Is It a Power Law? Eighteen Citation Counts

Department of Psychohistory | Intermediate | 45 minutes | English draft; human subject and learner review pending

Objectives and project question

You will read a rank-size list, see why a straight log-log line is not evidence of a power law, estimate an exponent by maximum likelihood above a stated cut, compare the power law with two simpler families, and say what so small a sample can distinguish. Project question: the department lists “power law dynamics” in artifact effectiveness; do the latest citation counts support a power law over simpler alternatives?

PSY-001 defines a compounding dimension, D_c = log(value_ratio) / log(scale_ratio) (its line 66): the exponent of a power-law relation between two quantities, whereas this lesson asks how many items have each size. Neither PSY-001 nor PSY-002 uses the term. PSY-009 measures concentration without assuming a family.

1. What the record says, and what the eighteen numbers are

Observed. The department record mentions power laws three times, always as a topic: the research areas “Fractal compounding analysis — measuring D_c, power laws, learning momentum” (line 10) and “Power law dynamics — Pareto/Zipf distributions in artifact effectiveness” (line 17), and the curriculum entry “Power Law Detection: identifying Pareto distributions in governance data” (line 26). None names a data set or a test. The specification that PSY-001 cites first names both: “If the top 20% of governance artifacts account for more than 60% of citations, a power law is present” (line 56), on the evolution records’ citation counts (line 60). That is a share threshold, not a fit.

A power law gives a value x a probability proportional to x^-alpha, with exponent alpha (Pareto and Zipf distributions are of this kind): values far above the typical one stay likely.

In the quality trend, “effectiveness” is a categorical label (PSY-005); the rule, and this lesson, use citation_count: the number of tracked files that mention one of the artifact’s citation paths, counted with git grep (scripts/evolution_metrics.py).

extract.py snapshots the latest run at commit 4b03838 (2026-10-13T18:43:54Z, 18 rows) into observed-input.json. Sorted, with a star on the two counts that were not measured:

65  55*  47  45  44  29  29  23  17  15  14  13  6  6  5  1  0  0*

The sum is 414. Three facts limit what follows:

Sixteen measurements, not eighteen. The starred 55 and zero have no citation_source and no citation path, which marks an artifact that “lives outside this repository (unmeasurable here)”; both kept a hand-typed value.
Dependent counts. The two 29s share nearly all their citing files: one artifact lists the other among its citation paths; and one file can cite several artifacts.
A match, not a use. The search is for a literal string (lines 76–86): a file that only lists an artifact’s path is counted like one that uses it.

A power law gives no probability to 0: the two zeros are set aside and 16 positive counts remain.

Practice Exercise 1

How many values can enter a log-log plot, how many of those were measured by git grep, and how many separate measurements is that at most?

Answer: 18 − 2 = 16 positive counts enter the plot. The starred 55 is hand-typed: 16 − 1 = 15 git-grep measurements. The two 29s count nearly the same files: at most 15 − 1 = 14 separate measurements.

2. A straight line on log-log paper

On a rank-size plot, log10(count) is drawn against log10(rank), largest count first. Under a power law with exponent alpha the points lie near a line of slope −1 / (alpha − 1). The tempting recipe: fit a least-squares line, read alpha = 1 + 1 / |slope|, and quote R², the share of variance the line accounts for. Computed by analyze.py on the 16 positive counts: slope −1.152, R² 0.669, implied exponent 1.868.

Synthetic. The script draws 2,000 samples of 16 values from each of the three families fitted in section 4 (scenarios.json stores their parameters as assumptions) and fits the same line:

Generating family	Median R²	R² at least 0.9
Power law	0.946	81.8%
Lognormal	0.88	40.0%
Exponential	0.776	7.3%

Clauset, Shalizi and Newman (2009, Appendix A) note that “in practice we rarely see a low r², regardless of the actual form of the distribution, so that the value of r² tells us little.” Here the low value is the informative one: 1,989 of the 2,000 power-law samples reach the recorded 0.669.

Practice Exercise 2

(a) The line has slope −1.152: compute the exponent it implies. (b) A colleague’s 16 counts give an R² above 0.9. What does the table say about their family?

Answer: (a) 1 + 1 / 1.152 = 1.868. (b) Very little: an R² of 0.9 or more occurs in 81.8% of the power-law samples, but also in 40.0% of the lognormal and 7.3% of the exponential ones. Only a low value discriminates: all but 2,000 − 1,989 = 11 power-law samples reach 0.669.

3. The exponent by maximum likelihood, above a stated cut

Maximum likelihood picks the exponent under which the observed values are most probable. A power law is fitted to a tail, the values at or above a cut x_min, and a fit must state the cut and how many values remain.

continuous:  alpha = 1 + n / sum of ln(x / x_min)      standard error = (alpha - 1) / sqrt(n)
discrete:    p(x) = x^-alpha / zeta(alpha, x_min)      alpha: model mean of ln x = sample mean of ln x

zeta(alpha, x_min) is the sum of k^-alpha over the integers k >= x_min. These counts are few, small integers, so this lesson fits the discrete form; the continuous formula treats integers as points on a continuum and can be computed by hand. It gives an exponent 0.055 higher at the cut 1 and 0.059 at the cut 13; no conclusion below depends on the choice.

Computed (KS, the Kolmogorov–Smirnov distance: the largest gap between the cumulative shares of the data and of the fitted law):

Cut `x_min`	Values kept	Exponent	Standard error	KS
1	16	1.297	0.075	0.395
13	12	2.21	0.349	0.214
44	5	7.602	2.955	0.174

Over the ten cuts tried the exponent runs from 1.297 to 7.602: it describes the cut as much as the data. The cited paper’s rule, the smallest KS distance, picks 44, a “tail” of five numbers. This lesson reports the cuts 1 (every positive count) and 13 (that rule among cuts keeping at least 10 values, a floor chosen after seeing the counts).

Synthetic. In 2,000 tails of 12 values drawn from a power law of exponent 2.21, 95% of the fitted exponents lie between 1.729 and 3.339.

Practice Exercise 3

At the cut 13, 12 values remain and the sum of ln(x / 13) is 9.459. Compute the continuous exponent and its standard error; compare with the discrete fit.

Answer: alpha = 1 + 12 / 9.459 = 2.269, and (2.269 − 1) / sqrt(12) = 0.366. The discrete fit gives 2.21 with a standard error of 0.349: the forms differ by 0.059, about a sixth of either standard error.

4. Simpler alternatives, and what eighteen counts can tell

Two simpler families live on the same integers: a discrete exponential (each extra unit multiplies the probability by a constant; no heavy tail) and a lognormal (ln x is normally distributed; rounded up to an integer here; two parameters). A fitted family’s log-likelihood is the logarithm of the probability it gives to the data; higher is better. R is one family’s log-likelihood minus another’s (negative: the second fits better), and Vuong’s test (1989, as the cited paper applies it) gives the chance of a ratio this far from zero if both fit equally well; the paper trusts the sign when p < 0.1, which this lesson calls decisive.

Computed log-likelihoods, from results.json:

Values	Power law	Exponential	Lognormal
All 16 positive counts	−80.97	−67.74	−69.93
The 12 counts of 13 or more	−49.96	−48.24	not fitted

Over all positive counts the power law is the worst of the three, decisively: R = −13.23 against the exponential (p < 0.0001) and −11.04 against the lognormal (p = 0.012). Exponential and lognormal are not separated (R = 2.19, p = 0.1779). A goodness-of-fit test agrees: the recorded KS distance, 0.395, exceeds that of all 2,000 synthetic samples from the fitted power law.

On the tail of 13 or more nothing is decided: R = −1.71 against the exponential (unrounded; p = 0.2557), and the power law is not ruled out either (501 of 2,000 synthetic tails lie at least as far from their fit, p = 0.2505, above the paper’s 0.1). Yet at the cuts 15 and 17 the exponential wins decisively (p = 0.0832 and 0.0364).

Synthetic: what can so few values distinguish? Among samples of 16 values, the generating family has the highest likelihood in 3,785 of 6,000, 63.1%. The recorded counts put the exponential on top; that happens in 2 of 2,000 samples from the power law. On the tail, with a power law (alpha 2.21) and an exponential (mean 33) from 13 up as generators:

Tail values	Power law: right sign / and decisive	Exponential: right sign / and decisive
12	66.2% / 27.0%	92.2% / 48.6%
50	96.4% / 66.0%	98.7% / 74.5%
200	100.0% / 99.3%	100.0% / 99.3%

Answer to the project question: no. Read as 16 independent draws, the counts rule out a power law that starts at 1; on the 12 counts of 13 or more, this sample cannot distinguish a power law from an exponential.

Practice Exercise 4

Classify the four comparisons at the cuts 1 and 13 with the 0.1 rule, then say which sentence the results support: “the counts follow an exponential law”, “the counts do not follow a power law over their full range”, or “the tail follows a power law”.

Answer: On all positive counts, power law against exponential (p < 0.0001) and against lognormal (p = 0.012): decisive, against the power law. The other two (p = 0.1779 and p = 0.2557): undecided. Only the second sentence is supported. The first fails because the lognormal stays in play; the third because “not ruled out” is weaker than “supported”.

The strongest objection to this reading is that it describes the instrument as much as the artifacts (PSY-011). Until 2026-09-13 the counts were typed by hand. In the last run before git grep the same artifacts read 55, 12, 11, 8, 5, 4, 4, 3, 3, 3, 2, 2, 1, 1, 1, 1, 0, 0: R² is 0.959, no likelihood comparison is decisive (against the exponential: R = 1.27, p = 0.7688), and the goodness-of-fit p-value is 0.0805, already under 0.1 (today: 0 of 2,000). The project’s own rule flips with the instrument: the three or four largest counts (20% of 18 is 3.6) held 67.2% or 74.1% of citations, a power law by its 60% line, and hold 40.3% or 51.2% now. Dropping the two hand-typed counts changes little (R = −12.08 against the exponential, p = 0.0001).

Practice Exercise 5

You may measure more artifacts. From the tail table, how many tail values do you ask for, and which results would make you accept “power law” for the tail?

Answer: About 200, calibrated against an exponential only: a decisive, correct verdict arrives in 66.0% and 74.5% of samples at 50 values, in 99.3% at 200. Accept only with a stated cut and tail size, a goodness-of-fit p-value above 0.1, and R favouring the power law over both alternatives with p < 0.1.

Sources, limits, and next check

Observed, pinned at Demerzel 4b03838800dcd4894f1e026461979bff1d985ca8: extract.py reads 27 files (five quality-trend files, the 18 records of state/evolution/, the department record, the specification, PSY-001 and PSY-002) and stops if one differs from its pinned Git blob. The README gives the commands, the 46 self-checks and the negative controls.

Method references. Clauset, A., Shalizi, C. R., and Newman, M. E. J. (2009), “Power-law distributions in empirical data”, SIAM Review 51(4), 661–703, arXiv:0706.1062: the method followed here; it warns that data sets under about 50 values “should be treated with caution”. Vuong, Q. H. (1989), “Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses”, Econometrica 57(2), 307–333.

Not done. The goodness-of-fit test holds the cut fixed, where the cited paper re-estimates it in every synthetic sample: 0.2505 is probably too lenient. Not fitted: a lognormal on the tail, a power law with a cut-off. Every p-value treats the counts as independent draws from one distribution; they are neither independent nor a sample of the governed files, only the 18 with an evolution record. Vuong’s p-value is a large-sample approximation, used here on 16 and 12 values. No human has reviewed the subject matter of this lesson and no learner has tried it.

Next check. Apply the git-grep rule to every tracked file and rerun. A tail of about 200 values that passes the goodness-of-fit test and beats both alternatives decisively would change the conclusion to “a power law is plausible in the tail”.