- Part 0 – Exploring relations between Pareto and Network Science
- Part 1 – Focusing on the power law
- Part 2 – Concepts of complex network and preferential attachment
- Part 3 – Statistical law or distribution?
- Part 4 – How network science bridges with statistics
This is part 5: What is traveling from statistical distributions to network science
In this last part of my close reading of Wikipedia on statistics and network science, I look at three transversal topics.
- Social inequalities, and notably the figure of Pareto
- Claims to pervasiveness (statements that a phenomenon is observed in many unrelated situations)
- Universality, a specific statistical concept imported from thermodynamics
These three micro analyses close the series of my empirical observations while reading my corpus of articles, and allow me to draw some conclusions on the rhetoric used in network science (the most recent field) to draw from statistics (the older field). My question here is what does it draw, and why.
As a post on my research blog, it is honest on what is actually done but at the cost of not-so-relevant material. I also follow the mantra of “release early, release often”, so please forgive the lack of finishing touches.
To understand these findings, we need a starting point. That starting point is a promise brought to existence by the recent (90s) discovery of a new type of mathematical structure, the complex network (whether you call it small world or scale-free does not matter much, as we have previously seen). This new discovery was/is an opportunity to make a scientific breakthrough, and for good reason. Indeed complex networks create an unprecedented bridge between empirical observations on the living and the social, and some new fascinating theories in physics. What if networks were a missing link to understand the laws of the social and the living? Such is the promise complex networks seem to make, and a necessary preliminary to understand the rhetoric of network science.
Vilfredo Pareto studied the question of inequalities, and more precisely “situations in which an equilibrium is found in the distribution of the ‘small’ to the ‘large'” (article on Pareto distribution). His work was published at the beginning of the 20th century, but he did not become such prominent figure at that time. A management consultant named J. M. Juran “stumbled across the work of Vilfredo Pareto and began to apply the Pareto principle to quality issues (for example, 80% of a problem is caused by 20% of the causes).” (article on Joseph M. Juran). It is Juran who coined the term of Pareto principle, and popularized his ideas. They seemed to have been influential in economics first, and not only about networks (Pareto efficiency for instance is related to games theory).
The figure of Pareto is an important figure in the rhetoric of network science, because he represents the empirical reach to social questions. Through him, the power law tells us something about how the social works. Inequalities are the archetypal example for the conceptual bridge between power-law and scale-free networks. This example is often declined in terms of number of citations for scientific papers, or hyperlink citations for web pages or websites, a context in which being cited is interpreted as a form of value. Preferential attachment, which generates power laws and may explain scale free networks, is also called “rich get richer”. Vilfredo Pareto and his legacy embody the empirical reach of the power law in the social world.
From the 50s to the 70s a fascinating new approach takes over the world of physics, from quantum mechanics to thermodynamics: renormalization group theory. It allows to reformulate old questions in a surprisingly fruitful way. It was remarkably successful in quantum mechanics, but also in thermodynamics where it allows to study phase transitions and solve an old mystery. Physicist observed that certain quantities follow power laws at the critical point of phase transitions. Puzzling observation, the exponents of these power laws are often the same, even in systems whose dynamics at a micro scale are completely different. This empirical fact obtained the pompous name of “universality”. The mathematical procedure of renormalization explained this universality by showing that dynamics at a micro scale do not matter at the critical point, and that dissimilar systems can have similar scaling dynamics. This theory, which also proves that phase transition give rise to power laws, is considered a masterpiece by physicists.
In the article on the power law we find what we call the rhetoric of universality, revolving around the idea that power laws are the sign of an underlying reality, as guaranteed by the theory of universality. Unfortunately this rationale is fallacious insofar as many other situations than phase transitions can give rise to power laws, and as universality is not a theory but an empirical observation, despite its unfortunate name. Phase transitions give rise to power laws, but power laws are not signs of phase transitions. The renormalization group theory can rarely be mobilized, universality does not apply, and finally there is no ground for stating that power laws are the symptom of an underlying mechanism. Credible sources explicitly discuss that fallacy, frame it as a misunderstanding and explain why it was prevalent. We hypothesize that the Wikipedia page on the power law reflects a past state of the scientific discussion.
The rhetoric of network science heavily relies on claims to pervasiveness. It constantly states that the different avatars of complex networks (scale-free/small-world networks, preferential attachment, power law…) are empirically observed in many unrelated situations. As we have already seen, pervasiveness is also a defining characteristic of scientific/statistical laws.
We observe that claims to pervasiveness as well as the (fallacious) rhetoric of universality are remarkably aligned with the belief that complex networks could lead to a scientific breakthrough similar to the discovery of a scientific law, but in the social and biological worlds. We hypothesize that the rhetoric of network science aims at sustaining the belief that complex networks are the key to scientific laws of a whole new empirical reach, even though it has not been the case so far.
We also conclude from our observation of that rhetoric that it is characteristic of structuralism.
What is traveling from statistical distributions to network science
Vilfredo Pareto and social inequalities
As we have seen, the Pareto distribution is an avatar, if not the main avatar, of the power law. It is named after an Italian scholar who studied inequalities in the late 19th / early 20th century.
Through his name, the question of inequalities connects to the concept of power law. The underlying mathematical reason is obvious: the power law is a model for the distribution of wealth.
“The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, […] is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena. […] Originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population […] Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income. […] This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the “small” to the “large”.”Pareto distribution
The article dedicated to the distribution of wealth also acknowledges the importance of the Pareto distribution for wealth inequalities.
“Pareto Distribution has often been used to mathematically quantify the distribution of wealth at the right tail (the wealth of very rich). In fact, the tail of wealth distribution, similar to the one of income distribution, behave like Pareto distribution but with ticker tail. […] The distribution of wealth throughout the population is often closely approximated by a Pareto distribution, with tails which decay as a power-law in wealth.”Distribution of wealth
In pages about statistics, income distribution is cited as an example of the power law, and economics are cited as an application of the statistics of the power law (or of complex networks), as in the following excerpt:
“A few notable examples of power laws are Pareto’s law of income distribution, structural self-similarity of fractals, and scaling laws in biological systems. Research on the origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including […] economics and more.”Power law
Another evidence for the connection between power law and inequalities is the “rich get richer” name for preferential attachment. In other words, inequalities come up as a name of the model bridging power-law and scale-free networks.
“The most widely known generative model for a subset of scale-free networks is Barabási and Albert’s (1999) rich get richer generative model in which each new Web page creates links to existing Web pages with a probability distribution which is not uniform, but proportional to the current in-degree of Web pages.”Scale-free network
Notice how despite the “rich get richer” mention, that example does not involve literal wealth, but only the links of web pages. Wealth is understood as a metaphor of having many connections. It is also possible that having links is considered literally as a form of value. The point could be perfectly legit, but the Wikipedia articles are not explicit on that matter. We still find multiple mentions of hyperlinks-as-value, for instance:
“The concept drew in part from a February 2003 essay by Clay Shirky, “Power Laws, Weblogs and Inequality”, which noted that a relative handful of weblogs have many links going into them but “the long tail” of millions of weblogs may have only a handful of links going into them.”Long tail
Also note that the “rich-get-richer” expression is not necessarily tied to the power law or scale-free networks. The statement has a life on its own in the economic sphere, and even has its own article.
“Thomas Piketty’s book Capital in the Twenty-First Century (2014) presents a body of empirical data spanning several hundred years that supports his central thesis that the owners of capital accumulate wealth more quickly than those who provide labour, a phenomenon widely described with the term “the rich-get-richer”.”The rich get richer and the poor get poorer
Conversely, inequalities of wealth or income are not systematic in the illustration of the power law, even in an economic context. See for instance below, a list of imbalances featuring GDP and healthcare but not income or wealth:
“The Pareto principle is a popular example of such a “law”. It states that roughly 80% of the effects come from 20% of the causes, and is thusly also known as the 80/20 rule. In business, the 80/20 rule says that 80% of your business comes from just 20% of your customers. In software engineering, it’s often said that 80% of the errors are caused by just 20% of the bugs. 20% of the world creates roughly 80% of worldwide GDP. 80% of healthcare expenses in the US are caused by 20% of the population.”Empirical statistical laws
Pareto is a key figure to the bridge between statistics and network science, between statistical laws and scale-free networks between distributions and complexity. This connection involves the question of social inequalities, not only as income and wealth (Pareto’s initial work) but also as number of connections. As the power law applies to scale-free networks via preferential attachment, theoretical elements on social inequalities move from statistics to network science.
Also note that the work of Pareto on inequalities is not limited to the statistical distribution. Another remarkable concept is the Pareto efficiency in the field of games theory.
“Pareto efficiency or Pareto optimality is a state of allocation of resources from which it is impossible to reallocate so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. […] “Pareto efficiency” is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is simply a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).”Pareto efficiency
Finally, we must also note that the multiple concepts named after Vilfredo Pareto were named as such and became popular much later, after World War II. The article on the Pareto principle states that “Management consultant Joseph M. Juran suggested the principle and named it after Italian economist Vilfredo Pareto”. In J. M. Juran’s own Wikipedia page we find confirmation:
“In 1941, Juran stumbled across the work of Vilfredo Pareto and began to apply the Pareto principle to quality issues (for example, 80% of a problem is caused by 20% of the causes). This is also known as “the vital few and the trivial many.” […] In a way, Pareto’s Principle puts numbers to the idea that in business, as in life, things are not evenly distributed. Pareto was studying land ownership in Switzerland. But Juran saw that it applied to business, as well.”Joseph M. Juran (Wikipedia article on)
Everywhere: claims to pervasiveness
Statistical laws and complex networks are often presented as pervasive. Our Wikipedia articles mention in multiple occasions that distribution X or phenomenon Y can be found everywhere. These statements of empirical ubiquity can take multiple forms and seems to play an important role in the rhetoric of statistics as well as network science. By looking at what it looks like, we can understood why it is so important. We observed three types of claims to pervasiveness:
- Enumeration, a list of empirical exemples.
- Distanced statement, eg. X is believed/said to be found everywhere.
- Direct statement, eg. X can be found everywhere.
Enumeration is the most common type of claim. It is generally presented as self-evident. The two examples below are about statistical distributions.
“The Pareto distribution […] is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena.”Pareto distribution
“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).”Power law
We have seen that in the argumentation, a major bridge between statistical distributions/laws and network science is through preferential attachment, a model that explains why in scale-free networks the number of links per node follows a power law distribution. These three elements are all subject to claims to pervasiveness: the power law / Pareto distribution (as shown above), complex/scale-free networks, and preferential attachment itself, as in the citation below:
“This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior. Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities, the wealth of extremely wealthy individuals, the number of citations received by learned publications, and the number of links to pages on the World Wide Web.”Preferential attachment
There is first a direct claim to empirical pervasiveness (“the species distribution and many other phenomena”) emphasized in the next sentence by an enumeration. Here pervasiveness is not only stated or implied, it is presented as “the primary reason for the historical interest”. The Wikipedia contributors are aware of the importance of pervasiveness even though it is not established as a fact. Note how cautious is the text, though.
Complex networks are also claimed to be pervasive, for instance in the article on scale-free networks, with a nice bullet-points list:
“A few examples of networks claimed to be scale-free include:Scale-free network
– Social networks, including collaboration networks. […]
– Many kinds of computer networks […]
– Software dependency graphs […]
– Some financial networks such as interbank payment networks
– Protein-protein interaction networks.
– Semantic networks.
– Airline networks.”
Note that the distanciation here is about scale-freeness, but not pervasiveness. It might look the same but the example below makes it clear that the doubt is about the criteria used to establish that a network is scale-free.
“Although many real-world networks are thought to be scale-free, the evidence often remains inconclusive, primarily due to the developing awareness of more rigorous data analysis techniques.”Scale-free network
The statement above is about beliefs. It displaces self-evidence from pervasiveness (many real-world networks are scale-free) to the state of beliefs (“many real-world networks are thought to be scale-free“). It claims that some believe in the pervasiveness of scale-free networks. It is similar to the “historical interest” in preferential attachment.
Sometimes the distanciation is not on pervasiveness itself but on its explanation. Note how, in the citation below, pervasiveness is assumed but its explanation is challenged and contextualized as a claim from Barabási.
“It is hypothesized by some researchers such as Barabási that the prevalence of small world networks in biological systems may reflect an evolutionary advantage of such an architecture.”Small-world network
And distanciation is not even always present. Often times it is implied or explicitly mentioned in all its self-evidence. For instance:
“Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology”Social network
The claim to pervasiveness lies in the word“most”. Here pervasiveness is casually assumed. Similarly but about a statistical law, note the “many” in the sentence below:
“Zipf’s law […] refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution.”Zipf’s law
Naturally, claims to pervasiveness are stronger in the case of statistical laws because empirical pervasiveness is their defining characteristic. In such situations we often find explicit claims such as those:
“An empirical statistical law or (in popular terminology) a law of statistics represents a type of behaviour that has been found across a number of datasets and, indeed, across a range of types of data sets.”Empirical statistical laws
“Scientific laws summarize and explain a large collection of facts determined by experiment”Scientific law
The rhetoric of pervasiveness shows how important is the cultural aspect of network science. Pervasiveness is a notion, it is hard to establish as a solid fact. Some sources are sometimes mentioned in Wikipedia, for instance Mark Buchanan’s book Ubiquity : The New Science of Universal Patterns. But most of the time the claim to pervasiveness is presented on the mode of self-evidence. We can hypothesize that pervasiveness is indeed obvious, but there is a problem. Observing a power law requires an apparatus. If pervasiveness is obvious, it is only in a certain context and for certain people: I am certainly not myself in position to observe it, and you are probably not either. In Wikipedia we are often asked to believe in a form of self-evidence that is inaccessible to us. In that sense, pervasiveness is a belief. And by reading Wikipedia we can at least establish that this belief is so common that it has become self-evident itself: it is obvious that complex networks, preferential attachment and the power law are thought to be pervasive. Because if they were not, such claims would not be disseminated everywhere in our corpus of articles.
The power law article features multiple claims to pervasiveness. Describing the series provides insights on the role of such statements. It all starts with a double direct claim:
“The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes.”Power law
The power law is not only followed by many phenomena, it is also followed by them over many orders of magnitude. Once again, the point relies on self-evidence (argument from Wikipedia’s authority). Follows a sophisticated statement on universality and the status of law:
“Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them. The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems; see also universality above.”Power law
Let us break down these two sentences. First of all we have “certain general classes of mechanisms.” In other words, a large group of phenomena. Those have something in common: they generate power-law relations with ease. Stating the existence of this large group of phenomena is a claim to pervasiveness, but more specific than the usual. Not just many things produce power-law, but “certain general classes” of phenomena, supposedly defined by other characteristics. And like for preferential attachment, this pervasiveness is the source of “scientific interest.”
But the second sentence makes a stronger point. It explains why the empirical observation (and “demonstration”) of the power law matters: because it hints at underlying mechanisms. The sentence states that some “deep” underlying mechanisms can produce similar effect (the power law) in seemingly “unrelated” phenomena. The power law would be a signature of a deeper mechanism, hence tracking the power law could allow investigating an underlying reality. As stated, this argument is that of “universality” (see next section).
Immediately following, the article makes a very explicit direct claim, using the term “ubiquity”, and followed by statements of quantity (“many”) and an enumeration of 3 “notable” examples and 7 scientific fields:
“The ubiquity of power-law relations in physics is partly due to dimensional constraints, while in complex systems, power laws are often thought to be signatures of hierarchy or of specific stochastic processes. A few notable examples of power laws are Pareto’s law of income distribution, structural self-similarity of fractals, and scaling laws in biological systems. Research on the origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics, computer science, linguistics, geophysics, neuroscience, sociology, economics and more.”Power law
The article also features a list of 51 examples of the power law, that constitutes a substantive piece of evidence for the pervasiveness claims:
“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). Among them are:
Astronomy […4 examples follow]
Physics […17 examples follow]
Biology […6 examples follow]
Meteorology […1 example follows]
General Science […12 examples follow]
Mathematics […8 examples follow]
Economics […3 examples follow]”Power law
The list in itself is not a scientific evidence but the article cited as a source for the list is one, and the list situates where and for whom pervasiveness appears. Although some of these claims to power law were challenged, as we have seen, they were not challenged as heavy-tail distributions. In that sense, even if the claim to pervasiveness were to shift from power law to heavy-tail distributions, it would ultimately remain.
Of the different articles we studied, the article on the power law is the one where the claims to pervasiveness are the most grounded, and where their importance is the most explicit: they support universality.
In its article, the power law has three main properties: scale invariance, lack of well-defined average value, and universality. What is universality?
The argument of universality is complicated. As a starting point, we will look into the universality section in the power law article, and unpack the series of statements. We will discuss the argumentation by explaining the necessary concepts and contextualizing the claims. We will mostly refer to what other Wikipedia articles mention, but we will also draw on other sources for the sake of clarity.
Universality in the power law article
The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation.”
The crucial role is played by the “equivalence of power laws”. This equivalence is a mathematical property also named scale invariance, and is the defining characteristic of the power law. As we have seen, power laws of the same exponent are indeed considered equivalent, which is a remarkable and specific property.
The argument of universality starts with this equivalence having a “deeper origin”. The rest of the Universality section develops this idea, starting with an example.
“In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the critical exponents of the system.”Power law
This sentence involves the concept of phase transition. It refers to what happens when a system moves from a certain stable state to another one, often brutally. The typical example is ice melting into water: ice is a stable state, water is a stable state, but many unusual things happen in between, during the melting. The transition between the phases has its own specific behavior.
Phase transitions in thermodynamics is not just a random example, it is the prototypical case that relates the power law to universality. The study of such physical phenomena motivated the development of a sophisticated mathematical apparatus that grounds the claim that power laws have a “deeper origin”. By design, this theory applies to the case of thermodynamic phase transitions.
The argument involves the variation of “certain quantities” involved in the process. These quantities are special because their evolution follows a power law. In the prototypical case, the studied system’s thermal capacity for instance is such a quantity. The sentence only states that phase transitions are “associated with” these quantities’ power law, but we can clarify the epistemic status of this association. Indeed the physics of phase transitions is mature and well established, theoretically and experimentally. It turns out that the association is both empirical and theoretical. Firstly, that a number of quantities follow a power law during a phase transition is an empirical fact. Secondly, theory predicts that certain quantities necessarily follow a power law at the crucial moment of the phase transition. In that sense, the scientific consensus seems to be that phase transitions give rise to power laws.
We must pay attention to the direction of the association between phase transitions and the power law. In a nutshell, phase transitions cause power laws. Strictly speaking, it is not really a cause-consequence relationship, so my statement is a little oversimplified. But it features the most important aspect: the “association” is from phase transition to power law. Though the statement does not precise this aspect, power laws are somehow a consequence of phase transitions, according the scientific consensus. And as we will see later, this point matters.
What are the “critical exponents of the system”? During phase transition, the special quantities follow a power law, but not necessarily the same: their exponents differ. As a reminder, the power law refers to a family of mathematical functions of the same general form but with different parameters. The general equation is f(x)=a.x^(-k) and its parameters are the factor a (generally considered unimportant) and the exponent k (characterizing the specific form of the power law). Here k is called the “critical exponent” of the system. The term critical is derived from the concept of critical point which refers to the key moment of the phase transition.
Why would the critical exponents be specific to the system, and not determined by empirical conditions? As surprising as it sounds, the independence of critical exponents to empirical conditions is both an empirical fact and a theoretical prediction, as stated just next.
“Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach criticality—can be shown, via renormalization group theory, to share the same fundamental dynamics.”Power law
In substance, this states that a “same fundamental dynamics” (an underlying reality) explains why multiple quantities follow a power law with the same critical exponent across multiple empirical situations. The evidence of that underlying reality is supposed to be provided by renormalization group theory. What is it?
Renormalization group theory (RG theory) is a mathematical apparatus that can be mobilized in a situation of scale invariance, when a phenomenon appears the same across many scales. It “was initially devised in particle physics” (Renormalization group in Wikipedia) but extends to other branches, notably quantum physics. It is worth noting that this theory is a massive body of knowledge and has played a major role in modern science since the 50s. It has a high credibility.
RG theory allows to reformulate a problem (for instance a physical model) in terms of scaling, in terms of what happens when you look at a system at a larger scale. In simpler words, when you zoom out. This procedure at the core of the theory is called renormalization. It transforms equations that describe a phenomenon at a known scale into equations that describe it when the scale changes. For instance imagine a pile of sand. If you look at the number of grains of sands, it tells you that if you zoom out you see more grains. Conversely if you look at the size of the grains, it tells you they look smaller as you zoom out. So when you zoom out, certain quantities go up (number of grains…) and others go down (apparent size of grains…). Remark that this is true at all scales: it does not matter how much you are currently zooming, if you zoom out you will see more grains and they will be smaller. A third kind of quantity also exists, that neither goes up or down but stays the same. Such quantities are the scale invariant ones. A classic example is the size of the avalanches in the pile of sand. Avalanches come up in all sizes and at all scales, from a few grains to huge chunks of the pile. They are a scale invariant phenomenon. When a quantity is invariant through renormalization, we can solve certain equations and get new informations on that quantity, directly derived from its scale-invariance.
Intuitively, RG theory looks at what happens during scale changes, and from that tells us which quantities are relevant or not at larger scales. It predicts why certain quantities are scale invariant at the critical point, and thus follow a power law. More importantly, by observing the scaling behavior of these quantities at the critical point, it predicts that they cannot be fully independent. More precisely, it predicts a limited number of defining variables, of degrees of freedom. The exponents of the power laws of the special quantities are called “critical exponents” and all depend only on these few degrees of freedom. One way to put it is to observe that, at the critical point, the singular dynamics of the system constrains all the special quantities not only to follow power laws, but also to have specific critical exponents. According to RG theory these critical exponents do not depend on the specifics of the system but only of the dynamic at the critical point. Empirical observations back this interpretation insofar as multiple systems that differ completely on the micro level share the same scaling dynamic at the critical point are repeatedly observed to have the same set of critical exponents. In this theory, such pervasive set of critical exponents is named a “universality class”.
“For instance, the behavior of water and CO2 at their boiling points fall in the same universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by a small set of universality classes.”Power law
We can now understand this sentence. Universality is not mentioned in the mundane sense, but “universality classes” refer to the similar behavior of certain physical systems despite their different microphysic structures.
Of course, universality classes get their name from the pervasiveness of empirical observations. As we will see the pervasiveness of these sets of critical exponents across seemingly unrelated cases has puzzled physicist for a long time, which is one of the reasons why RG theory is considered a remarkable achievement. Note however that this pervasiveness is not absolute. Some empirical observations do not match theory for unknown reasons, and there is no general consensus on how many universality classes there are. The theory has validity conditions outside which it does not work. This is nothing unusual and I do not mention it to downplay the remarkable success of RG theory (notably in quantum physics) but to make it clear the universality classes are not as universal as their name suggests. They are pervasive, but not universal.
“Similar observations have been made, though not as comprehensively, for various self-organized critical systems, where the critical point of the system is an attractor. Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.”Power law
The same point again, essentially, with an important detail: for the first time it defines the concept of “universality” in itself. And so ends the section on universality in the power law article.
At the end of this sophisticated rhetoric, universality is the “sharing of dynamics” between “various […] critical systems”, hence the “fundamental dynamics” “associated with the emergence of power-law distributions”. Back to the original statement, universality is the “deeper origin”, the underlying reality giving rise to power laws. In addition, the highly respected renormalization group theory is convoked as a guarantee for the statement. Leveraging our previous explanations on RG theory to simplify the central claim, we can reduce it to this sentence:
Power laws reveal the underlying presence of a phase transition, a phenomenon known as universality.
Unfortunately, that statement is false.
Debunking the universality rhetoric
First of all we must clarify that universality is indeed the name physicists give to the empirical pervasiveness of sets of critical exponents (universality classes). The observation finds a theoretical explanation by RG theory: microscopic details do not matter at the critical point, so that only the scaling dynamic determines the system’s behavior. But the name refers to an empirical observation, not a theoretical claim. Universality is not a statement according to which RG theory proves that power laws are pervasive, universal, or the sign of a phase transition. Despite its pompous name, universality just refers to an observation. An intriguing one, an important one, but still a simple observation.
“In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system.”Universality (dynamical systems)
Moreover, power laws are simply not the sign of a phase transition. Though it is true that phase transitions give rise to power laws, the reverse is not. The presence of a phase transition implies that of a power law, but power laws imply phase transitions only in if you know for sure that no other factor can. Without any knowledge on that possibility, one cannot rule out the possibility that something else is generating the power law. So we cannot conclude that a power law implies a phase transition. The rationale of universality, as stated in the article on the power law, is bogus. But the situation is worse, because we actually know that a large number of unrelated situations can generate power laws. In that sense, it is largely established that power laws are not a sign of a phase transition.
As a reference for the “association” between phase transitions and the power law, I will use a recent source from outside Wikipedia. My rationale for bringing this source is that it provides both authority and clarity on that specific matter. It is reliable because it has been produced by the renowned Santa Fe Institute, a world class institution for the study of complexity. It brings clarity because it is intended as a pedagogical document, as a part of the Complexity Explorer online lessons. Also note that it is recent (March 2019). This 7 minutes 35 seconds video is available on YouTube and is titled: How many power laws are from phase transitions? It features professor Dave Feldman half-reluctantly explaining why he thinks almost no power laws observed in complex systems arise from phase transitions. The full text transcript is available as an appendix (but you should rather watch the video).
“I should mention that much more than some of the other videos, I’ll be giving some opinions, and less an accounting of mathematical or empirical facts. I think the position that I’m going to carve out is pretty much a standard one, within the study of complex systems, but there’s certainly some room for disagreement.
So, what fraction of the power laws that we observe in complex systems arise from phase transitions? So I think the answer is: almost none. That phase transitions likely are not an explanation for the vast majority of power laws we see in complex systems. […]
But phase transitions and power laws have often been closely linked, more so in the past, less so these days, and I want to offer some thoughts on why that may be. I think a lot of it has to do with the culture and habits of mind of physicists. […] Within physics, and I say this as a physicist, the theory of phase transitions […] has everything that physicists would get excited about. […]
So a lot of the claims that linked power laws and other areas of complex systems with phase transitions, were originated by physicists who are accustomed to associating power laws and phase transitions. […] But as we’ve seen in this unit, power laws also arise from many many other types of situations that have nothing to do with a phase transition. And this is something that many physicists, I think, weren’t aware of. And so, when they saw power law behavior, they were quick to say: “oh, this must indicate some sort of a phase transition, that the system is poised between two different states, a state of order, and disorder”. […]
So… to sum up, phase transitions most definitely give rise to power laws. Of that, there is no dispute. But there are many other mechanisms that give rise to power laws as well. So I think that the physicists’ understandable fascination with phase transitions and power laws has sometimes, maybe, extended a little bit too far into the field of complex systems. That phase transitions are beautiful physics, and a really impressing theory, but maybe not always so useful in the study of complex systems.”Prof. Dave Feldman, in this YouTube video
In substance, D. Feldman frames the universality issue as a misunderstanding. He states that the misunderstanding used to be widespread among physicists, but that nowadays, in the study of complex system, it has been mostly debunked. We have seen however that it remains present inside Wikipedia. We can hypothesize that there is a delay in knowledge dissemination from research to Wikipedia (the video was published recently, in March 2019). In any case, Wikipedia reproduces a rhetoric that used to be quite standard in network science. If the argument was fallacious from start, it is important to understand why it did not appear as such.
D. Feldman suggests two hypotheses to excuse the fallacy. I just want to remind first that this fallacy is not universality in itself, but the belief that all power laws are related to universality. In addition, it is also important to remark that, as he puts it, that theory “has everything that physicists would get excited about.” In this context, D. Feldman suggests:
- A methodological bias: “physicists [were] accustomed to associating power laws and phase transitions.”
- A weakness in the “culture of physicists, that maybe suggests why there is a little bit of hype”: “physicists tend to not learn much statistics.”
It seems that the physicists studying complexity jumped to conclusions and convoked universality on the basis of superficial signs. The bridge was the power law, and a weak understanding of statistics allowed the misunderstanding of universality to stay. But what was the reason for the “hype”, the motivation for jumping to conclusions? What was universality doing for network science?
Universality was a structuralist argument. It provided a structure to complex networks, it made them comparable to a scientific law. A.-L. Barabási, famous champion of network sciences, has continuously pursued a structuralist agenda, culminating with the title of his last book: The Formula: The Universal Laws of Success.
The repeated mention of a “deeper connection”, “same fundamental dynamics” or other “underlying mechanism” is a signature of structuralism. As theorist Alison Assiter is quoted in the Wikipedia page on structuralism, “structures are the “real things” that lie beneath the surface or the appearance of meaning.” The rhetoric of pervasiveness and universality tries to confer power law and scale-free networks a status close to those of a scientific law. It sees universality as a structure, and formulates it as such. In that it differs from thermodynamics, from where the concept is imported, which has an empiricist formulation of universality.
Empiricist and structuralist versions of universality
In the articles related to the study of phase transitions, universality is constantly presented as an empirical fact, a repeated observation. The structuralist understanding of universality is an invention of scholars studying complexity (network science included). Physicists who study phase transitions in thermodynamics have an empiricist understanding of universality.
“In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system.”Universality (dynamical systems)
“It is a remarkable fact that phase transitions arising in different systems often possess the same set of critical exponents. This phenomenon is known as universality.”Phase transition
“This intriguing phenomenon, called universality, is explained, qualitatively and also quantitatively, by the renormalization group.”Critical phenomena
Even when universality is presented as explained by theory, it remains framed as an empirical phenomenon, for instance in the article on RG theory:
“This coincidence of critical exponents for ostensibly quite different physical systems is called universality − and is now successfully explained by the RG”Renormalization group
That being said, the empirical grounding is not incompatible with a more mundane notion of universality, in the structuralist sense of a mathematical or physical law. However these two conceptions of universality (empiricist and structuralist) are never confused in the field of physics. Take for instance the following passage:
“Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features.”Critical exponent
The word “universal” is used here in a structuralist sense, but then it is framed as a conjecture and not an empirical fact. We must remark however that the empiricist framing is always at risk to slip to a structuralist framing, at least because they share the exact same words.
The article on scale invariance presents an interesting mix of empiricist and structuralist statements. The following statement is for instance aligned with those we have just seen:
“Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition.”Scale invariance
In the first sentence of the article however, we find the following statement, whose exact signification is unclear:
“scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, thus represent a universality.”Scale invariance
I interpret this statement as structuralist. Note however that the rest of the article adopts the empiricist perspective, notably in a whole section dedicated to universality.
In the article on percolation, we find an explicitly structuralist version of universality, presented as a “principle”:
“The universality principle states that the numerical value of p_c is determined by the local structure of the graph”Percolation theory
The other explicit structuralist statements accumulate in the article on the power law, the conceptual bridge between network science and statistics. We have seen those already. There are however implicit statements of universality, for instance:
“Six degrees of separation is the idea that all living things and everything else in the world are six or fewer steps away from each other”Six degrees of separation
The idea of six degrees of separation is powerful. But why is it necessarily about “all living things and everything else in the world?” Could it not have a domain of validity? Would the idea be less powerful? I hypothesize that the idea is formulated this way because historically, it came from a structuralist intuition. But without a strong theory to back it, universality falls back to just pervasiveness.
We have seen countless claims to pervasiveness, and we can now understand them as a toned-down version of universality. Those pose as important empirical fact, but as we have seen about the power law, without a theory to unify the multiple empirical cases, those might as well be unrelated. Despite what the structuralist rhetoric we studied suggests, pervasiveness cannot be considered a piece of evidence for an underlying reality. Pervasiveness does not require universality. But universality of course requires pervasiveness, and even got its name from it.
“Universality gets its name because it is seen in a large variety of physical systems.”
Universality (dynamical systems)
Text content of the video Fractals and Scaling: How many power laws are from phase transitions? produced for the Santa Fe Institute’s Complexity Explorer lessons.
“So we’ve seen that phase transitions give rise to power laws. At the critical point, at the point where the system goes from one phase to another, where the transition occurs, many quantities of interest are distributed according to a power law. And that’s not the case on either side of that transition. So power laws and phase transitions are closely linked. The question then is: what fraction of the power laws we observe, more generally in the study of complex systems, can be said to be due to a phase transition-like behavior of some sort? So that’s a question that I want to address from a number of different angles on this video. And I should mention that much more than some of the other videos, I’ll be giving some opinions, and less an accounting of mathematical or empirical facts. I think the position that I’m going to carve out is pretty much a standard one, within the study of complex systems, but there’s certainly some room for disagreement.
So, what fraction of the power laws that we observe in complex systems arise from phase transitions? So I think the answer is: almost none, that phase transitions likely are not an explanation for the vast majority of power laws we see in complex systems.
So what do I say that? Well, a phase transition is a very unusual state of affairs. The phase transition it’s a critical point, a critical temperature, a critical transition probability, only one out of a whole number of different things. So, it’s very unlikely that we sort of encounter that by chance. In the physical world, things are very rarely poised right at the critical point, right between liquid and solid, or magnet and non-magnet. Things tend to be solid or liquid or gas, and not poised right at that point in-between the two.
So because phase transitions by definition occur at a very narrow, at a very particular point, a particular set of parameters, it seems unlikely that that could be a generic explanation for power laws that we see quite commonly throughout a whole range of social and biological and technological phenomena.
But phase transitions and power laws have often been closely linked, more so in the past, less so these days, and I want to offer some thoughts on why that may be. I think a lot of it has to do with the culture and habits of mind of physicists. So let me explain what I mean by this.
So, within physics, and I say this as a physicist, the theory of phase transitions, also named as the theory of critical phenomena, is a fantastic theory. It has everything that physicists would get excited about. It explains a broad range of phenomena in fairly simple terms, right, so this is the idea of universality, that many different transitions, even in systems that seem very different, are characterized by the same exponents. So it collects a lot phenomena together, into a similar quantitative framework. There is some nontrivial mathematics behind it, renormalization group explaining why some of this is so… So it’s the sort of things that the physicists, most physicists, just love. It’s a great theory, it deservedly got a lot of attention in… the 70s and the 80s I would say. So the theory of critical phenomena is a significant accomplishment within the study of physics, both theoretical and experimental.
So a lot of the claims that linked power laws and other areas of complex systems, with phase transitions, were originated by physicists who are accustomed to associating power laws and phase transitions. And again, phase transitions are seen as a very interesting thing in physics, it’s an unusual state of affairs, difficult to explain but then, it can be understood with some mathematics.
So there is an interest, a fascination one is drawn towards phase transitions, and phase transitions give rise to power laws, and so in the mind of many physicists the two are closely linked. And of course that’s right, it is indeed the case that power laws arise from phase transitions. But as we’ve seen in this unit, power laws also arise from many many other types of situations that have nothing to do with a phase transition. And this is something that many physicists, I think, weren’t aware of. And so, when they saw power law behavior, they were quick to say: “oh, this must indicate some sort of a phase transition, that the system is poised between two different states, a state of order, and disorder”. So I think physicists first tended to be drawn towards power laws in the first place, because power laws are associated with phase transitions which are interesting, and then seeing power laws and immediately associate them with phase transitions and critical phenomena.
There is one other reason having to do with the culture of physicists, that maybe suggest why there is a little bit of hype, or maybe alleged power laws that turned out upon reexamination not be that well-described by a power law. And that is that physicists tend to not learn much statistics. It is rarely a degree requirement for physicists. My background is in statistical physics but I was never required to take a statistics class, and insofar as we do learn things like statistics, it is more about error propagation in experiments, and less about testing hypotheses and model verification.
So those are skills that are taught more often in the sciences, I think in biology and economics, and actually less in physics, and of course it’s taught in statistics all the time. So all that is to say is that physicists didn’t necessarily know about some of the more advanced or modern data analysis techniques that I described in unit 5, leading some to claim that power laws were present in light of, maybe, not such convincing evidence.
So… to sum up, phase transitions most definitely give rise to power laws. Of that, there is no dispute. But there are many other mechanisms that give rise to power laws as well. So I think that the physicists’ understandable fascination with phase transitions and power laws has sometimes, maybe, extended a little bit too far into the field of complex systems. That phase transitions are beautiful physics, and a really impressing theory, but maybe not always so useful in the study of complex systems.”