Fitting to the Power-Law Distribution

Michel L. Goldstein, Steven A. Morris, Gary G. Yen School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, OK 74078 (Receipt date: 02/11/2004)

This paper reviews and compares methods of fitting power-law distributions and methods to test goodness-of-fit of power-law models. It is shown that the maximum likelihood estimation (MLE) and Bayesian methods are far more reliable for estimation than using graphical fitting on log-log transformed data, which is the most commonly used fitting technique. The Kolmogorov-Smirnoff (KS) goodness-of-fit test is explained and a table of KS values designed for the power-law distribution is given. The techniques presented here will advance the application of complex network theory by allowing reliable estimation of power-law models from data and further allowing quantitative assessment of goodness-of-fit of proposed power-law models to empirical data.

PACS Number(s): 02.50.Ng, 05.10.Ln, 89.75.-k

I. INTRODUCTION

In recent years, a significant amount of research focused on showing that many physical and social phenomena follow a power-law distribution. Some examples of these phenomena are the World Wide Web [1], metabolic networks [2], Internet router connections [3], journal paper reference networks [4], and sexual contact networks [5]. Often, simple graphical methods are used for establishing the fit of empirical data to a power-law distribution. Such graphical analysis can be erroneous, especially for data plotted on a log-log scale. In this scale, a pure power law distribution appears as a straight line in the plot with a constant slope. The pure power-law distribution, known as the zeta distribution, or discrete Pareto distribution [6] is expressed as:

1 k −γ pk()= (1) ζ (γ ) where: k is an integer usually measuring some variable of interest, e.g., number of links per network node p(k) is the probability of observing the value k; γ is the power-law exponent; ζ(γ) is the Riemann zeta function. Without a quantitative measure of goodness of fit, it is difficult to make final conclusions about how well the data approximates a power-law distribution. Moreover, a quantitative analysis of the goodness of fit enables the identification of possible interesting external phenomena that could be causing the distribution to deviate from a power-law. In some cases the underlying process may not actually generate power-law distributed data, but outside influences, such as data collection techniques, may cause the data to appear as power-law distributed. Quantitative assessment the goodness-of-fit for the power-law distribution can assist on identifying these cases. In the remainder of this paper, Section II discusses the methods for fitting a power law. Section III presents two candidate goodness-of-fit tests of a power-law distribution, the Kolmogorov-Smirnov test, and the χ2 goodness-of-fit test. Section IV illustrates the application of fitting and goodness-of-fit testing to analysis of a series of collections of journal papers. Finally, Section V presents conclusions about the problem of fitting power-law distributions and discusses some possible further analysis that can be implemented.

II. FITTING POWER-LAW DISTRIBUTIONS

Many methods exist in the theory of parameter estimation that can be used for estimating the exponent of the power-law distribution [7]. This section overviews three methods, namely maximum likelihood estimation (MLE), Bayesian Estimation, and linear regression-based methods.

2 In some cases the head of the distribution may deviate from a power-law, while the tail appears to be a power-law. A good example of this is the distribution of outbound links on a webpage [8]. Most have few links only, but some do have a larger amount of links, especially pages that give a list of interesting pages, also called hubs [9]. On a log- log plot, the number of outbound links in the tail appears to be linear, suggesting a “power-law tail.” It is necessary to have a strict definition of a power-law tail, and define estimators and tests for this distribution. It is important to note that the tail usually contains only a small fraction of the data. Thus, no statistical methods may be available to accurately estimate the power-law exponent, or even determine that the distribution has a power-law tail. An analysis of this uncertainty was recently performed by Jones and Handcock [10]. The scope of this paper is limited to analyzing the fitting of power-law functions and to an entire distribution and applying goodness-of-fit tests for validation and comparison. A deeper analysis for tail distributions would require first an analysis of the basis of a tail- only distribution and is beyond the scope of this paper.

A. Maximum likelihood estimation (MLE) MLE is often used for estimating the exponent of a power-law distribution [6]. It is based on finding the maximum value of the likelihood function:

N x−γ lx()γ | = ∏ i i=1 ζγ() (2) NN Lx()γ | ==logl()γγ| x∑∑()−log xii−logζ(γ)=−γlog x−Nlogζ()γ ii==11 where: l(γ|x) is called the likelihood function of γ given the data x L(γ|x) is the log-likelihood function. The log-likelihood is used because it simplifies the calculation and, because the log function is a monotonically increasing function, which does not disturb the point where the maximum is obtained. This maximum can be obtained theoretically for the zeta distribution by finding the root of the derivative of the log-likelihood function:

ddN 1 Lx()γζ|l= −−∑ ogxi N ()γ=0 ddγζi=1 ()γγ (3) ζγ′() 1 N ⇒− = ∑log xi ζγ() N i=1 where: ζ ′(γ ) is the derivative of the Riemann Zeta function. A table with the value of the ratio ζ ′(γ ) ζ (γ ) can be obtained in [11] or values can be generated on most modern mathematical and engineering calculation programs. Calculation of MLE is very fast and robust; however it only offers a single estimate of the power-law exponent without information to define the confidence interval of that estimate. In order to deal with this deficiency, a Bayesian estimator, discussed below, can be used.

B. Bayesian estimation A Bayesian estimator, derived from MLE, differs from MLE in the meaning of the parameter estimate [12]. In a Bayesian approach, the unknown parameter is not a single value, but a distribution, called the posterior distribution. Moreover, this approach incorporates what is known about the values of the parameter before any data is analyzed, speeding up convergence and assuring that the final estimate is constrained to values that are deemed as reasonable. This is done by the definition of a prior distribution p(θ). The final posterior distribution is given by a normalized multiplication of the prior and the likelihood. For the power-law distribution:

N −γ px()||γγ⋅⋅p( ) l(γx) p(γ) xi px()γ |.==∞∞∝∏ p(γ ) (4) i=1 ζγ() ∫∫px()||γγ⋅⋅p()dγl()γx p(γ)dγ −∞ −∞

The choice of the prior, as mentioned, relates to the range and, possibly, the distribution in this range. Moreover, the prior also defines possible discretization of the estimated parameter. A discrete prior always generates a discrete posterior distribution.

4 Another good feature of the Bayesian estimation process is that it is naturally adaptable to iterative estimation, where one sample or a subset of the samples is analyzed at a time. This is particularly useful if it is interesting to analyze the influence of each of the samples, and if the amount of memory available for implementation is low (the only thing that has to be stored is the posterior at each step).

C. Linear regression-based estimators For power-law exponent estimation, linear regression is an often used estimation procedure [13]. Different variations of this technique are all based on the same principle: a linear fit is made to the data that is plotted on a log-log scale. Actually, with reasonable accuracy, the linear fit can be made by hand on a log-log plot of the distribution. However, the linear fitting does not take into consideration that almost all of the data observed is on the first few points of the distribution. For example, for an exponent, γ, of 3.0, 93.6% of the data is expected to have k=1 or k=2. Therefore, an estimation method that does not incorporate this fact will fit to the “noise” in the tail, where very few observations occur [14]. Because of this, two modifications to direct linear fitting were proposed: 1) the use of only the first 5 points for regression and, 2) the use of logarithmically binned data. The first variation is straightforward to implement. The second variation is based on adding all values that fall into bins that are logarithmically spaced (same size in the logarithmic scale), and then performing the linear regression on the log of the quantity of these groups of data. This method is similar to binning methods used for curve estimation [1]. The advantage of this method is that, by grouping the data points the noise is reduced. The reduction of noise is dependent on the size chosen for the bins. However, this method only generates a graph that is approximately linear even when the distribution is a power-law, as can be seen in Figure 1 for “log2” bins, i.e., bin boundaries at (0, 1, 2, 4, 8, ...). The linearization error decreases the estimation accuracy. More importantly, the slope obtained is not directly the value of the power-law exponent. This can be observed by plotting the exponent of the power-law distribution and the slope obtained by simulating this distribution and using the method explained above. Figure 2 shows this

5 plot for “log2” bins. Approximating the relation to a line, the following transformation equation was obtained:

b = 1.094⋅−γ 0.963 (5) where: b is the measured exponent γ is the actual exponent. Another common method of linear estimation is of using 5 bins per decade. Using the same method, the transformation equation obtained using 10 bins (2 decades) was:

b = 1.026⋅−γ 0.931. (6)

The most important observation about the linear fitting methods is that they are not tied to the definition of a probability distribution. Thus, the integration of the fit line may not be unity. By using the slope given by this fitted line and forcing an adjustment to the intercept in order for the fitted function to be a probability distribution, the final function may end up visually distant from the empirical distribution. Examples in Section IV will assist on illustrating this behavior.

III. GOODNESS-OF-FIT TESTS

In statistical analysis, many methods for assessing the goodness-of-fit of a distribution have been proposed. Among these methods, the most commonly used for general distributions are Pearson’s χ2 goodness-of-fit test, and the Kolmogorov-Smirnov (KS) type test.

A. Pearson’s χ2 goodness-of-fit test Pearson’s χ2 test is the most commonly used test for large samples. It was introduced by Pearson in 1900 [15] and is defined as the following test when hypothesizing for a specific distribution:

6 2 C ()OEii− 2 Q = ∑ ~& χC−1 (7) i=1 Ei where: C is the total number of classes

Oi is the observed value related to class i

Ei is the expected value of class i. When the distribution is independent on the data, the number of degrees of freedom of the χ2 distribution is, as shown in Equation (7), C-1. However, when the distribution in the hypothesis has some parameters that are estimated using the same data to which the test is going to be applied, the number of degrees of freedom decreases [16]. More specifically, the number of degrees of freedom of the χ2 distribution is C-s-1, where s is the number of parameters that were obtained from the data. This decrease in the number of degrees of freedom assumes that the parameters were obtained using MLE method. Using other methods may cause this number do decrease even more. Thus, it is only possible to apply the χ2 test when MLE is performed. For MLE, the degrees of freedom used for testing for the power-law distribution is C-2. Another important decision for the χ2 test is on the number of classes to use. Later analysis of the χ2 test has shown that the test is not valid when the expected value of the quantity in any of the classes is less than 5 [16]. Therefore, it is necessary to sum all values from the tail of the distribution into a class whose total expected value is greater than 5. For example, in a dataset with 5,000 samples and a γ of 3.0, there would be 10 classes; a class for integers 1 to 9 and a tail class for all points whose frequency is greater than 9. Nicchols [17] points out that the need for all class expected values to be greater than 5 is a rule of thumb, it has a purely heuristic reason and is the main criticism researchers have about using this test. For example, another possible solution would be to use the smallest classes possible for the tail, i.e., instead of grouping all tail values into one class, they would be grouped into classes such that each of the class has an expected value as close to 5 as possible. By choosing this heuristic solution, which does not have conflicts with any of the test assumptions, the χ2 test statistic may vary considerably. Because of this, most analyses tend to employ the Kolmogorov-Smirnov test.

B. Kolmogorov-Smirnov-type test The KS-type test has recently been applied to testing goodness-of-fit when total sample size is small. The test is based on the following value:

KF=sup *( x) −S( x) (8) x where: F*(x) is the hypothesized cumulative distribution function S(x) is the empirical distribution function based on the sampled data. Kolmogorov [18] first supplied a table for the quantiles of this distribution for the case where the probability function was independent on the data points. However, when there is dependence, other tables must be used. This limitation was not taken into consideration by Pao and Nicholls in their application [17, 19] of the KS test to power- laws. Without correcting for this factor, the KS test gives a rejection rate lower than what is expected [20]. Lilifoers later introduced tables for using the KS test with other distributions, such as normal and exponential [21, 22]. These tables were obtained using a Monte Carlo method, which is based on the generation of a large number of distributions with random parameters and calculating the test statistic for each of the test cases. From these tests, empirical values for the quantiles can be extracted. The same procedure was used to obtain these values for the power-law distribution. For each of the logarithmically spaced sample sizes, 10,000 power-law distributions were simulated, with random exponent from 1.5 to 4.0. Statistics were collected from these simulations to generate the KS table, shown in Table I. This table was created assuming that the estimation method used was the MLE. If other estimation methods are used, it would be necessary to construct a new KS table. A step by step example of how to apply the KS test for determining the goodness-of- fit to a power-law is presented in the next section.

8 IV. EXAMPLES

First a simple example will be given on how the KS table can be used for determining the goodness-of-fit of an empirical distribution to a power-law distribution. Using data from a small collection of 131 papers and 359 authors that cover the topic of MEMS RF switches, the process of using the KS goodness-of-fit test for the papers per author distribution follows four steps: 1) Use the MLE method for estimating the power-law exponent. In this case, the estimated exponent was 2.76. 2) Generate the hypothesized cumulative distribution F*(x) using the cumulative sum of equation (1) and build a table showing side by side the values of F*(x) and S(x) where there were values observed in the dataset. This table is shown in Table II. 3) Calculate the absolute difference between each pair of values and find the maximum. This is the KS test statistic. The absolute differences can be seen in Table II. The value in bold is largest difference for this dataset. 4) On the table in the Appendix, the largest value, 0.0313, is compared with the values in the row with the closest number of points. For a more conservative approach, where it is better to accept the hypothesized distribution when there is a doubt, the row with the lower number of points should be used. In this case, use the row for 100 points. This row shows that for 90% of the cases when the distribution was a power-law, the KS statistic was 0.0580 or below. The maximum observed KS statistic for this example was much lower than this. In other words, the p value, or Observed Significance Level (OSL) is greater than 10%. Thus, using a confidence level of 5%, there is no statistical evidence to support that this distribution is not power-law. This simple example shows two important details about any goodness-of-fit tests: the result of the test does not prove that a sample actually comes from a power-law distribution, it can only suggest when the chance of being a power-law is low. As would be expected, higher chance of the sample being tested of being from a power-law distribution is suggested by a higher the p value. The latter can be used as a method to

9 compare samples to infer which are more likely to be generated by a power-law distribution. A second example is a collection of papers covering the topic of vibrating sandpiles, containing 368 papers with 6272 references. The power-law exponent of the paper per reference distribution was extracted using the four methods discussed above (the Bayesian method was not used because it generates results that are not easily comparable to the other results): MLE, linear regression in log-log scale, linear regression using the first 5 points only and linear regression using logarithmically binned data. A third example estimates the power-law exponent of the authors per paper distribution of a collection 336 papers and 422 authors from a collaboration network associated with researchers at the University of Maryland Psychiatric Research Center. Figure 3 shows a comparison of the different results for this dataset. As discussed, the linear regression on log-log transformed data fits all points with equal weight, and greatly underestimates the exponent. Using the first 5 points, the method over-estimated the exponent, because it does not take into consideration the tail, while the log2 binned data underestimated the exponent because it places too much emphasis on the tail points. Table III shows a summary of the estimated exponents obtained using each of the fitting methods applied to 27 different collections of journal papers from 27 different research topics. These collections were gathered from the Institute for Scientific Information Web of Science product over a period of two years from queries and seed references and were used for research in information visualization and knowledge domain mapping. The characteristics of these collections are summarized in Table IV. For paper collections two distributions are usually claimed as power-laws: papers per author (Lotka’s Law [23]) and papers per reference ([24]). The number of papers in each collection varies from 131 to 14,211 and the MLE power-law exponents vary from 1.99 to 3.71 for the distribution of papers per author and 1.98 to 3.93 for the distribution of papers per reference. Using the MLE, the two goodness-of-fit tests discussed above were used to analyze all 27 datasets. Table V shows the overall result of the number of distributions that were actually accepted as power laws using both goodness of fit tests described in the previous section using a 95% confidence level.

10 These results support the idea that it is not possible to assume in all datasets that these distributions are actually power-law. Papers per author distributions experience a 56% acceptance rate using the KS test and appear to be more likely to be accepted as actual power-laws, . However, using the KS test of power-law fit on the papers per reference distribution the acceptance rate was only 7%, so that it is very unlikely to be an actual power law, as suggested by Naranan [24]. Further analysis on the ability to define power- law tail distributions would be needed to test if the paper-per-reference distribution is power-law tail only as reported by Redner [4]. The two collections where paper per reference distribution was accepted as power-law by both χ2 and KS tests are two small datasets containing 148 papers and 3,767 references, and 131 papers and 1,573 references, respectively. In Figure 4 to Figure 7, some examples of distributions and their actual goodness of fit test values are shown. In these examples, it is easy to observe that visual inspection in a log-log plot is not accurate enough to determine if these distributions are actual power- laws or not.

V. CONCLUSIONS

This paper presents an analysis of the extent to which empirical data can be assumed to be power-law distributed. First a brief discussion of possible fitness methods was presented. Then two well-known goodness-of-fit measurements were used for the analysis: Pearson’s χ2 test and the Kolmogorov-Smirnov test. A KS table for testing the fitness of MLE estimated power-law was provided. Using these goodness-of-fit tests on 27 collections of journal papers and testing them for two distributions that are usually believed as power-laws, the papers per author and papers per reference distributions, it was shown that caution must be taken when assuming power-law distributions. Especially on the papers per reference distributions, it was observed that in many cases the power-law distribution could not be substantiated. Importantly, the usual method for observing power-law distributions, that is, plotting on log-log scale, offers no support on actually identifying poor goodness of fit.

11 Most of the fitness problems may be caused by external effects that usually affect the initial points of the distribution only (the head of the distribution). Further analysis would be required to test if the tail of the distribution is a power-law. However, when testing for the fit of the tail, it is wise to be cautious about the extreme paucity of sample points that generate the tail of the distribution. The power of goodness-of-fit tests decreases when fewer points are sampled. Therefore, it becomes much more difficult to confirm that the distribution of the tail is power-law and not any other distribution. Another possible analysis that could be performed with this data is a quantitative analysis of the modifying external effects. For example, if it is known that, in some collections of journal paper there may be some survey papers that reference many papers that were never referenced before and that are actually external from the dataset, this may cause an unexpected increase in the number of references appearing in only one paper (the first value in the distribution). With a goodness of fit test it is possible to establish some hypothesis on the amount of external references that were added to the database and, possibly, remove them from further analyses. Overall, the evaluation of these tests is simple and does not add much to the overall processing complexity. The insightful understanding of goodness of fit measurements when testing for power-law distributions enhances the capabilities of analysis of datasets that may show highly skewed distributions. It is a vital process in order to confirm assumptions and make meaning full comparisons when modeling of the datasets.

Table I - KS test table for power-law distributions Quantile # samples 0.9 0.95 0.99 0.999 10 0.1765 0.2103 0.2835 0.3874 20 0.1257 0.1486 0.2003 0.2696 30 0.1048 0.1239 0.1627 0.2127 40 0.0920 0.1075 0.1439 0.1857 50 0.0826 0.0979 0.1281 0.1719 100 0.0580 0.0692 0.0922 0.1164 500 0.0258 0.0307 0.0412 0.0550 1000 0.0186 0.0216 0.0283 0.0358 2000 0.0129 0.0151 0.0197 0.0246 3000 0.0102 0.0118 0.0155 0.0202 4000 0.0087 0.0101 0.0131 0.0172 5000 0.0073 0.0086 0.0113 0.0147 10000 0.0059 0.0069 0.0089 0.0117 50000 0.0025 0.0034 0.0061 0.0077

Table II - Sample results for using the KS goodness-of-fit test x S(x) F*(x) |F*(x) - S(x)| 1 0.7647 0.7960 0.0313 2 0.9188 0.9132 0.0056 3 0.9692 0.9513 0.0178 4 0.9860 0.9686 0.0174 5 0.9916 0.9779 0.0137 6 0.9972 0.9835 0.0137 17 1.0000 0.9971 0.0029

Table III – Sample statistics for power-law fitting of all datasets varying the fitting method Papers per author Papers per reference

µ σ µ σ MLE 2.63 0.47 MLE 2.78 0.51 Linear 2.17 0.48 Linear 2.04 0.46 Linear (5p) 2.59 0.51 Linear (5p) 2.77 0.49 Log2 bins 2.73 0.55 Log2 bins 2.60 0.46

Table IV - Summary table of a series of 27 paper collections used in for demonstrating power-law fitting and goodness of fit testing.

no. of no. of no. of Index topic papers references authors 1 agent based models 148 3767 259 2 angiogenesis 453 8246 1590 3 anthrax 2472 25010 4493 4 atrial ablation 3095 22670 6574 5 biosensors 5892 32767 11034 6 botox 1560 20819 3521 7 cocition and bibliographic coupling 550 13010 492 8 complex networks 902 19185 1665 9 distance education 1391 16603 2472 10 econophysics 482 6281 588 11 ht supercon 1631 29044 3001 12 info science 14211 119289 9413 13 information visualization 2450 56912 5545 14 mems RF switch 131 1573 359 15 milgrams 404 6791 465 16 molecular imprinting 513 5717 785 17 nerve agents 407 8293 1064 18 neuroimaging 671 25279 2042 19 ontology 224 6501 456 20 schizophrenia 513 20422 1477 21 scientometrics 3468 70117 2928 22 self organized criticality 1634 27622 2176 23 silicon on insulator semiconductor 2383 23041 4902 24 superstring 6652 53568 4813 25 TQM 1893 28216 2875 26 U of Maryland 336 5890 422 27 vibrating sandpiles 368 6272 547

Table V - Overall results for the goodness of fit for all datasets Papers per author Papers per reference 2 2 χ test KS test χ test KS test # rejected 10 (37%) 12 (44%) # rejected 26 (96%) 25 (93%) # accepted 17 (63%) 15 (56%) # accepted 1 (4%) 2 (7%)

Figure 1 - Log-2 bin results for a theoretical power-law distribution with N=1000 and γ=2.0

Figure 2 - Empirical transformation between the slope of the log-2 binned data and the power-law exponent

Figure 3 - Results of fitting using the different fitting methods for papers per reference distribution for the vibrating sandpiles dataset.

Figure 4 –Papers per author distribution for the University of Maryland dataset. The circles represent the actual empirical distribution, the line is the Maximum Likelihood fit (gamma = 2.02).

The database has 336 papers, 422 authors, Q = 3.24, pQ = 0.7785, K = 0.0158, pK > 0.1.

Figure 5 – Papers per author distribution for the atrial ablation dataset. The circles represent the actual empirical distribution, the line is the Maximum Likelihood fit (gamma = 2.11). The database -5 has 3,095 papers, 6,574 authors, Q = 62.7, pQ = 1.5·10 , K = 0.0125, pK < 0.01.

Figure 6 – Reference distribution for the superstring dataset. The circles represent the actual empirical distribution, the line is the Maximum Likelihood fit (gamma = 1.98). The database has

6,652 papers, 53,568 references, 208,119 citations, Q = 604, pQ = 0, K = 0.0139, pK < 0.001.

Figure 7 – Reference distribution for the angiogenesis dataset. The circles represent the actual empirical distribution, the line is the Maximum Likelihood fit (gamma = 2.33). The database has 453 -5 papers, 8,246 references, 18,818 citations, Q = 1.74, pQ = 1.2·10 , K = 0.0107, pK < 0.01.

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