Power-Law Distributions in Empirical Data

Power-Law Distributions in Empirical Data Aaron Clauset Cosma Rohilla Shalizi M. E. J. Newman

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Power-Law Distributions in Empirical Data 2009 SIAM Review / arXiv 10.1137/070710111 0706.1062 clauset2009-power-law-data clauset2009-power-law-data.txt, the project's staged copy a5d664b496d08419e9bb9a40b818dbcb553f8bb0 arxiv-open statistics

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them method Quotation withheld because this record is not cleared for redistribution. power-law distributions in empirical data Quotation withheld because this record is not cleared for redistribution. simulation simulation Quotation withheld because this record is not cleared for redistribution. mechanisms the lower bound of the power-law range is chosen by minimising the Kolmogorov-Smirnov distance between the empirical distribution and the best-fit model above that bound - too high a bound shrinks the data and too low a one includes non-power-law points, so the minimum lies between Quotation withheld because this record is not cleared for redistribution. mechanisms each synthetic data set is scored against ITS OWN best-fit power law rather than against the distribution it was drawn from, which is what makes the resulting p-value unbiased: the synthetic sets undergo exactly the calculation the real data underwent, fitting included Quotation withheld because this record is not cleared for redistribution. mechanisms synthetic data are generated semiparametrically - drawn from the fitted power law with probability equal to the observed tail fraction, and otherwise resampled from the observed points below the cut-off - so the synthetic sets share the empirical body while following the power law in the tail Quotation withheld because this record is not cleared for redistribution. mechanisms the test distribution is constructed to be hard rather than convenient: it is exponential below the cut-off and a power law above, and its slope is CONTINUOUS at the cut-off, so it deviates only gently from the power law below that point - the authors' word is challenging Quotation withheld because this record is not cleared for redistribution. mechanisms uncertainty on the fitted parameters is obtained by a nonparametric bootstrap - drawing a fresh sequence of n points uniformly at random from the original data and re-estimating both the lower bound and the scaling parameter on each surrogate set Quotation withheld because this record is not cleared for redistribution. mechanisms the alternative goodness-of-fit statistics are tested rather than assumed: reweighted KS and Kuiper behave like plain KS, while Anderson-Darling returns cut-offs an order of magnitude too large, and the cost of that conservatism is stated as fewer tail observations, larger error on the scaling parameter and less ability to validate the model Quotation withheld because this record is not cleared for redistribution. mechanisms the number of synthetic data sets is fixed by the precision wanted in the p-value rather than by convention, derived from the test's expected worst-case performance about this quote: 2026-08-16: THIS QUOTE IS TRUNCATED BECAUSE THE FORMULA IT LEADS TO IS UNREADABLE IN THE STAGED TEXT. The pdftext extraction flattens the expression for the required number of synthetic data sets out of order - the staged body reads 'at least 1 2 synthetic data sets' with '4 e-' appearing separately - so the rule the sentence states cannot be quoted verbatim without quoting nonsense. The span stops at the last readable words. The paper's own worked example (accuracy to two decimal digits requiring about 2500 sets) is likewise in the damaged region and is deliberately NOT reproduced here, because I would be typing it rather than cutting it. Quotation withheld because this record is not cleared for redistribution. mechanisms the direction of the p-value is inverted relative to its ordinary use: the quantity being tested is the model the authors want to assess rather than a null, so a HIGH p-value is the favourable one and a low value rules the power law out Quotation withheld because this record is not cleared for redistribution. one can never confirm a power law, only find data consistent with it what this limits: every power-law claim anywhere: Quotation withheld - the hypothesis cannot be confirmed Quotation withheld because this record is not cleared for redistribution. few empirical phenomena obey power laws across the whole range of x what this limits: the range over which any fitted power law holds: Quotation withheld Quotation withheld because this record is not cleared for redistribution. Known sampling biases in several of the reanalysed data sets were left uncorrected. what this limits: the reanalyses in this paper: WE HAVE NOT ATTEMPTED TO CORRECT ANY BIASES IN OUR ANALYSIS Quotation withheld because this record is not cleared for redistribution. Many of the reanalysed data sets are partial samples of larger populations. what this limits: the datasets reanalysed: MANY OF THESE DATA SETS ARE ONLY SUBSETS OF MUCH LARGER ENTITIES (SUCH AS THE WEB SITES, WHICH ARE ONLY A SMALL FRACTION OF THE ENTIRE WEB) Quotation withheld because this record is not cleared for redistribution. The framework is presented only for the power law; no extension to other distributional forms is given. what this limits: the applicability of the framework: IT COULD ALSO BE EXTENDED TO OTHER, NON-POWER-LAW DISTRIBUTIONS ... ALTHOUGH WE HAVE NOT GIVEN SUCH AN EXTENSION HERE Quotation withheld because this record is not cleared for redistribution. The procedure often cannot separate power-law from log-normal behaviour. what this limits: any claim that a power law rather than a log-normal has been found: IT IS EXTREMELY DIFFICULT TO TELL THE DIFFERENCE BETWEEN LOG-NORMAL AND POWER-LAW BEHAVIOR Quotation withheld because this record is not cleared for redistribution. Two model forms indistinguishable on the observed data can give enormously different extrapolations into the tail, which is where such fits are usually applied. what this limits: extrapolating any of these fits to large rare events: the paper states conclusions from different fitted forms can differ enormously while being indistinguishable on the data Quotation withheld because this record is not cleared for redistribution. Seven of the twenty-four data sets previously conjectured to be power laws are firmly ruled out. what this limits: any of the seven named data sets as power-law distributed: HTTP connections, earthquakes, web links, fires, wealth, web hits and the metabolic network are ruled out here Quotation withheld because this record is not cleared for redistribution. Two published scaling parameters for the same citation data disagree with each other and with this paper's maximum-likelihood estimate. what this limits: a published scaling exponent for a data set: for these citation data three different values are in print, and the paper says the earlier two are incompatible with the maximum-likelihood estimate Quotation withheld because this record is not cleared for redistribution. The accept/reject verdict rests on a stated p-value threshold of 0.1. what this limits: the power-law verdicts in this paper: they follow a p-value threshold of 0.1, and cases near it are decided by that choice Quotation withheld because this record is not cleared for redistribution. The range fitted is chosen, and that choice controls how much data the test sees. what this limits: the sample size behind any of these fits: xmin is estimated, and it determines how many points enter the test Quotation withheld because this record is not cleared for redistribution. Errors run in both directions on small samples: a power law can look like a good fit to non-power-law data, and true power laws cannot be separated from alternatives. what this limits: a power-law fit on a small sample in either direction: the paper states both false acceptance and failure to discriminate occur there Quotation withheld because this record is not cleared for redistribution. the fitting procedure is deliberately applied across twenty-four data sets from unrelated branches of enquiry, so that the method rather than any one field's data is what is tested Quotation withheld because this record is not cleared for redistribution. a goodness-of-fit p-value computed for the power law AND for each plausible competing distribution, and the two compared eliminates: that the power law fits because heavy-tailed data fits almost any heavy-tailed form - a log-normal or a stretched exponential may fit as well or better, and only comparing them shows which · outcome (stated by the tagger, no quotation): held Quotation withheld because this record is not cleared for redistribution. an explicit rejection rule fixed in advance: p greater than 0.1 keeps the power-law hypothesis, otherwise it is rejected eliminates: that the decision to accept a power law is made after seeing how good the fit looks · outcome (stated by the tagger, no quotation): held Quotation withheld because this record is not cleared for redistribution. the whole procedure was run on SYNTHETIC data drawn from known distributions eliminates: that the method itself is wrong - synthetic data has a known answer, so the procedure can be checked against ground truth before it is trusted on real data · outcome (stated by the tagger, no quotation): held Quotation withheld because this record is not cleared for redistribution. power law power law Quotation withheld because this record is not cleared for redistribution. maximum likelihood maximum likelihood Quotation withheld because this record is not cleared for redistribution. scaling parameter scaling parameter Quotation withheld because this record is not cleared for redistribution. xmin xmin Quotation withheld because this record is not cleared for redistribution. likelihood ratio test likelihood ratio test Quotation withheld because this record is not cleared for redistribution. log-normal log-normal Quotation withheld because this record is not cleared for redistribution. power law - a hypothesis one can only fail to reject; the paper says an observed quantity can rarely if ever be certified as power-law distributed Quotation withheld because this record is not cleared for redistribution. power law - a property of the tail only, holding above a chosen minimum rather than across the whole range Quotation withheld because this record is not cleared for redistribution. scaling parameter - a quantity whose published values for one data set disagree, because they depend on the estimator used Quotation withheld because this record is not cleared for redistribution. contradicts → previously claimed power laws in the literature Quotation withheld because this record is not cleared for redistribution. contradicts → previously reported scaling parameters for the citation distribution of [48], namely 2.9 in [62] and 2.5 in [33] Quotation withheld because this record is not cleared for redistribution. builds-on → Ijiri and Simon [28] and Mitzenmacher [42] on the limits of characterising empirical distributions Quotation withheld because this record is not cleared for redistribution. power law —contrasted-with→ log-normal power law log-normal strength of the relation: 1 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the p-value supports a classification as power-law, log-normal or undecided; contrasted-with is loose. Quotation withheld because this record is not cleared for redistribution. scaling parameter —operationalised-as→ maximum likelihood scaling parameter maximum likelihood strength of the relation: 1 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span compares reported scaling parameters with 'our maximum likelihood figure'; operationalised-as is loose. Quotation withheld because this record is not cleared for redistribution. xmin —part-of→ power law xmin power law strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. 24 — real-world data sets tested for power-law fit Quotation withheld because this record is not cleared for redistribution. 2 < α < 3 — typical range of the scaling parameter Quotation withheld because this record is not cleared for redistribution. 8226 — average US city/town/village population (2000 Census) Quotation withheld because this record is not cleared for redistribution. 180 — typical adult male American height (near-Gaussian example) Quotation withheld because this record is not cleared for redistribution. 2 — largest deviations in a well-behaved distribution Quotation withheld because this record is not cleared for redistribution. least-squares fitting yields substantially inaccurate power-law parameters Quotation withheld because this record is not cleared for redistribution. their framework combines maximum likelihood with KS goodness-of-fit Quotation withheld because this record is not cleared for redistribution. some conjectured power laws are ruled out by the data Quotation withheld because this record is not cleared for redistribution. power laws usually hold only above a minimum value xmin Quotation withheld because this record is not cleared for redistribution. scaling parameter scaling parameter Quotation withheld because this record is not cleared for redistribution. lower bound xmin lower bound xmin Quotation withheld because this record is not cleared for redistribution. goodness-of-fit statistic goodness-of-fit statistic Quotation withheld because this record is not cleared for redistribution. quantity x quantity x Quotation withheld because this record is not cleared for redistribution. power law power law Quotation withheld because this record is not cleared for redistribution. scaling parameter scaling parameter Quotation withheld because this record is not cleared for redistribution. tail of the distribution tail of the distribution Quotation withheld because this record is not cleared for redistribution. Bayesian Bayesian Quotation withheld because this record is not cleared for redistribution. confidence-interval confidence-interval Quotation withheld because this record is not cleared for redistribution. hypothesis-testing hypothesis-testing Quotation withheld because this record is not cleared for redistribution. mean mean Quotation withheld because this record is not cleared for redistribution. null-hypothesis null-hypothesis Quotation withheld because this record is not cleared for redistribution. p-value p-value Quotation withheld because this record is not cleared for redistribution. regression regression Quotation withheld because this record is not cleared for redistribution. sample-size sample-size about this quote: QUOTE REPAIRED: the extractor split a hyphen-space and the quote was recorded faithfully with the damage in it, which made it unanchorable. The repaired string is an EXACT SUBSTRING of the staged text - that is why it was accepted, not because a rule produced it. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution. standard-deviation standard-deviation Quotation withheld because this record is not cleared for redistribution. standard-error standard-error about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. 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