This exploration searches for traces of the circulation of structuralist claims from statistics to network science, from Pareto to Barabási. Barabási’s claims are indeed reminiscent of the foundations of statistics, when Quetelet or Galton aimed at unveiling the laws of the social. Where does it come from? I saw a possible bridge in the concept of power law, for four reasons:
- It is a statistical distribution, comparable to the normal distribution and many others.
- It ties to network science insofar as it characterizes scale-free networks (ie. the distribution of node degrees).
- As a “statistical law” it carries a structuralist point, a claim to universality.
- It also carries a political meaning since it roots in Pareto’s study of inequalities.
Using Wikipedia as a documentary source, I had already observed that the power law and scale-free networks were bridging the domain of statistics and the domain of network science (see my previous blog post on the subject). To challenge my hypothesis and dig deeper, I engaged with the qualitative reading of the relevant Wikipedia pages. From that work I expect to get an overview of the conceptual landscape and to identify hot spots to be leveraged as landmarks for a further study of the academic literature.
From my distant reading of the question, looking at the hyperlinks network, I obtained a list of about 100 Wikipedia pages. This list was too long for the time I had, so I reduced it to the most cited pages in each of my three categories, “Pareto power law”, “Network science”, and the “bridge” between them. I quickly realized that important pages were not listed, so I added them. Of the 45 articles I read, 13 were added in a second time. The distant reading (the network exploration) put me on the right tracks but as expected, the close reading (the qualitative inquiry) was more accurate at delineating the question.
Note: we will not systematically explain the concepts, since it would be make the text heavier for a minor benefit. The general argument is about the relations between the concepts, and should be understandable even without being familiar with them. If not, then of course we suggest to read the corresponding Wikipedia pages.
This is part 1: focusing on the power law
Just defining the concepts we need to understand the field requires an effort. Not because it is complicated, though it might also be, but because each concept appears under different names, with different meanings, and do not always relate consistently. We will focus first on bringing clarity to the main concepts. This part looks only into the power law and a group of closely related concepts that we may call its family – there is quite a lot to write about. The next part will look into other concepts, and then we will inquire further specific kinds of claims and arguments.
As this is a post on my research blog, it tends to expose more than a typical paper would. I publish it early, in the spirit of open research, and following the open source software mantra “release early, release often”. So the text is quite long, but you can the findings summarized below.
The power law is often referred to as a the Pareto distribution, despite minor differences. The log-normal distribution is often presented as an alternative to the power law, bust also sometimes as an equivalent. There is a controversy about which is better fitting empirical cases, which has failed to reach closure for two decades. Both have crucial similarities but come with widely different narratives. It turns out that the tail of distributions is generally their relevant part, and it does not allow to differentiate our two competitors. As a way out of the controversy, they can be framed as instances of the more general heavy-tail distributions, which also questions the narratives convoked to interpret empirical data.
The power law and its family
- Power law
- Pareto distribution / principle
- Yule–Simon distribution
- Log-normal distribution
- Heavy-tailed distribution
- 80-20 rule
- Zipf’s law
- Long tail
- Matthew effect
A dozen concepts occupy a pretty narrow theoretical space. As we will see, they are close enough that they are sometimes convoked as if they were equivalent. If we had to pick one representative, in the context of Wikipedia it would be the power law, but the Pareto distribution is also very important as we will see. We start by looking at those and from there we give an overview of this space and clarify the relations between these different concepts, and analyze a few key observations.
“In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another.”Power law
In Wikipedia, the power law is posing as the mother of all of its kind. Not only is it defined in a relatively simple and generic manner, independently from any other law or distribution, but it also pretends to subsume all others. In its dedicated page, the section Examples opens with a typical claim about the pervasiveness of the power law, of a sort of claim we will investigate further in a moment: “More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income).” Follows a massive list of 50 cases organized in 7 sections, among which we find several familiar figures: “Pareto distribution and the Pareto principle also called the “80–20 rule””, “Zipf’s law”, “Yule–Simon distribution (discrete)”, and “The scale-free network model”.
Other distributions also acknowledge the generality of the power law. For instance the Pareto distribution, in its own article, is presented as a culturally situated version of the power law.
“The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena.”Pareto distribution
So what makes the Pareto distribution something more specific than the power law? The quote above, which is the first sentence of the article, states that the power law takes the “Pareto” name in a certain cultural context. However a second point immediately follows that precises that the power law takes the “Pareto” name for a specific value of a parameter of the power law, producing a specific “80-20” pattern.
“The Pareto distribution has colloquially become known and referred to as the Pareto principle, or “80-20 rule”, and is sometimes called the “Matthew principle”. This rule states that, for example, 80% of the wealth of a society is held by 20% of its population.”Pareto distribution
Note that the “Pareto principle” and the “Matthew principle” have distinct dedicated articles, and that the “80-20 Rule” redirects not to “Pareto distribution” but to “Pareto principle”. Doublons are not uncommon in Wikipedia and they often make sense insofar as the different pages bring different perspectives on the same thing. Those doublons sometimes coexist ignoring each other, but it is not the case here since the different concepts refer to each other explicitly, and link to each other. However not all variations of the power law state equivalence or relation to all others.
Overview of the conceptual space
The following diagram exposes how each article positions itself vis-à-vis other concepts. Each arrow states how the Wikipedia article at the source of the arrow describes its relation with the concept at the target. The absence of a link means the absence of a position (all pairs have been systematically explored).
Download the source file as a CSV matrix
(RIGHT CLICK > SAVE AS or equivalent)
By exploring this landscape we find different types of relations: inclusion (ie. one concept being more generic and the other one more specific), equivalence (ie. two different names for the same thing), and difference. If we omit the details to only look at the type of relations, if we also omit the missing links, and we generalize the relations of inclusion, the network’s structure appears quite close to an ontology. This coherence shows a general consensus, in Wikipedia, about this otherwise tight conceptual space. The diagram below shows the relations of inclusion as boxes containing each other, which greatly simplifies the image. Five relations are not of inclusion, and are thus remarkable:
- Difference between log-normal distribution and the power law
- Equivalence between the log-normal and Pareto distributions
- Equivalence between the Pareto distribution and the Matthew effect
- Equivalence between the Pareto principle and the 80-20 rule
- The long tail is a feature of the power law and its more specific versions
The general landscape seems quite classic, except for a paradoxical relation of equivalence between the Pareto distribution and the log-normal distribution, as in the following quote:
“The Pareto distribution and log-normal distribution are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the exponential distribution and normal distribution.”Pareto distribution
This quote is key because it hints at the specific position of the Pareto distribution in this conceptual landscape. On the one hand, the central concept is the power law (PL), and not the Pareto distribution (PD). The PL is more generic (both concepts agree on that), is claiming to subsume other laws (which PD is not), and has the more open definition. It is also strictly speaking the proper statistical concept to refer to in most situations, for instance to characterize what the long tail is a feature of. But on the other hand the PD is often mentioned along with, or even instead of, the PL. For instance in the following quote, the Zipf distribution uses both the PL and the PD as points of comparison, both times marking its key feature of being discrete (non-continuous).
“Zipf’s law […] can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions. […] The Zipf distribution is sometimes called the discrete Pareto distribution because it is analogous to the continuous Pareto distribution in the same way that the discrete uniform distribution is analogous to the continuous uniform distribution.”Zipf’s law
In this example we can see that the concept of power law is so vague that it raises issues when it is necessary to refer to it. Because of its extremely broad definition (“a functional relationship between two quantities, where […] one quantity varies as a power of another”) the power law plays the role of a group of notions (“family”) or a general principle. It does not have a precise equation and strictly speaking, cannot be called a distribution. This role is better played by the Pareto distribution, which is the archetype of the Power law. We hypothesize that it might actually be the prototype of the power law, insofar as the latter might have been forged as a generalization of the former. Anyway this relation leads to metonymies where the Pareto distribution is invoked in place of the power law when the point requires a precise distribution.
The Pareto distribution and the power law nevertheless have a paradoxical situation. Despite being largely equivalent, they seem in opposite situations towards the log-normal distribution: the power law is mentioned as different, while the Pareto distribution is mentioned as equivalent. How could they be, if they are more or less the same thing? As we will see, this paradox is just an instance of a more general situation where the log-normal controversy is mentioned both as equivalent and as different to the power law. This seems to be the particular form taken in Wikipedia by a controversy about the pervasiveness of these two notions. And as we will see, the Pareto distribution plays a role in it.
Taking a look at the concept map above, you can notice that the log-normal distribution only has inbound links, and no outbound links. It does not state any relation with any other concept (of our list), even though other concepts state relations with it. Contrary to the power law and similarly to the Pareto distribution, it is a distribution with a precise equation. It is related to the most central and important of all statistical distribution, the normal distribution, of which it is the logarithmic version. It also relate to the very important central limit theorem. Finally, it also takes the name of an important historical figure of statistics, Francis Galton.
“In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. […] The distribution is occasionally referred to as the Galton distribution or Galton’s distribution, after Francis Galton. The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas. […] A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain.”Log-normal distribution
The log-normal distribution, like the power law, has a claim to pervasiveness. The section “Occurrence and applications” of its article lists 21 cases and starts this way:
“The log-normal distribution is important in the description of natural phenomena. This follows, because many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale.”Log-normal distribution
Log-normal/power law controversy
Despite their different origins and characterizations, the log-normal and power law distributions have important similitudes, in particular when used to fit empirical data. A reader could not guess it by just reading the article on the log-normal distribution, since it does not relate to the characterization of the power law and this similitude is not mentioned. However it is extensively evoked in the article about the power law:
“Few empirical distributions fit a power law for all their values, but rather follow a power law in the tail. […] all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f(x) and x, and the straight-line on the log–log plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws (e.g., if the generating process of some data follows a Log-normal distribution). Thus, accurately fitting and validating power-law models is an active area of research in statistics […] For example, log-normal distributions are often mistaken for power-law distributions”Power law
To understand what is happening here we need to take a few steps outside Wikipedia. There is indeed a controversy about which of the two distributions better fit empirical data, but unfortunately there is no dedicated Wikipedia page. I realized that a quick investigation in the academic literature was fruitful to understand some Wikipedia statements. A complete study would certainly be worth it but to keep us focused I choose to compromise. I dedicated a few hours to a quick-and-dirty mini literature review, whose findings are not robust but sufficient to help us. In the following paragraphs we will see why a controversy emerged and how it leads to a specific kind of statement in Wikipedia.
On that topic, the Wikipedia article has for reference A Brief History of Generative Models for Power Law and Lognormal Distributions by M. Mitzenmacher, published in 2004. This paper “intended to be accessible to a general audience” provides an excellent overview of the question and summarizes that “lognormal and power law distributions connect quite naturally, and hence, it is not surprising that lognormal distributions have arisen as a possible alternative to power law distributions across many fields.” For a less academic and more dramatic opinion on the controversy, let us cite a 2005 post on the 3 Quark Daily blog where an exchange of emails between
C. Shalizi (champion of the log-normal) and A.-L. Barabási (defender of the power law) sets off the following comment: “This is actually a new trend — take someone’s claim that something is a power law/lognormal and then claim it is actually distributed the other way. Frankly, the two distributions are very close…” Wikipedia does not mention it, but it is important to acknowledge here that there has been a “trend” of claims of this type: “empirical data X, previously considered to follow a power law, are in fact better fit by a log-normal distribution” – or the other way around. At the core of the controversy lies this long lasting dispute about which of the two distributions is a better fit for empirical data (M. Mitzenmacher could trace it back to the 1950s), but the dispute itself is not the controversy.
The log-normal controversy is a dispute about the dispute. On the level of the core dispute itself (power law versus log-normal distribution), at least actors agree on their disagreement. But on the state of the debate, there is no consensus. In particular, multiple authors have successively tried to put an end to the dispute, without any success.
The most common attempt to reach closure states that (1) considering the kinship between the two models, and (2) the inherent gap between any model and the empirical data it fits, then (3) the dispute does not have enough relevance to justify the attention it gets. For commodity, let us call this point the pragmatist closure. In his 2004 paper M. Mitzenmacher already argues for it and suggests that “from a more pragmatic point of view, it might be reasonable to use whichever distribution makes it easier to obtain results.” In 2009, A. Clauset, C. Shalizi and M. E. J. Newman conclude their paper Power-law distributions in empirical data this way “In closing, we echo comments made by Ijiri and Simon more than thirty years ago and similar thoughts expressed more recently by Mitzenmacher. They argue that the characterization of empirical distributions is only a part of the challenge that faces us in explaining the causes and roles of power laws in the sciences. […] We hope that the methods given here will prove useful in all of these endeavors, and that these long-held hopes will at last be fulfilled.” Are their hopes fulfilled now? In a paper published in 2012 and titled Pareto or log-normal? A recursive-truncation approach to the distribution of (all) cities, G. Fazio and M. Modica once again attempted without success to reach a pragmatist closure, observing that “repeating [their analysis] confirms the difficulty of distinguishing a Pareto tail from the tail of a log-normal and, in turn, identifying the city size distribution as a false or a weak Pareto law”. Note in this citation how the key element is the tail of the distribution.
Other authors disagree with the pragmatist closure, presumably because they believe that the core dispute is actually relevant and deserves to be decided. Such authors have tried to reach closure by proposing better models. An often cited 2000 paper by W. J. Reed titled The Pareto Law of Incomes – an Explanation and an Extension argues in favor of the “double Pareto” distribution (followed by other papers of the same author). As M. Mitzenmarcher summarizes, “an appropriate double Pareto distribution can closely match the body of a lognormal distribution and the tail of a Pareto distribution” but obviously, this new model did not close the dispute. A more recent attempt at closure by better model can be found in a 2016 paper by J. Luckstead and S. Devadoss titled Pareto tails and lognormal body of US cities size distribution. “The purpose of [their] study is to propose a distribution […] to model lower and upper tails with Pareto and middle range with lognormal […]. We denote this distribution as Pareto-tails lognormal (PTLN).” M. Bee tried a similar move in 2015 with a paper whose title will suffice: Estimation of the lognormal-Pareto distribution using probability weighted moments and maximum likelihood. There is even a Wikipedia page on the Modified lognormal power-law distribution. Note that in these examples as well, the tail is the key element.
We hypothesize that the inability to reach closure is an issue for Wikipedia contributors who cannot meet a firm ground to settle a statement on the core dispute. It is unclear if the differences between the power law and the log-normal distribution are relevant, and it is unclear if one of the two models, or any other model, is actually better. If a contributor, according to a precautionary or neutrality principle, would undertake to represent the different positions of the controversy, she would corner herself in both (1) mentioning the dispute and (2) contesting its relevance. This double movement to simultaneously acknowledge and undermine the dispute is characteristic to certain statements or sets of statements we find in Wikipedia. For instance the Power law article states that “log-normal distributions are often mistaken for power-law distributions”, while the Pareto distribution article states that “The Pareto distribution and log-normal distribution are alternative distributions for describing the same types of quantities.” The article on Gibrat’s law (a law related to the log-normal distribution even though it does not appear in the diagram presented before) offers another example of this double movement, once again involving distribution tails:
“While the city size distribution is often associated with Zipf’s law, this holds only in the upper tail, because empirically the tail of a log-normal distribution cannot be distinguished from Zipf’s law. A study […] finds that the entire distribution of cities, not just the largest ones, is log-normal. But this last claim that the lognormal distribution cannot be rejected has been shown to be the result of a statistics with little power: the uniformly most powerful unbiased test comparing the lognormal to the power law shows unambiguously that the largest 1000 cities are distinctly in the power law regime.”Gibrat’s law
As we have had multiple occasions to note, tails matter. There is indeed a whole science of distribution tails and, for clarity, it is worth noting that even though there is a notional use of “long tail”, there also is a specific statistical meaning. In the articles we have analyzed the notional use is common but not universal. The other important type of tail is the “heavy tail” which appears to be the largest category, subsuming both the power law and the log-normal distribution.
“The term [long tail] is often used loosely, with no definition or arbitrary definition, but precise definitions are possible. […] In statistics, the term long-tailed distribution has a narrow technical meaning, and is a subtype of heavy-tailed distribution. […] Note that there is no sense of the “long tail” of a distribution, but only the property of a distribution being long-tailed.”Long tail
The multiple mentions of distribution tails we have seen so far come from the fact that the power law and the log-normal distribution have identical tails. The tail is naturally central to the pragmatist closure of the log-normal/power law controversy, since it is where the dispute dissolves.
“Few empirical distributions fit a power law for all their values, but rather follow a power law in the tail.”Power law
“While the city size distribution is often associated with Zipf’s law, this holds only in the upper tail, because empirically the tail of a log-normal distribution cannot be distinguished from Zipf’s law.”Gibrat’s law
But the tail is also central to the closure by better model, since many of them achieve to fit empirical data by compositing different power laws. Since the log-normal/power law dispute only happens in the head (of the distributions), this strategy is very efficient. Indeed the head is smaller than the tail and fitting a different law for the head greatly improves the model quality. The “double Pareto” distribution for instance takes its name from being the composite of two different Pareto distributions, separated by a single breaking point. We can find an illustration of such “broken power law” in the article about Zipf’s law.
Compositing power laws displaces the debate to discussing where and how to determine breaking points. Since it keeps using power laws, it allows to retain some of their features and strengthens the corresponding rationale. The log-normal distribution and the power law come with different rationales and whichever better fits empirical data can claim its own explanation for the observed phenomenon. Unfortunately Wikipedia is not rich enough to investigate this aspect of the controversy, so we will just flag this sub-controversy for further inquiry in the academic literature. We can nevertheless mention this passage from the Rank-size distribution article:
“Most simply and commonly, a distribution may be split in two, termed the head and tail. If a distribution is broken into three pieces, the third (middle) piece has several terms, generically middle, also belly, torso, and body. These frequently have some adjectives added, most significantly long tail, also fat belly, chunky middle, etc. In more traditional terms, these may be called top-tier, mid-tier, and bottom-tier. […] Purely quantitatively, a conventional way of splitting a distribution into head and tail is to consider the head to be the first p portion of ranks, which account for 1-p of the overall population, as in the 80:20 Pareto principle, where the top 20% (head) comprises 80% of the overall population. The exact cutoff depends on the distribution – each distribution has a single such cutoff point—and for power laws can be computed from the Pareto index. […] The Yule–Simon distribution that results from preferential attachment (intuitively, “the rich get richer” and “success breeds success”) simulates a broken power law and has been shown to “very well capture” word frequency versus rank distributions. It originated from trying to explain the population verses rank in different species. It has also been show to fit city population versus rank better.”Rank-size distribution
Finally, we must note that tail issues are also, and possibly mainly, about the size. This is why heavy-tailed distribution is the most generic category of our conceptual landscape. Vilfredo Pareto was “was fascinated by problems of power and wealth”, as mentioned in its own article. The Pareto distribution indeed formalizes a deeply rooted inequality. “It is a social law”, he wrote. The existence of a long tail is enough to characterize this inequality. The log-normal distribution may have a different head, it embeds as much inequality. But the inequality can be evaluated according to the thickness of the tail, hence the relevance of the general group of heavy-tailed distributions, who all embed this relation to inequality.
As a conclusion to this long section about the power law and its related concepts, and as a hint to the next part of our inquiry, network science, this quote from the article on complex networks, shows that the heavy-tailed distribution matters more than the power law:
“Most of these reported “power laws” fail when challenged with rigorous statistical testing, but the more general idea of heavy-tailed degree distributions—which many of these networks do genuinely exhibit (before finite-size effects occur)—are very different from what one would expect if edges existed independently and at random”Complex network