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Re: [igraph] Possible error in documentation of KS.p for R/igraph fit_po

From: Tamas Nepusz
Subject: Re: [igraph] Possible error in documentation of KS.p for R/igraph fit_power_law (or in my understanding)
Date: Tue, 1 Oct 2019 13:21:48 +0200

The ‘plfit’ implementation also uses the maximum likelihood principle to determine alpha for a given xmin; When xmin is not given in advance, the algorithm will attempt to find itsoptimal value for which the p-value of a Kolmogorov-Smirnov test between the fitted distribution and the original sample is the largest.
This is not true; plfit does the opposite and looks for the _smallest_ value of the _KS test statistic_ instead (not the p-value); see this line in the plfit code:


Basically, the test statistic of the Kolmogorov-Smirnov test is the largest absolute difference between the observed and the fitted CDF along the Y axis. A small test statistic means a good fit. However, it is probably true that a smaller test statistic means a larger p-value, so in some sense the documentation is correct, but it does not describe exactly what's going on behind the scenes in the algorithm.

KS.p    Numeric scalar, the p-value of the Kolmogorov-Smirnov test. Small p-values (less than 0.05) indicate that the test rejected the hypothesis that the original data could have been drawn from the fitted power-law distribution.
This suggests that large KS.p means greater likelihood that the distribution could have come from the power-law distribution.

Let me explain what the underlined part means; I think it is correct.

The KS test goes like this. You have a null hypothesis that the observed sample was drawn from a certain power-law distribution (whose parameters we have determined with the fitting process). You calculate the test statistic D, which is constructed in a way that smaller D values mean that the observed sample is "more similar" to the CDF of the fitted power law.

The p-value is then the probability that, given that the null hypothesis is true, the test statistic is larger than or equal to the test statistic that we have observed. So, if you test statistic is, say, 0.02 and the corresponding p-value is, say 0.04, it means that _if_ I draw a sample from the fitted distribution, the probability of seeing a test statistic that is larger than or equal to 0.02 is less than 0.04 (in other words, unlikely). It does _not_ say anything about whether the sample was really drawn from the fitted distribution; in other words, it does _not_ say anything about whether the fitted distribution is correct or not. It simply says: _if_ the null hypothesis is true, it is unlikely that you would have achieved a result that is at least as extreme as the one that you have seen.

The typical interpretation of the p-value is that small p-values mean that your null hypothesis is most likely not true (it was "disproven" by the test), while large p-values mean that your null hypothesis could either be true or false, and the test could not disprove it. A statistical test can never "confirm" your null hypothesis, but usually it is not the goal anyway because the null hypothesis usually tends to be something "uninteresting".

The power-law fit is an odd beast, though: here, first you perform a _fitting_ of the parameters of the power-law distribution to your observed data, and _then_ perform a test where the null hypothesis is that the fit is good. In this case, the null hypothesis is _exactly_ what you are looking for, and a small p-value mean that the test "refuted" the null hypothesis, hence your fit is not good. Large p-values are good. Small p-values mean that no matter how hard you try to fit a power-law to your observed sample, it is most likely not a power-law distributed sample.

In a complete graph, each of N vertices has degree N-1; definitely not a power-law. Yet: [...]

[1] 1

If the explanation of KS.p is correct, this suggests a strong fit to power law,
No, it does not. Large p-values do not mean anything; it is as if the test was throwing its hands in the air and say something like "I have no idea whether your data is a power-law or not".
However, looking at the other extreme, let's generate a distribution expected to follow the power law: 

> sfp <- sample_fitness_pl(1000, 50000, 2.2)

1000 vertices is probably too small to observe a "real" power law; your sample will suffer from finite size effects, and I think that's why the test says that it is probably not a power-law. Another problem could be the number of edges; it means that your mean degree will be around 50, which is not very typical for "real" power-laws. Discrete power-law distributions like the Yule-Simon or the zeta distribution have means closer to 1; for instance, for the Yule-Simon distribution, it is (s-1) / s where s is its shape parameter.


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