137: 5132: 5130: 5129: 5117: 5116: 5100: 5098: 5097: 5092: 5087: 5086: 5048: 5046: 5045: 5040: 5035: 5034: 5013: 5007: 5006: 4999: 4998: 4990: 4986: 4967: 4965: 4964: 4959: 4954: 4953: 4928: 4926: 4925: 4920: 4918: 4917: 4905: 4904: 4888: 4886: 4885: 4880: 4868: 4866: 4865: 4860: 4855: 4854: 4832: 4830: 4829: 4824: 4822: 4821: 4813: 4809: 4808: 4806: 4805: 4796: 4795: 4786: 4777: 4772: 4739: 4738: 4730: 4717: 4715: 4714: 4709: 4694: 4692: 4691: 4686: 4684: 4683: 4675: 4671: 4669: 4668: 4659: 4658: 4649: 4642: 4640: 4639: 4630: 4619: 4616: 4611: 4578: 4577: 4561: 4559: 4558: 4553: 4551: 4550: 4530: 4528: 4527: 4522: 4520: 4518: 4517: 4508: 4497: 4488: 4486: 4485: 4480: 4478: 4477: 4452: 4450: 4449: 4444: 4442: 4441: 4433: 4429: 4427: 4426: 4414: 4407: 4405: 4404: 4395: 4384: 4325: 4323: 4322: 4317: 4315: 4311: 4310: 4302: 4291: 4286: 4275: 4270: 4265: 4254: 4234: 4232: 4231: 4226: 4211: 4209: 4208: 4203: 4198: 4197: 4177: 4173: 4171: 4170: 4158: 4145: 4139: 4138: 4125: 4120: 4111: 4108: 4094: 4085: 4074: 4066: 4047: 4042: 3984: 3982: 3981: 3976: 3959: 3909: 3907: 3906: 3901: 3843: 3841: 3840: 3835: 3758: 3756: 3755: 3750: 3732: 3730: 3729: 3724: 3712: 3710: 3709: 3704: 3702: 3701: 3690: 3674: 3672: 3671: 3666: 3661: 3660: 3649: 3643: 3642: 3585: 3583: 3582: 3577: 3553: 3551: 3550: 3545: 3527: 3525: 3524: 3519: 3495: 3493: 3492: 3487: 3460: 3458: 3457: 3452: 3449: 3444: 3435: 3433: 3416: 3405: 3397: 3379: 3378: 3368: 3363: 3362: 3361: 3344: 3340: 3339: 3326: 3303: 3301: 3300: 3295: 3293: 3291: 3290: 3281: 3270: 3258: 3256: 3255: 3250: 3245: 3244: 3236: 3232: 3230: 3229: 3217: 3210: 3208: 3207: 3198: 3187: 3157: 3155: 3154: 3149: 3137: 3135: 3134: 3129: 3127: 3126: 3125: 3102: 3100: 3099: 3094: 3082: 3080: 3079: 3074: 3053: 3051: 3050: 3045: 3024: 3022: 3021: 3016: 2995: 2993: 2992: 2987: 2966: 2964: 2963: 2958: 2946: 2944: 2943: 2938: 2918: 2896: 2870: 2868: 2867: 2862: 2841: 2839: 2838: 2833: 2812: 2810: 2809: 2804: 2802: 2801: 2734: 2732: 2731: 2726: 2702: 2700: 2699: 2694: 2692: 2691: 2648:Curved power law 2643: 2641: 2640: 2635: 2630: 2629: 2614: 2613: 2568: 2566: 2565: 2560: 2536: 2534: 2533: 2528: 2526: 2525: 2509: 2507: 2506: 2501: 2499: 2498: 2474: 2473: 2451: 2449: 2448: 2443: 2441: 2440: 2416: 2415: 2403: 2402: 2386: 2384: 2383: 2378: 2360: 2358: 2357: 2352: 2341: 2340: 2331: 2307: 2305: 2304: 2299: 2277: 2275: 2274: 2269: 2267: 2266: 2248: 2247: 2235: 2234: 2212: 2210: 2209: 2204: 2202: 2198: 2197: 2196: 2195: 2194: 2184: 2180: 2178: 2177: 2165: 2141: 2139: 2138: 2129: 2128: 2127: 2109: 2108: 2098: 2095: 2090: 2072: 2068: 2066: 2065: 2053: 2041: 2040: 2028: 2024: 2017: 2015: 2014: 2002: 1970: 1968: 1967: 1962: 1960: 1959: 1956: 1944: 1941: 1939: 1938: 1937: 1936: 1921: 1920: 1919: 1907: 1906: 1896: 1893: 1864: 1862: 1861: 1856: 1851: 1850: 1847: 1828: 1826: 1825: 1820: 1818: 1817: 1816: 1815: 1757:Broken power law 1654:Gamma correction 1560:in complex media 1496:Pareto principle 1403:spectral density 1378:90–9–1 principle 1331:and exponential 1250:species richness 1205:M–sigma relation 1153: 1151: 1150: 1145: 1109: 1107: 1106: 1101: 1099: 1098: 1095: 1082: 1080: 1079: 1074: 1058: 1056: 1055: 1050: 1038: 1036: 1035: 1030: 1028: 1027: 1024: 1005: 1003: 1002: 997: 995: 994: 950: 948: 947: 942: 930: 929: 901: 899: 898: 893: 881: 879: 878: 873: 868: 867: 794:computer science 691: 689: 687: 686: 681: 662:, with exponent 623: 621: 620: 615: 593: 591: 590: 585: 567: 565: 564: 559: 525: 523: 522: 517: 515: 514: 474: 472: 471: 466: 454: 452: 451: 446: 425: 423: 422: 417: 415: 414: 395: 393: 392: 387: 371: 369: 368: 363: 348: 346: 345: 340: 307: 306: 291: 290: 238: 236: 235: 230: 218: 216: 215: 210: 198: 196: 195: 190: 188: 187: 149:scale invariance 143:Scale invariance 116:species richness 43: 36: 21: 9034: 9033: 9029: 9028: 9027: 9025: 9024: 9023: 8994: 8993: 8984:Wayback Machine 8904:Wayback Machine 8893: 8836: 8806:10.2307/2333389 8783: 8737: 8632: 8613: 8581: 8580: 8572: 8565: 8557: 8553: 8545: 8541: 8498: 8494: 8461: 8457: 8408: 8404: 8381: 8377: 8354: 8350: 8327: 8323: 8289: 8283: 8279: 8270: 8266: 8243: 8239: 8216: 8212: 8155: 8151: 8141: 8139: 8131: 8130: 8126: 8097: 8093: 8078:10.2307/1390871 8062: 8051: 8042: 8038: 8029: 8025: 7985: 7975: 7971: 7932: 7928: 7923: 7919: 7910: 7908: 7899: 7898: 7894: 7877: 7873: 7824: 7820: 7810: 7808: 7804: 7793: 7787: 7783: 7749: 7743: 7739: 7700: 7696: 7657: 7653: 7596: 7592: 7543: 7539: 7508: 7504: 7472: 7468: 7459: 7457: 7447: 7443: 7410: 7409: 7405: 7365: 7359: 7355: 7306: 7302: 7285: 7281: 7271: 7269: 7259: 7255: 7244: 7240: 7233: 7217: 7213: 7206: 7190: 7186: 7155: 7151: 7142: 7138: 7129: 7127: 7119: 7118: 7114: 7079: 7072: 7032: 7026: 7022: 6991: 6987: 6938: 6934: 6905: 6901: 6840: 6836: 6813:10.1038/470163a 6779: 6775: 6764: 6760: 6718: 6714: 6683: 6679: 6671: 6667: 6659: 6655: 6646: 6644: 6633: 6629: 6566: 6562: 6547: 6546: 6539: 6531: 6510: 6477: 6473: 6442: 6438: 6377: 6373: 6365: 6361: 6306: 6297: 6249: 6243: 6239: 6194: 6187: 6136: 6125: 6115: 6113: 6106: 6102: 6092: 6087: 6068:Power-law fluid 6028: 6003: 5999: 5975: 5953:applies, where 5930: 5909: 5906: 5905: 5879: 5875: 5873: 5870: 5869: 5836: 5835: 5831: 5829: 5826: 5825: 5806: 5791: 5787: 5762: 5761: 5757: 5752: 5746: 5733: 5729: 5727: 5724: 5723: 5700: 5696: 5669: 5667: 5653: 5652: 5650: 5647: 5646: 5627: 5624: 5623: 5612: 5575: 5572: 5571: 5554: 5550: 5548: 5545: 5544: 5518: 5517: 5515: 5512: 5511: 5489: 5488: 5486: 5483: 5482: 5465: 5461: 5459: 5456: 5455: 5430: 5427: 5426: 5409: 5405: 5403: 5400: 5399: 5374: 5373: 5371: 5368: 5367: 5336: 5335: 5331: 5317: 5314: 5313: 5291: 5287: 5281: 5277: 5275: 5263: 5252: 5238: 5220: 5216: 5202: 5201: 5194: 5184: 5180: 5166: 5165: 5154: 5153: 5151: 5149: 5146: 5145: 5125: 5121: 5112: 5108: 5106: 5103: 5102: 5082: 5078: 5073: 5070: 5069: 5027: 5023: 4989: 4988: 4987: 4985: 4977: 4974: 4973: 4946: 4942: 4934: 4931: 4930: 4913: 4909: 4900: 4896: 4894: 4891: 4890: 4874: 4871: 4870: 4850: 4846: 4841: 4838: 4837: 4814: 4801: 4797: 4791: 4787: 4785: 4773: 4762: 4757: 4753: 4752: 4729: 4728: 4726: 4723: 4722: 4703: 4700: 4699: 4676: 4664: 4660: 4654: 4650: 4648: 4644: 4643: 4635: 4631: 4620: 4618: 4612: 4601: 4573: 4572: 4570: 4567: 4566: 4546: 4542: 4540: 4537: 4536: 4513: 4509: 4498: 4496: 4494: 4491: 4490: 4473: 4469: 4461: 4458: 4457: 4434: 4422: 4418: 4413: 4409: 4408: 4400: 4396: 4385: 4383: 4366: 4363: 4362: 4352: 4336: 4301: 4276: 4274: 4255: 4253: 4246: 4242: 4240: 4237: 4236: 4220: 4217: 4216: 4178: 4166: 4162: 4157: 4153: 4152: 4141: 4131: 4127: 4121: 4116: 4095: 4090: 4075: 4073: 4062: 4043: 4038: 3993: 3990: 3989: 3952: 3935: 3932: 3931: 3916: 3895: 3892: 3891: 3829: 3826: 3825: 3805: 3738: 3735: 3734: 3718: 3715: 3714: 3691: 3686: 3685: 3683: 3680: 3679: 3650: 3645: 3644: 3635: 3631: 3602: 3599: 3598: 3559: 3556: 3555: 3533: 3530: 3529: 3501: 3498: 3497: 3469: 3466: 3465: 3445: 3440: 3417: 3406: 3404: 3393: 3374: 3370: 3364: 3357: 3353: 3352: 3335: 3331: 3327: 3322: 3320: 3317: 3316: 3286: 3282: 3271: 3269: 3267: 3264: 3263: 3237: 3225: 3221: 3216: 3212: 3211: 3203: 3199: 3188: 3186: 3169: 3166: 3165: 3143: 3140: 3139: 3115: 3114: 3110: 3108: 3105: 3104: 3088: 3085: 3084: 3059: 3056: 3055: 3030: 3027: 3026: 3001: 2998: 2997: 2972: 2969: 2968: 2952: 2949: 2948: 2914: 2886: 2880: 2877: 2876: 2847: 2844: 2843: 2821: 2818: 2817: 2782: 2778: 2743: 2740: 2739: 2720: 2717: 2716: 2709: 2678: 2674: 2657: 2654: 2653: 2650: 2619: 2615: 2606: 2602: 2585: 2582: 2581: 2575: 2542: 2539: 2538: 2521: 2517: 2515: 2512: 2511: 2494: 2490: 2469: 2465: 2457: 2454: 2453: 2452:, separated at 2436: 2432: 2411: 2407: 2398: 2394: 2392: 2389: 2388: 2366: 2363: 2362: 2336: 2332: 2327: 2313: 2310: 2309: 2287: 2284: 2283: 2281: 2262: 2258: 2243: 2239: 2230: 2226: 2218: 2215: 2214: 2190: 2186: 2185: 2173: 2169: 2164: 2160: 2159: 2152: 2148: 2134: 2130: 2117: 2113: 2104: 2100: 2099: 2097: 2091: 2080: 2061: 2057: 2052: 2048: 2036: 2032: 2010: 2006: 2001: 2000: 1996: 1988: 1985: 1984: 1978: 1955: 1951: 1942: for 1940: 1932: 1928: 1927: 1923: 1915: 1911: 1902: 1898: 1897: 1892: 1871: 1868: 1867: 1846: 1842: 1834: 1831: 1830: 1811: 1807: 1806: 1802: 1785: 1782: 1781: 1759: 1754: 1732: 1719: 1677:state of matter 1625:, respectively. 1589:Square–cube law 1546: 1483: 1456:venture capital 1451: 1421:in a region or 1415: 1318: 1316:General science 1295: 1293:Climate science 1283: 1226:allometric laws 1218: 1181: 1169: 1160: 1127: 1124: 1123: 1094: 1090: 1088: 1085: 1084: 1068: 1065: 1064: 1044: 1041: 1040: 1023: 1019: 1011: 1008: 1007: 987: 983: 963: 960: 959: 925: 921: 910: 907: 906: 887: 884: 883: 863: 859: 851: 848: 847: 770:complex systems 757: 733: 712: 705: 698: 669: 666: 665: 663: 603: 600: 599: 573: 570: 569: 535: 532: 531: 507: 503: 501: 498: 497: 494: 460: 457: 456: 431: 428: 427: 407: 403: 401: 398: 397: 381: 378: 377: 357: 354: 353: 299: 295: 283: 279: 247: 244: 243: 224: 221: 220: 204: 201: 200: 180: 176: 156: 153: 152: 145: 140: 96: 83:relative change 51: 44: 37: 30: 28: 23: 22: 15: 12: 11: 5: 9032: 9022: 9021: 9016: 9011: 9006: 8992: 8991: 8974: 8968: 8956: 8948: 8941: 8935: 8932:The New Yorker 8921: 8911: 8906: 8892: 8891:External links 8889: 8888: 8887: 8840: 8834: 8818: 8787: 8781: 8768: 8748:(2): 226–251. 8730: 8694:(4): 525–539. 8683: 8647:(4): 661–703. 8636: 8630: 8617: 8611: 8598: 8579: 8578: 8563: 8551: 8539: 8506:Tectonophysics 8492: 8455: 8418:(2): 167–173. 8402: 8391:(3): 537–546. 8375: 8364:(2): 224–251. 8348: 8337:(2): 392–399. 8321: 8300:(3): 327–337. 8277: 8264: 8253:(3): 668–678. 8237: 8226:(2): 391–396. 8210: 8149: 8124: 8091: 8072:(3): 493–509. 8049: 8036: 8023: 7969: 7926: 7917: 7892: 7871: 7858:10.1086/503294 7818: 7781: 7760:(4): 305–320. 7737: 7694: 7651: 7590: 7553:(4): 599–614. 7537: 7502: 7466: 7441: 7403: 7353: 7300: 7279: 7253: 7238: 7231: 7211: 7204: 7184: 7149: 7136: 7112: 7070: 7020: 7001:(2): 683–684. 6985: 6948:(9): 693–696. 6942:Nature Physics 6932: 6899: 6834: 6773: 6758: 6712: 6677: 6665: 6653: 6627: 6580:(2): 413–422. 6560: 6537: 6508: 6487:(4): 305–320. 6471: 6452:(3): 153–181. 6436: 6371: 6359: 6295: 6237: 6208:(1): 109–122. 6185: 6148:(5): 323–351. 6123: 6099: 6098: 6091: 6088: 6086: 6085: 6080: 6075: 6070: 6065: 6060: 6055: 6050: 6045: 6040: 6035: 6029: 6027: 6024: 6001: 5974: 5971: 5941:of a property 5929: 5926: 5913: 5893: 5890: 5887: 5882: 5878: 5857: 5854: 5851: 5845: 5842: 5839: 5834: 5822: 5821: 5809: 5805: 5802: 5799: 5794: 5790: 5786: 5783: 5780: 5777: 5771: 5768: 5765: 5760: 5755: 5749: 5745: 5741: 5736: 5732: 5717: 5716: 5703: 5699: 5692: 5688: 5685: 5682: 5678: 5675: 5672: 5666: 5660: 5657: 5631: 5611: 5608: 5588: 5585: 5582: 5579: 5557: 5553: 5525: 5522: 5496: 5493: 5468: 5464: 5443: 5440: 5437: 5434: 5412: 5408: 5381: 5378: 5351: 5345: 5342: 5339: 5334: 5330: 5327: 5324: 5321: 5310: 5309: 5294: 5290: 5284: 5280: 5274: 5271: 5266: 5261: 5258: 5255: 5251: 5245: 5242: 5237: 5234: 5228: 5223: 5219: 5215: 5209: 5206: 5200: 5197: 5192: 5187: 5183: 5179: 5173: 5170: 5164: 5160: 5157: 5128: 5124: 5120: 5115: 5111: 5090: 5085: 5081: 5077: 5051:Hill estimator 5038: 5033: 5030: 5026: 5022: 5019: 5016: 5010: 5005: 5002: 4996: 4993: 4984: 4981: 4957: 4952: 4949: 4945: 4941: 4938: 4916: 4912: 4908: 4903: 4899: 4878: 4858: 4853: 4849: 4845: 4834: 4833: 4820: 4817: 4812: 4804: 4800: 4794: 4790: 4784: 4781: 4776: 4771: 4768: 4765: 4761: 4756: 4751: 4748: 4745: 4742: 4736: 4733: 4707: 4696: 4695: 4682: 4679: 4674: 4667: 4663: 4657: 4653: 4647: 4638: 4634: 4629: 4626: 4623: 4615: 4610: 4607: 4604: 4600: 4596: 4593: 4590: 4587: 4584: 4581: 4576: 4549: 4545: 4516: 4512: 4507: 4504: 4501: 4476: 4472: 4468: 4465: 4454: 4453: 4440: 4437: 4432: 4425: 4421: 4417: 4412: 4403: 4399: 4394: 4391: 4388: 4382: 4379: 4376: 4373: 4370: 4351: 4348: 4335: 4332: 4314: 4308: 4305: 4300: 4297: 4294: 4289: 4285: 4282: 4279: 4273: 4268: 4264: 4261: 4258: 4252: 4249: 4245: 4224: 4213: 4212: 4201: 4196: 4193: 4190: 4187: 4184: 4181: 4176: 4169: 4165: 4161: 4156: 4151: 4148: 4144: 4137: 4134: 4130: 4124: 4119: 4115: 4107: 4104: 4101: 4098: 4093: 4089: 4084: 4081: 4078: 4072: 4069: 4065: 4060: 4057: 4054: 4051: 4046: 4041: 4037: 4033: 4030: 4027: 4024: 4021: 4018: 4015: 4012: 4009: 4006: 4003: 4000: 3997: 3974: 3971: 3968: 3965: 3962: 3958: 3955: 3951: 3948: 3945: 3942: 3939: 3915: 3912: 3899: 3833: 3804: 3801: 3748: 3745: 3742: 3722: 3700: 3697: 3694: 3689: 3676: 3675: 3664: 3659: 3656: 3653: 3648: 3641: 3638: 3634: 3630: 3627: 3624: 3621: 3618: 3615: 3612: 3609: 3606: 3588:central moment 3575: 3572: 3569: 3566: 3563: 3543: 3540: 3537: 3528:diverge: when 3517: 3514: 3511: 3508: 3505: 3485: 3482: 3479: 3476: 3473: 3462: 3461: 3448: 3443: 3439: 3432: 3429: 3426: 3423: 3420: 3415: 3412: 3409: 3403: 3400: 3396: 3391: 3388: 3385: 3382: 3377: 3373: 3367: 3360: 3356: 3351: 3347: 3343: 3338: 3334: 3330: 3325: 3289: 3285: 3280: 3277: 3274: 3260: 3259: 3248: 3243: 3240: 3235: 3228: 3224: 3220: 3215: 3206: 3202: 3197: 3194: 3191: 3185: 3182: 3179: 3176: 3173: 3147: 3124: 3121: 3118: 3113: 3092: 3072: 3069: 3066: 3063: 3043: 3040: 3037: 3034: 3014: 3011: 3008: 3005: 2985: 2982: 2979: 2976: 2956: 2936: 2933: 2930: 2927: 2924: 2921: 2917: 2913: 2910: 2906: 2903: 2900: 2895: 2892: 2889: 2885: 2860: 2857: 2854: 2851: 2831: 2828: 2825: 2814: 2813: 2800: 2797: 2794: 2791: 2788: 2785: 2781: 2777: 2774: 2771: 2768: 2765: 2762: 2759: 2756: 2753: 2750: 2747: 2724: 2708: 2705: 2704: 2703: 2690: 2687: 2684: 2681: 2677: 2673: 2670: 2667: 2664: 2661: 2649: 2646: 2645: 2644: 2633: 2628: 2625: 2622: 2618: 2612: 2609: 2605: 2601: 2598: 2595: 2592: 2589: 2574: 2571: 2558: 2555: 2552: 2549: 2546: 2524: 2520: 2497: 2493: 2489: 2486: 2483: 2480: 2477: 2472: 2468: 2464: 2461: 2439: 2435: 2431: 2428: 2425: 2422: 2419: 2414: 2410: 2406: 2401: 2397: 2376: 2373: 2370: 2350: 2347: 2344: 2339: 2335: 2330: 2326: 2323: 2320: 2317: 2297: 2294: 2291: 2265: 2261: 2257: 2254: 2251: 2246: 2242: 2238: 2233: 2229: 2225: 2222: 2201: 2193: 2189: 2183: 2176: 2172: 2168: 2163: 2158: 2155: 2151: 2147: 2144: 2137: 2133: 2126: 2123: 2120: 2116: 2112: 2107: 2103: 2094: 2089: 2086: 2083: 2079: 2075: 2071: 2064: 2060: 2056: 2051: 2047: 2044: 2039: 2035: 2031: 2027: 2023: 2020: 2013: 2009: 2005: 1999: 1995: 1992: 1977: 1974: 1973: 1972: 1954: 1950: 1947: 1935: 1931: 1926: 1918: 1914: 1910: 1905: 1901: 1891: 1887: 1884: 1881: 1878: 1875: 1865: 1854: 1845: 1841: 1838: 1814: 1810: 1805: 1801: 1798: 1795: 1792: 1789: 1758: 1755: 1753: 1750: 1749: 1748: 1743: 1731: 1728: 1727: 1726: 1718: 1715: 1714: 1713: 1702: 1699: 1692: 1675:Supercritical 1673: 1666: 1657: 1651: 1645: 1639: 1633:critical point 1626: 1615:electrostatics 1603: 1592: 1586: 1577:approximate a 1567: 1561: 1554: 1545: 1542: 1541: 1540: 1533: 1528: 1518: 1512: 1506: 1499: 1489: 1482: 1479: 1478: 1477: 1471: 1465: 1462: 1459: 1450: 1447: 1446: 1445: 1442: 1439: 1433: 1430: 1414: 1411: 1410: 1409: 1399: 1392: 1389: 1375: 1365: 1358: 1351: 1346: 1340: 1335: 1325: 1317: 1314: 1313: 1312: 1305: 1302: 1299: 1294: 1291: 1290: 1289: 1282: 1279: 1278: 1277: 1270: 1267:forest patches 1263: 1253: 1246: 1243: 1236: 1229: 1217: 1214: 1213: 1212: 1207: 1201: 1194: 1187: 1180: 1177: 1176: 1175: 1168: 1165: 1159: 1156: 1143: 1140: 1137: 1134: 1131: 1093: 1072: 1059:(Greek letter 1048: 1022: 1018: 1015: 993: 990: 986: 982: 979: 976: 973: 970: 967: 956:power function 952: 951: 939: 936: 933: 928: 924: 920: 917: 914: 891: 871: 866: 862: 858: 855: 756: 753: 731: 711: 708: 703: 696: 679: 676: 673: 613: 610: 607: 583: 580: 577: 557: 554: 551: 548: 545: 542: 539: 513: 510: 506: 493: 490: 464: 444: 441: 438: 435: 413: 410: 406: 385: 361: 350: 349: 337: 334: 331: 328: 325: 322: 319: 316: 313: 310: 305: 302: 298: 294: 289: 286: 282: 278: 275: 272: 269: 266: 263: 260: 257: 254: 251: 228: 208: 186: 183: 179: 175: 172: 169: 166: 163: 160: 144: 141: 139: 136: 95: 92: 26: 9: 6: 4: 3: 2: 9031: 9020: 9017: 9015: 9012: 9010: 9007: 9005: 9002: 9001: 8999: 8989: 8988:download page 8985: 8981: 8978: 8975: 8972: 8969: 8966: 8965:Cosma Shalizi 8962: 8961: 8957: 8954: 8953: 8949: 8946: 8943:Philip Ball: 8942: 8940: 8936: 8933: 8929: 8925: 8922: 8919: 8915: 8912: 8910: 8907: 8905: 8901: 8898: 8895: 8894: 8884: 8880: 8876: 8872: 8868: 8864: 8860: 8856: 8852: 8848: 8847: 8841: 8837: 8831: 8827: 8823: 8819: 8815: 8811: 8807: 8803: 8799: 8795: 8794: 8788: 8784: 8778: 8774: 8769: 8765: 8761: 8756: 8751: 8747: 8743: 8736: 8731: 8727: 8723: 8719: 8715: 8711: 8707: 8702: 8697: 8693: 8689: 8684: 8680: 8676: 8672: 8668: 8664: 8660: 8655: 8650: 8646: 8642: 8637: 8633: 8631:0-297-64376-2 8627: 8623: 8618: 8614: 8612:0-19-850164-1 8608: 8604: 8599: 8595: 8594: 8588: 8587: 8586: 8585: 8575: 8570: 8568: 8560: 8555: 8548: 8543: 8535: 8531: 8527: 8523: 8519: 8515: 8511: 8507: 8503: 8496: 8488: 8484: 8480: 8476: 8472: 8468: 8467: 8459: 8451: 8447: 8443: 8439: 8435: 8431: 8426: 8421: 8417: 8413: 8406: 8398: 8394: 8390: 8386: 8379: 8371: 8367: 8363: 8359: 8352: 8344: 8340: 8336: 8332: 8325: 8316: 8315:2027.42/30190 8311: 8307: 8303: 8299: 8295: 8288: 8281: 8274: 8268: 8260: 8256: 8252: 8248: 8241: 8233: 8229: 8225: 8221: 8214: 8206: 8202: 8198: 8194: 8190: 8186: 8182: 8178: 8173: 8168: 8164: 8160: 8153: 8138: 8134: 8128: 8119: 8114: 8110: 8106: 8102: 8095: 8087: 8083: 8079: 8075: 8071: 8067: 8060: 8058: 8056: 8054: 8046: 8040: 8033: 8027: 8019: 8015: 8011: 8007: 8003: 7999: 7996:(6): 066120. 7995: 7991: 7984: 7982: 7973: 7965: 7961: 7957: 7953: 7949: 7945: 7942:(6): 066115. 7941: 7937: 7930: 7921: 7907:on 2016-02-08 7906: 7902: 7896: 7887: 7882: 7875: 7867: 7863: 7859: 7855: 7851: 7847: 7842: 7837: 7833: 7829: 7822: 7803: 7799: 7792: 7785: 7776: 7771: 7767: 7763: 7759: 7755: 7748: 7741: 7733: 7729: 7725: 7721: 7717: 7713: 7709: 7705: 7698: 7690: 7686: 7682: 7678: 7674: 7670: 7666: 7662: 7655: 7647: 7643: 7639: 7635: 7631: 7627: 7623: 7619: 7614: 7609: 7605: 7601: 7594: 7586: 7582: 7578: 7574: 7570: 7566: 7561: 7556: 7552: 7548: 7541: 7533: 7529: 7525: 7521: 7517: 7513: 7506: 7498: 7494: 7490: 7486: 7482: 7478: 7470: 7456: 7455:ReactionWheel 7452: 7445: 7437: 7433: 7429: 7425: 7421: 7417: 7413: 7407: 7399: 7395: 7391: 7387: 7383: 7379: 7375: 7371: 7364: 7357: 7349: 7345: 7340: 7335: 7331: 7327: 7323: 7319: 7315: 7311: 7304: 7295: 7290: 7283: 7268: 7264: 7257: 7249: 7242: 7234: 7228: 7224: 7223: 7215: 7207: 7201: 7197: 7196: 7188: 7180: 7176: 7172: 7168: 7164: 7160: 7153: 7146: 7140: 7126: 7122: 7116: 7108: 7104: 7100: 7096: 7092: 7088: 7084: 7077: 7075: 7066: 7062: 7058: 7054: 7050: 7046: 7043:(6): 067103. 7042: 7038: 7031: 7024: 7016: 7012: 7008: 7004: 7000: 6996: 6989: 6981: 6977: 6973: 6969: 6965: 6961: 6956: 6951: 6947: 6943: 6936: 6927: 6922: 6918: 6914: 6910: 6903: 6895: 6891: 6886: 6881: 6877: 6873: 6869: 6865: 6861: 6857: 6853: 6849: 6845: 6838: 6830: 6826: 6822: 6818: 6814: 6810: 6806: 6802: 6797: 6792: 6788: 6784: 6777: 6769: 6762: 6754: 6750: 6746: 6742: 6738: 6734: 6730: 6726: 6722: 6716: 6708: 6704: 6700: 6696: 6692: 6688: 6681: 6674: 6669: 6662: 6661:Sornette 2006 6657: 6643:on 2015-03-18 6642: 6638: 6631: 6623: 6619: 6615: 6611: 6606: 6601: 6597: 6593: 6588: 6583: 6579: 6575: 6571: 6564: 6556: 6555: 6550: 6544: 6542: 6534: 6529: 6527: 6525: 6523: 6521: 6519: 6517: 6515: 6513: 6503: 6498: 6494: 6490: 6486: 6482: 6475: 6467: 6463: 6459: 6455: 6451: 6447: 6440: 6432: 6428: 6423: 6418: 6414: 6410: 6406: 6402: 6398: 6394: 6390: 6386: 6382: 6375: 6368: 6363: 6355: 6351: 6346: 6341: 6336: 6331: 6327: 6323: 6320:(5). e19779. 6319: 6315: 6311: 6304: 6302: 6300: 6291: 6287: 6283: 6279: 6275: 6271: 6267: 6263: 6259: 6255: 6248: 6241: 6233: 6229: 6224: 6219: 6215: 6211: 6207: 6203: 6199: 6192: 6190: 6181: 6177: 6173: 6169: 6165: 6161: 6156: 6151: 6147: 6143: 6142: 6134: 6132: 6130: 6128: 6111: 6104: 6100: 6097: 6096: 6084: 6081: 6079: 6076: 6074: 6071: 6069: 6066: 6064: 6061: 6059: 6056: 6054: 6051: 6049: 6046: 6044: 6041: 6039: 6036: 6034: 6031: 6030: 6023: 6019: 6017: 6012: 6007: 5997: 5992: 5989:For example, 5987: 5984: 5981:For example, 5979: 5970: 5968: 5964: 5960: 5956: 5952: 5948: 5944: 5940: 5936: 5925: 5911: 5888: 5880: 5876: 5852: 5832: 5800: 5792: 5788: 5784: 5778: 5758: 5747: 5739: 5734: 5730: 5722: 5721: 5720: 5701: 5697: 5690: 5664: 5655: 5645: 5644: 5643: 5629: 5621: 5617: 5607: 5605: 5600: 5583: 5577: 5551: 5542: 5520: 5491: 5462: 5438: 5432: 5406: 5396: 5376: 5365: 5332: 5328: 5325: 5319: 5288: 5282: 5278: 5272: 5269: 5264: 5259: 5256: 5253: 5249: 5243: 5240: 5235: 5232: 5217: 5213: 5204: 5195: 5181: 5177: 5168: 5158: 5155: 5144: 5143: 5142: 5122: 5118: 5113: 5109: 5083: 5079: 5067: 5064:For a set of 5062: 5060: 5056: 5052: 5031: 5028: 5024: 5017: 5014: 5008: 5003: 5000: 4991: 4982: 4979: 4971: 4950: 4947: 4943: 4936: 4910: 4906: 4901: 4897: 4876: 4851: 4847: 4818: 4815: 4810: 4798: 4792: 4788: 4782: 4779: 4774: 4769: 4766: 4763: 4759: 4754: 4749: 4746: 4743: 4740: 4731: 4721: 4720: 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