Despite its popularity, the power law has not been without its failures and has rather come under criticism. In the paper ‘Scale-dependent price fluctuations for the Indian stock market’, Matia K, Pal M, Salunkay H, Stanley HE (2004), the authors explained how Indian stock market may belong to a universality of class different than that observed in developed markets.
In her book, Complexity: A Guided Tour, Melanie Mitchell, 2009 mentions how too many phenomena are being described as power law or scale-free. Data used by Laszlo Barabasi and colleagues for analyzing metabolic networks came from a web-based database to which biologists from all over the world contributed information. Such biological databases, while invaluable to research are invariably incomplete and error-ridden. A number of networks previously identified to be “scale-free” using curve fitting techniques have later been shown to in fact to have non-scale free distributions. Considerable controversies over which real-world networks are scale-free.
Evelyn Fox Keller mentioned that the current assessments of the commonality of power laws are probably overestimates. According to Cosma Shalizi,“Our tendency to hallucinate power laws is a disgrace…Preferential attachment is not necessarily the one that naturally occurs in nature. There turn out to be nine and sixty ways of constructing power laws, and every single of them is right. It’s not obvious how to decide which ones are the mechanisms that are actually causing the power law mechanism in the real world.” Even for networks that are actually scale-free, there are many possible causes for power-law degree distributions in networks.
Normality cannot be junked
Non-Normality facts and fallacies, Esch, JOIM, 2010
David N Esch in the Journal of Investment Management addresses the non-normality facts and fallacies. The author reinitiates the century-old debate by suggesting that normal efficient models can’t be simply rejected i.e. market rationality can’t be just junked.
Power-law distributions are not alone
The power law is, as we have seen, impressively ubiquitous, but they are not the only form of broad distribution. Gaussian distributions tend to prevail when events are completely independent of each other. As soon as you introduce the assumption of interdependence across events, Paretian distributions tend to surface because positive feedback loops tend to amplify small initial events. For example, the fact that a website has a lot of links increases the likelihood that others will also link to this website.
Beyond Gaussian averages: Redirecting organization science toward extreme events and power laws, Andriani, McKelvey 2007
“Gaussian distributions can morph into Paretian distributions under two conditions – when tension increases and when the cost of connections decreases. In our globalizing economy, tension rises as competitive intensity increases and as business landscapes evolve faster than the capacity of most organizations to adapt. At the same time, costs of connections are rapidly decreasing as public policy shifts towards the freer movement of goods, money and ideas and rapid improvements in the price-performance of IT infrastructures dramatically reduce the cost of information transmission. Bottom line: Paretian distributions become even more prevalent.”
In the papers, ‘Power laws, Pareto distributions and Zipf's law, M. E. J. Newman (2006) and N. Jan, L. Moseley, T. Ray, and D. Stauffer is the fossil record indicative of a critical system? Adv. Complex Syst. 2, 137–141 (1999), the authors explain how Pareto and Galton could be reconciled.
The two distributions can be reconciled. For example, if we consider one of the most famous systems in theoretical physics, the Ising model of a magnet. In its paramagnetic phase, the Ising model has a magnetization that fluctuates around zero. Suppose we measure the magnetization ‘m’ at uniform intervals and calculate the fractional change ‘δ = (∆m)/m’ between each successive pair of measurements. The change ‘∆m’ is roughly normally distributed and has a typical size set by the width of that normal distribution. The 1/m, on the other hand, produces a power-law tail when small values of m coincide with large values of ∆m.
Natural systems are replete with phase changes like Ising model. And since aspects of the same natural systems can exhibit the two respective distributions, this suggests that both distributions could not only co-exist in natural systems but also could be linked with its dynamic nature.