A Quick Empirical Reproof of the Asymptotic Normality of the Hirsch Citation Index (First proved by Canfield, Corteel and Savage)

Shalosh B. EKHAD and Doron ZEILBERGER

Abstract: Inspired by Alexander Yong’s recent critique of the Hirsch Citation index, we give an empirical (yet very convincing!) reproof of the asymptotic normality of the Hirsch citation index (alias size of Durfee square) with respect to the uniform distribution on the “sample space” of integer partitions of n. This result was first proved rigorously (but with much greater effort!) by the humans Rodney Canfield, Sylvie Corteel, and Carla Savage. In particular, we confirm the Canfield-Corteel-Savage rigorous evaluation of the average: (^{\frac{\sqrt{6} log 2}{π}})\sqrt{n} + O(1) = 0.5404446 \sqrt{n} + O(1), and estimate the variance, numerically, as 0.0811 \sqrt{n} + O(1), and get estimates extremely close to those of the standard Normal Distribution for the first 12 standardized moments. We also observe that what Yong calls “Rodney Canfield’s concentration conjecture”, that asserts that most of the “mass” is close to the average, follows immediately from the Canfield-Corteel-Savage 1998 result, since the variance is proportional to the average (with a rather small proportionality constant, namely 0.0811/0.5404446, that is approximately 0.1501). All the results in this article were obtained via straightforward symbol- and number- crunching, by the aid of a Maple package called HIRSCH, that

is available free of charge from http://www.math.rutgers.edu/~zeilberg/tokhniot/HIRSCH .

The Cinderella Story of the Size of the Durfee Square

Once upon a time there was an esoteric and specialized notion, called “size of the Durfee square”, of interest to at most 100 specialists in the whole world. Then it was kissed by a prince called Jorge Hirsch ([H]), and became the famous (and to quite a few people, infamous) h-index, of interest to every scientist, and scholar, since it tells you how productive a scientist (or scholar) you are!

When Rodney Canfield, Sylvie Corteel, and Carla Savage wrote their beautiful article, [CCS], proving, rigorously, by a very deep and intricate analysis, the asymptotic normality of the random variable “size of Durfee square” defined on integer-partitions of n (as n → ∞), with precise asymptotics for the mean and variance, they did not dream that one day their result should be of interest to anyone who has ever published a paper.

Alexander Yong’s Critique of the h-index

In the latest issue of the Notices of the American Mathematical Society, Alexander Yong contributed ([Y]) a very insightful critique of the Hirsch citation index([H]). Yong mentioned [CCS], but apparently missed its full significance. In particular, what Yong calls “Canfield’s concentration conjecture” is an immediate consequence of the fact, proved in [CCS], that what now is called the h-index is asymptotically normal, and the fact, also proved there, that the asymptotic variance is proportional to the asymptotic average.

Why this Redux?

To be honest, we don’t have the patience to follow the intricate analysis of [CCS], and while we trust them completely, it is nice to find out things by ourselves. More importantly, we want to describe, via this case-study, how one can get much quicker, the same mathematical knowledge, by combining number-crunching and naive symbol-crunching to get (empirically, but very reliably) limiting distributions of many families of combinatorial ‘statistics’ (or random-variables). Often this method can be used to get fully rigorous results, (see [Z1][Z2]), but with much lesser effort, one can get empirical proofs. It is also very easy to come up with examples where a fully rigorous proof is completely beyond the scope of humans, or even computers, and when it is, it is not worth the efforts!

What is the h-index (alias Durfee Square)

Recall that the size of the Durfee square, alias h-index, is defined as follows. For a partition of a positive integer, n, λ = (λ_{1}, …, λ_{k}), (where λ_{1} ≥ … ≥ λ_{k} ≥ 1, and λ_{1} + … + λ_{k} = n), h(λ) is the largest i such that λ_{i} ≥ i.

In this note we reprove the above-mentioned [CCS] result about the asymptotic normality of the h-index empirically, that immediately implies the concentration-about-the-mean property, (what Yong erroneously thought was only a conjecture, but was in fact a theorem).

Symbolic Moment Calculus

In [Z1] (see also [Z2] and [CJZ]) we initiated a symbolic-computational method for the automatic (and rigorous!) proof of limit laws for many families of combinatorial random variables. But with much lesser effort, one can always derive the same results empirically by a combination of number-crunching and symbol-crunching using the very naive approach that we will briefly recall.

Let X_{n} be an infinite sequence of combinatorial families (for example, {0, 1}^{n}), and let f(x) be a random variable (for example, the sum of the entries, alias the number of 1’s).

Define a sequence of polynomials, C_{n}(t), in a variable t, by

C_{n}(t) := ∑_{x∈X_{n}} t^{f(x)} ,

(in the above example C_{n}(t) = (1 + t)^{n}), called the combinatorial generating functions. Under the uniform distribution, this turns into probability generating functions, by dividing by C_{n}(1) (alias |X_{n}|)

P_{n}(t) := \frac{C_{n}(t)}{C_{n}(1)} .

(in the above example C_{n}(t) = (1 + t)^{n}/2^{n}), To get the expectation, E_{n}(f), let’s call it a_{n}, one simply computes \frac{d}{dt}P_{n}(t)|_{t=1} (in the above example a_{n} = n/2). To get the centralized version, one divides P_{n}(t) by t^{a_{n}}, getting

Q_{n}(t) := \frac{P_{n}(t)}{t^{a_{n}}} .

The variance, m_{2}(n), is given by

m_{2}(n) = (t\frac{d}{dt})^{2}Q_{n}(t)|_{t=1} ,

and the higher moments, k ≥ 3, by

m_{k}(n) = (t\frac{d}{dt})^{k}Q_{n}(t)|_{t=1} .

Finally, the standardized moments, α_{k}(n), are given by

α_{k}(n) = \frac{m_{k}(n)}{m_{2}(n)^{k/2}} .

The random variable f has a limiting distribution if for every k > 2,

β_{k} := lim_{n→∞} α_{k}(n) ,

exists. This is usually a continuous probability distribution, and very often happens to be the good old normal distribution \frac{1}{\sqrt{2π}} e^{−x^{2}/2}, whose moments are 0 for k odd, and \frac{k!}{(k/2)!2^{k/2}} for k even. In that case our family of combinatorial random variables is called asymptotically normal.

In many cases, this approach can be used to teach the computer to prove rigorous results, by getting either closed-form, or recursive descriptions, of the leading asymptotics, in n, of the moments m_{k}(n), for symbolic n and k, and from which one can easily get the leading asymptotics for the standardized moments, α_{k}(n). This happens when the C_{n}(t) are either given explicitly, or via a decent grand generating function

F (q, t) := ∑_{n=0}^{∞} C_{n}(t)q^{n} ,

where F (q, t) is more-or-less explicit.

But what if F (q, t) is not so nice? Then we can abandon (alleged) ‘rigor’, and do things empirically. Use F (q, t) to crank out the first 10000 or whatever, polynomials C_{n}(t), and do all the above steps up to, say, the 14-th moment, and estimate asymptotics of α_{k}(n) for k ≤ 14 or whatever. If the leading terms seem to agree with those of the (standard) normal distribution: 1, 0, 3, 0, 15, 0, 105, 0, 945, … we have a very convincing empirical proof of asymptotic normality. Also, as a bonus we can numerically estimate asymptotic expressions for the average a_{n}, and the variance m_{2}(n).

The Hirsch (formerly Durfee) Polynomials

Alexander Yong reminds us (Eq. 1 of [A]), about the famous Euler-Gauss identity

∏_{i=1}^{∞} \frac{1}{1 − q^{i}} = ∑_{k=0}^{∞} \frac{q^{k^{2}}}{∏_{j=1}^{k}(1 − q^{j})^{2}} .

This lovely identity was given a pretty combinatorial proof by the Sylvester school (Durfee was Sylvester’s graduate student), that could be found, for example, in George Andrews’ partition bible([A], pp. 27-28). That proof immediately implies that the grand generating function

∑_{n=0}^{∞} C_{n}(t)q^{n} ,

equals

∑_{k=0}^{∞} \frac{q^{k^{2}}t^{k}}{∏_{j=1}^{k}(1 − q^{j})^{2}} .

(Note that in order to get the first N^{2} members of the sequence C_{n}(t) we only need to take the sum up to k = N and then taylor it up to q^{N^{2}}.) This generating function is not so easy to handle, but Maple can be easily used to crank-out the first 10000 terms. The first 6400 members of the probability generating functions (i.e. C_{n}(t)/C_{n}(1) = C_{n}(t)/p_{n}) can be found in Maple input format, suitable for computer-experimentation, in the file

http://www.math.rutgers.edu/~zeilberg/tokhniot/oHIRSCH1 .

The first 100 terms of the subsequence consisting of perfect squares, i.e. the list of C_{i^{2}}(t), for 1 ≤ i ≤ 100 can be found here:

http://www.math.rutgers.edu/~zeilberg/tokhniot/oHIRSCH2 .

The statistical analysis, as outlined above, can be found in the file

http://www.math.rutgers.edu/~zeilberg/tokhniot/oHIRSCH3 .

Let’s summarize our findings (Warning: these are non-rigorous, (but reliable), estimates).

a_{n} = 0.5404446395 \sqrt{n} + 0.085691 + 0.0374788 \frac{1}{\sqrt{n}} + O(\frac{1}{n}) .

m_{2}(n) = 0.081057 \sqrt{n} + 0.018459 − 0.018015 \frac{1}{\sqrt{n}} + O(\frac{1}{n}) .

Finally for the fourth through the eighth standardized even moments, we have:

α_{4}(n) = 3.000000000 − .084847493 \frac{1}{\sqrt{n}} − 0.1071813 \frac{1}{n} + O(\frac{1}{n^{3/2}} ) ,

α_{6}(n) = 15.0000000 − 12.60947794 \frac{1}{\sqrt{n}} + 2.080133651 \frac{1}{2} + O(\frac{1}{n^{3/2}} ) ,

α_{8}(n) = 105.0000000 − 174.8856 \frac{1}{\sqrt{n}} + 104.0909 \frac{1}{n} + O(\frac{1}{n^{3/2}} ) ,

etc. etc. Very convincing!.

As already noted in the abstract, the variance is proportional (asymptotically) to the average. It follows that, since the h-index is asymptotically normal, that there is concentration about the mean.

The Maple Package HIRSCH

Readers are welcome to continue to explore, and generalize, by downloading the Maple package HIRSCH, already mentioned in the abstract, whose url is:

http://www.math.rutgers.edu/~zeilberg/tokhniot/HIRSCH .

References

[A] G. Andrews, “The Theory of Partitions”, Cambridge University Press, 1998. Originally published by Addison-Wesley, 1976.

[CCS] E.R. Canfield, S. Corteel and C.D. Savage, Durfee polynomials, Elec. J. Comb. 5 (1998), R32. (21 pages). Available on-line from:

http://www.combinatorics.org/ojs/index.php/eljc/article/view/v5i1r32/pdf .

[CJZ] E.R. Canfield, S. Janson, and D. Zeilberger, The Mahonian Probability Distribution on Words is Asymptotically Normal, Advances in Applied Mathematics 46 (2011), 109-124. [Special issue in honor of Dennis Stanton]. Available on-line from:

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/mahon.html .

[H] J. E. Hirsch, An index to quantify an individual’s scientific research output, Proceedings of the National Academy of Science, 102 # 46 (Sept. 1, 2005), 16569-16572. Available on-line from:

http://www.pnas.org/content/102/46/16569.full .

[Y] A. Yong, Critique of Hirsch’s Citation Index: a combinatorial Fermi problem, Notices of the American Mathematical Society 61 #9 (Oct. 2014), 1040-1050. Available on-line from:

www.ams.org/notices/201409/rnoti-p1040.pdf .

[Z1] D. Zeilberger, The Automatic Central Limit Theorems Generator (and Much More!), “Advances in Combinatorial Mathematics: Proceedings of the Waterloo Workshop in Computer Algebra 2008 in honor of Georgy P. Egorychev”, chapter 8, pp. 165-174, (I.Kotsireas, E.Zima, eds. Springer Verlag, 2009.) Available on-line from:

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/georgy.html .

[Z2] D. Zeilberger, HISTABRUT: A Maple Package for Symbol-Crunching in Probability theory, The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, Aug. 25, 2010. Available on-line from:

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/histabrut.html .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

zeilberg at math dot rutgers dot edu ; http://www.math.rutgers.edu/~zeilberg/ .

Shalosh B. Ekhad, c/o D. Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

EXCLUSIVELY PUBLISHED IN : http://www.math.rutgers.edu/~zeilberg/pj.html and
arxiv.org.

Oct. 31, 2014