## Power-law distributions in empirical data (2009)

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Venue: | ISSN 00361445. doi: 10.1137/ 070710111. URL http://dx.doi.org/10.1137/070710111 |

Citations: | 200 - 3 self |

### BibTeX

@INPROCEEDINGS{Clauset09power-lawdistributions,

author = {Aaron Clauset and Cosma Rohilla Shalizi and M. E. J. Newman},

title = {Power-law distributions in empirical data},

booktitle = {ISSN 00361445. doi: 10.1137/ 070710111. URL http://dx.doi.org/10.1137/070710111},

year = {2009}

}

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### Abstract

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the empirical detection and characterization of power laws is made difficult by the large fluctuations that occur in the tail of the distribution. In particular, standard methods such as least-squares fitting are known to produce systematically biased estimates of parameters for power-law distributions and should not be used in most circumstances. Here we describe statistical techniques for making accurate parameter estimates for power-law data, based on maximum likelihood methods and the Kolmogorov-Smirnov statistic. We also show how to tell whether the data follow a power-law distribution at all, defining quantitative measures that indicate when the power law is a reasonable fit to the data and when it is not. We demonstrate these methods by applying them to twentyfour real-world data sets from a range of different disciplines. Each of the data sets has been conjectured previously to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out.

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Citation Context ... as a parameter because we know its value automatically once we are given a list of the other parameters—it is just the length of that list.Power-law distributions in empirical data 11 the evidence) =-=[30, 35]-=-, i.e., the likelihood of the data given the number of model parameters, integrated over the parameters’ possible values. Unfortunately, the integral cannot usually be performed analytically, but one ... |

200 | Goodness-of-Fit Techniques - D’Agostino, Stephens - 1986 |

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Citation Context ...ect them. Similar considerations apply for the PDF, which must integrate to 1 over the range from xmin to ∞. Standard methods exist to incorporate constraints like these into the regression analysis (=-=Weisberg, 1985-=-), but they are not used to any significant extent in the literature on power laws. APPENDIX B: Maximum likelihood estimators for the power law In this section we give derivations of the maximum likel... |

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Citation Context ...eral other established and statistically principled approaches for model comparison, such as a fully Bayesian approach [32], a cross-validation approach [59], or a minimum description length approach =-=[20]-=-, although none of these methods are described here. In the discrete case, x can take only a discrete set of values. In this paper we consider only the case of integer values with a probability distri... |

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Citation Context ...l. Finally, as with our estimate of the scaling parameter, we would like to quantify the uncertainty in our estimate for xmin. One way to do this is to make use of a nonparametric “bootstrap” method (=-=Efron and Tibshirani, 1993-=-). Given our n measurements, we generate a synthetic data set with a similar distribution to the original by drawing a new sequence of points xi, i = 1 . . . n uniformly at random from the original da... |

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Citation Context ... perhaps, the longest history. We give only a brief summary of this material here; readers interested in pursuing the topic further are encouraged to consult the books by Adler et al. [4] and Resnick =-=[49]-=- for a more thorough explanation. 6 In the statistical literature, researchers often consider a family of distributions of the form p(x) ∝ L(x)x −α , (3.12) where L(x) is some slowly varying function,... |

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Citation Context ...stributions has, perhaps, the longest history. We give only a brief summary of this material here; readers interested in pursuing the topic further are encouraged to consult the books by Adler et al. =-=[4]-=- and Resnick [49] for a more thorough explanation. 6 In the statistical literature, researchers often consider a family of distributions of the form p(x) ∝ L(x)x −α , (3.12) where L(x) is some slowly ... |

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Citation Context ...ons to observed data is the method of maximum likelihood, which provably gives accurate (asymptotically normal) parameter estimates in the limit of large sample size (Barndorff-Nielsen and Cox, 1995; =-=Wasserman, 2003-=-). Assuming that our data are drawn from a distribution that follows a power law exactly for x ≥ xmin, we can derive maximum likelihood estimators (MLEs) of the scaling parameter for both the discrete... |

18 | Currency and commodity metabolites: their identification and relation to the modularity of metabolic networks - Huss, Holme - 2007 |

9 | Dynamics of bayesian updating with dependent data and misspecified models
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Citation Context ...ithout the p-value to tell us when the results are significant. The Bayesian estimation used is equivalent to a smoothing, which to some extent buffers the results against the effects of fluctuations =-=[53]-=-, but the method is not capable, itself, of saying whether the results could be due to chance [39, 65].Power-law distributions in empirical data 21 1 0.9 0.8 (a) 0.012 0.010 (b) error rate 0.7 0.6 0.... |

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Citation Context ...lution of Internet structure or traffic patterns, then it may matter greatly whether the observed quantity follows a power law or some other form. In closing, we echo comments made by Ijiri and Simon =-=[28]-=- more than thirty years ago and similar thoughts expressed more recently by Mitzenmacher [42]. They argue that the characterization of empirical distributions is only a part of the challenge that face... |

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Citation Context ...nt to a smoothing of the MLE, which buffers the results against fluctuations to some extent (Shalizi, 2007), but the method is incapable, itself, of saying whether the results could be due to chance (=-=Mayo, 1996-=-; Wasserman, 2006).14 P(x) P(x) P(x) P(x) 10 0 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 0 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 0 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 0 10 −3 10 −2 10 −1 10 0 (a) words... |

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5 |
Finance 4
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Citation Context ... et al. (2006). An alternative approach, quite common in the economics literature, is simply to limit the analysis to the largest observed samples only, such as the largest √ n or 1 10n observations (=-=Farmer et al., 2004-=-). The methods we describe in Section III offer several advantages over these visual or heuristic techniques. In particular, the goodness-of-fit-based approach gives accurate estimates of xmin with sy... |

3 |
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Citation Context ... regularity conditions, if the data are independent, identically-distributed draws from a distribution with parameter α, then as the sample size n → ∞, ˆα → α almost surely. Proof. See, for instance, =-=[46]-=-. Proposition B.2 ([43]). The maximum likelihood estimator ˆα of the continuous power law converges almost surely on the true α. Proof. It is easily verified that ln(x/xmin) has an exponential distrib... |

2 |
Finance 5
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Citation Context ...er nonstatistical argument favoring one distribution or another. The specific problem of the indistinguishability of power laws and stretched exponentials has also been discussed by Malevergne et al. =-=[36]-=-. In some other cases the likelihood ratio tests do give conclusive answers. For instance, the stretched exponential is ruled out for the book sales, telephone calls, and citation counts, but is stron... |

1 |
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(Show Context)
Citation Context ...the largest or smallest values generated by probability distributions, values that assume some importance in studies of, for instance, earthquakes, other natural disasters, and the risks thereof— see =-=[24]-=-.Power-law distributions in empirical data 15 hold up under closer scrutiny. Consider Fig. 4.1a, which shows the CDFs of three small data sets (n = 100) drawn from a power-law distribution with α = 2... |

1 |
Performance Evaluation 42
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Citation Context ...than any other heavy-tailed distribution. (In such cases, non-parametric estimates of the distribution may be useful, though making such estimates for heavy-tailed data presents special difficulties (=-=Markovitch and Krieger, 2000-=-).) If, on the other hand, our goal is to infer plausible mechanisms that might underlie the formation and evolution of Internet structure or traffic patterns, then it may matter greatly whether the o... |

1 |
2006, in Optimality: The Second Erich L. Lehmann Symposium, edited by
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(Show Context)
Citation Context ...re, by contrast, we use the p-value as a measure of the hypothesis we are trying to verify, and hence high values, not low, are “good.” For a general discussion of the interpretation of p-values, see =-=[40]-=-.18 A. Clauset, C. R. Shalizi and M. E. J. Newman which we discuss in Section 5. Second, as mentioned above, it is possible for small values of n that the empirical distribution will follow a power l... |

1 |
Bayesian learning, evolutionary dynamics, and information theory
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(Show Context)
Citation Context ...t to the likelihood ratio test under reasonable conditions. Bayesian estimation in this context is equivalent to a smoothing of the MLE, which buffers the results against fluctuations to some extent (=-=Shalizi, 2007-=-), but the method is incapable, itself, of saying whether the results could be due to chance (Mayo, 1996; Wasserman, 2006).14 P(x) P(x) P(x) P(x) 10 0 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 0 10 −5 10... |

1 |
Estimating heavy-tail exponents through distributions in empirical data 43 max self-similarity, Preprint math/0609163
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(Show Context)
Citation Context ...n and identify a point beyond which the value appears relatively stable. But these approaches are clearly subjective and can be sensitive to noise or fluctuations in the tail of the distribution— see =-=[58]-=- and references therein. A more objective and principled approach is desirable. Here we review two such methods, one that is specific to discrete data and is based on a so-called marginal likelihood, ... |