The Pareto distribution and Price's law

As detailed in the previous post, the ratio $f$ of the top authors that publish a fraction $v$ of all publications is independent from the total number of authors $N_0$. Of course, this result is incompatible with Price's law (that for $v=0.5$, $f = 1/\sqrt{N_0}$). This issue has been discussed by Price and co-workers [1], but I will take here a slightly different approach.

I had assumed in my derivation that he domain of the distribution was unbound above ($H = \infty$), and that the exponent $\alpha$ was higher than 1. One can relax these assumptions and check their effect on $f$ by:

1. imposing a finite upper bound $H$ and
2. by setting $\alpha = 1$. Note that 2. also requires 1. 

Role of the upper bound

In the finite $H$ case one must use the full expressions (containing $H$ and $L$) for the various quantities. In this section, we will continue to assume that $\alpha > 1$. Since $L$ acts everywhere as a scale factor for $x$ (and $H$) I will set it to 1 in the following. It is also reasonable to assume that the least productive authors have one publication (why truncate at a higher value?!) Consequently, all results will also depend on $H$, but presumably not explicitly on $N_0$, which is a prefactor for the PDF and should cancel out of all expectation calculations. It is, however, quite likely that $H$ itself will depend on $N_0$, since more authors will lead to a higher maximum publication number!

In my opinion, the most reasonable assumption is that there is only one author with $H$ publications, so that $N_0 p(H) = 1 \Rightarrow H \simeq (N_0 \alpha)^{\frac{1}{\alpha + 1}}$, neglecting the normalization prefactor of $p(x)$.

The threshold number $x_f$ is easy to obtain directly from $S(x)$:

$$x_f = \left [ f + (1-f) H^{-\alpha}\right ]^{-1/\alpha}$$

From its definition, the fraction $v$ is given by: $v = \dfrac{\alpha}{\mu} \dfrac{1}{1-H^{-\alpha}} \dfrac{1}{\alpha - 1} \left ( x_f^{1-\alpha} - H^{1-\alpha} \right )$. Note that we need here the complete expression for the mean [2]:

$$\mu = \dfrac{\alpha}{\alpha - 1} L \dfrac{1-H^{1-\alpha}}{1-H^{-\alpha}}$$

Plugging $x_f$ and $\mu$ in the definition of $v$ and setting $v = 1/2$ yields:

$$\begin{equation} f = f_{\infty} \dfrac{\left ( 1 + H^{1-\alpha}\right )^{\frac{\alpha}{\alpha - 1}} - 2^{\frac{\alpha}{\alpha - 1}}H^{-\alpha}}{1-H^{-\alpha}}, \quad \text{with } f_{\infty} = \left( \dfrac{1}{2} \right )^{\frac{\alpha}{\alpha - 1}},\end{equation}$$

and we assume that the upper bound is given by:

$$\begin{equation} H = (N_0 \alpha)^{\frac{1}{\alpha + 1}}. \end{equation}$$

Exponent $\alpha = 1$

Let us rewrite the PDF, CDF and survival function in this particular case:

$$p(x) = \dfrac{1}{1 - H^{-1}} \dfrac{1}{x^2}; \, F(x) = \dfrac{1- x^{-1}}{1 - H^{-1}} ; \, S(x) = 1 - F(x) = \dfrac{x^{-1}- H^{-1}}{1 - H^{-1}}$$

$$x_f = S^{-1}(f) = \dfrac{1}{f + (1-f) H^{-1}}$$

$$v = \dfrac{1}{2} = 1 - \dfrac{\ln(x_f)}{\ln(H)} \Rightarrow x_f = \sqrt{H} \quad \text{and, since } H = \sqrt{N_0}, \, x_f = N_0^{1/4}$$

Putting it all together yields $f = \dfrac{N_0^{1/4} - 1}{N_0^{1/2} - 1}$ and, in the high $N_0$ limit, $f \sim N_0^{-1/4}$, so the number of "prolific" authors $N_p = f N_0 = N_0^{3/4}$, a result also obtained by Price et al. [1] using the discrete distribution. They also showed that other power laws (from $N_0^{1/2}$ to $N_0^{1}$) can be obtained, depending on the exact dependence of $H$ on $N_0$.

Fraction $f$ of the most prolific authors that contribute $v = 1/2$ of the total output, as a function of the total number of authors, $N_0$, for various exponents $\alpha$. The unbound limit $f (H \rightarrow \infty)$, calculated in the previous post is also shown for $\alpha > 1$. With my choice for the relation between $N_0$ and $H$, this also corresponds to $N_0 \rightarrow \infty$. The particular value $\alpha = 1.16$ yields the 20/80 rule, but also the 0.6/50 rule shown as solid black line. Note that the curve for $\alpha = 1$ is computed using a different formula than the others and does not reach a plateau: its asymptotic regime $f \sim N_0^{-1/4}$ is shown as dotted line.

The graph above summarizes all these results: for $\alpha = 1$, $f$ reaches the asymptotic regime $f \sim N_0^{-1/4}$ very quickly ($N_0 \simeq 100$). For $\alpha > 1$, $f$ leaves this asymptote and saturates at its unbound limit $f (H \rightarrow \infty)$, calculated in the previous post. This regime change is very slow for $\alpha  < 2$: the plateau is reached for $N_0 > 10^6$.

In conclusion, an attenuated version of Price's law is indeed obtained for $\alpha = 1$(where it holds for any $N_0$) but also for reasonably low $\alpha > 1$, in particular for $\alpha = 1.16$ (of 20/80 fame) where it applies for any practical number of authors! As soon as $\alpha$ exceeds about 1.5, the decay is shallow and saturates quickly so $f$ is relatively flat.

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¹ Allison, P. D. et al., Lotka's Law: A Problem in Its Interpretation and Application Social Studies of Science 6, 269-276, (1976).↩

² https://en.wikipedia.org/wiki/Pareto_distribution ↩