Every human being asks the question “why,” though at different levels, to explain the unexplained. It follows a pattern of questions, like “Why did the financial crisis in the U.S. happen in August 2008?” “Why did the space shuttle Challenger explode?” “Why did the terrorists commit the September 11 attacks?” In statistical terms, such unexpected events are named “outliers,” however, they are part of the system and among the components constituting the overall system’s complex behavior. Thus, they need to be part of the explanation in order for the explanation to be complete. We are naturally tempted to come up with universal explanations of the complexity behind these major events so that we can be ready when a similar thing happens again. Though simple mathematical equations or relationships relate to us better and provide a universal explanation, they are typically practical only when the outliers are excluded from the system behavior. Statistics help us greatly in quantifying and characterizing the outliers, especially in the form of probabilistic expressions, such as “there is a 30% chance of a hurricane next week.”

Understanding the complexity around us involves the development of a model that is simple enough for us to comprehend but yet universal enough to capture most of the dynamics of the complexity. The simpler and the more universal the model, the more powerful it is. The universality of a model, however, is hindered by the potential inability to capture something unexpected. The tradeoff between simplicity and universality exists in all modeling efforts; and the models finding the delicate balance in this tradeoff are the most effective ones. A simple mathematical relationship known as “the power law” has been used extensively to characterize and model various natural and social phenomena.

What is the Power Law?

The “power law” does not refer to a misconception that “whoever has power will rule,” but rather it refers to a particular way of characterizing dependency between two quantities. When the number or frequency of an object or event varies as a power of some attribute of that object (e.g., its size), the number or frequency is said to follow a power law. In more general terms, there exists a power law relationship between x and y if y is growing or reducing polynomially when x is growing linearly (y _͌ x^–α). Mathematically speaking, this means that the relationship between y and x is mainly characterized by the exponent -a. An exponent is simply shorthand for multiplying that number of identical factors. So, 4³ is the same as 4x4x4; that is three identical factors of 4. As shown in Figure 1, a quantity with an exponent has three components: the base, the exponent, and the coefficient. So, for 4³, the base is 4, the exponent is 3, and the coefficient is an implicit 1.

Figure 1: Description of an exponent in a power law relationship.

Figure 2: Sample power law relationships between x and y, where y = x^–α.

The power law relationships are traditionally expressed with a negative exponent, which simply means the inverse of the quantity. That is, y _͌ x^–α is equivalent to y _͌ 1/x^α. For example, when a is 2, y will reduce from 1/4 (i.e. 0.25) to 1/9 (i.e. ~0.11) if x grows from 2 to 3. Likewise, when a is 0.5, y will reduce from 1/2 (i.e. 0.5) to 1/3 (i.e. 0.33) if x grows from 4 to 9. For those who enjoy graphs, Figure 2 illustrates these mathematical relationships in linear and logarithmic scales.

Power law on different scales: Atoms to planets

To start with, gravitation, acoustics, electrostatics, and light and electromagnetic radiation, all exhibit a form of power law in that physical quantity or strength that is inversely proportional to the square of the distance, which corresponds to a power exponent of 2. [1] Gravitational force between two particles, the electrostatic force of attraction between two electrically charged particles, the intensity of sound signals coming from a source, and finally the intensity of light or electromagnetic field coming from a source all follow a power law with respect to the distance.

What makes the power law relationships more interesting is their independence from scale or size of the measures being related to each other. This is why we sometimes call power law relationships as “scale-free” relationships or “scale-invariance.” For example, the gravitational force between two spherical particles decays with a power exponent of 2 regardless of the sizes of the particles though the actual force is certainly dependent on the particle sizes. So, the particles can be at nano scales (e.g. a group of atoms) or macro scales (e.g. a planet), but the relationship stays the same!

Power law in frequency: Wealth, terror, and earthquakes

A common usage of power law relationships has been to model and understand frequency of a varying measure. A power law typically very well represents the distribution of wealth in a society. [2] According to a recent study, the distribution of wealth in China during the years 2003–2005 follows a power law with an exponent ranging from 1.758 to 2.285. If we consider an average exponent of 2 for Chinese wealth distribution, this means that if there are 1 million Chinese people who owned $1000 there were 1000 that owned $1M. Thus, the power law essentially expresses how skewed the distribution of a frequency is (see Figure 2). The larger the power exponent, the more skewed the distribution. In this case, a larger power exponent means a more imbalanced wealth distribution while a power exponent of 1 refers to an evenly distributed wealth.

Many other social patterns exhibit power law. A recent study showed that it exists even in terror events! The number of casualties per insurgent event and the number of insurgent events per day follow a power law. [3] Historical data for the last two centuries show further that the number of casualties per war or a terror attack follows a power law distribution. What is even more interesting is that the number of casualties and the number of attacks within an insurgent conflict both follow power law. That is, when only a particular conflict between two countries or ethnic groups is considered, the number of casualties per insurgent event and the number of insurgent events per day follow the power law. This suggests a “self-similar” pattern. Likewise, traffic measurements for many systems show power law distributions of size. For instance, if one observes the data traffic on an Internet connection and counts the number of bytes being transmitted per hour over that connection, a power law distribution of the count of bytes will emerge. Further, if this counting is done per minute instead of per hour, a similar distribution will still emerge – again showing a self-similar pattern. [4]

The power law has been observed in several natural phenomena as well. The frequency of earthquake magnitudes follows a power law. [5] This refers to the intuitive notion that the number of earthquakes with small magnitudes (which humans do not even feel) is much larger than the number of earthquakes with large magnitudes, (which can kill many humans). Small earthquakes are the norm while large ones the outliers. However, without the outliers, there is no power law distribution! Thus, the power law distribution of a quantity comes with an interesting observation: If a quantity is indeed following a power law distribution, then the likelihood of an outlier event increases as the time goes by without an outlier event. This is why geoscientists would make comments like “The region X is due for a major earthquake!” indicating that the region X has not been receiving a major earthquake (i.e. an outlier) for several years. The issue, though, is determining the threshold for an outlier is typically ambiguous and may require many years of measurements and data, which may be impractical.

Power law in growth: Rich get richer

Growth of systems also exhibit power law in various ways. Social growth follows power law due to the well-known “rich get richer” rule, which refers to the intuition that “important” people in the society attract more of the attention of newcomers. This dynamic situation is observed, for example, in the growth of the Internet. Several studies [6] showed that the connections between Internet Service Providers (ISPs) (e.g., AOL, Yahoo!, AT&T, Sprint) follow a power law distribution in that the number of connections per ISP (which shows how well an ISP is connected to the rest of the world) is represented by power law. In other words, there are few ISPs with many connections to other ISPs while most ISPs have a few connections to the others. This is believed to be due to the “rich get richer” rule since an existing ISP with many connections is more likely to gain the business of a new ISP who is joining to the Internet. So, it is somewhat an economic pattern too.

If economics (or the money) is taken out of the picture, social growth still exhibits power law. Online social networks such as Facebook, LinkedIn, and Flickr are clearly following a power law distribution. It is found that the power exponents are in the range of 2.5 to 3.7, indicating a highly imbalanced social growth pattern where few people are at the “center” of the social network with hundreds or thousands of friends, and many people have only one or two friends. [7] Again, the typical explanation for this growth pattern has been the “rich get richer” rule, but “richness” refers to the number of existing friends in this context rather than money.

Physical growth shows power law too in many ways. For instance, roughness of a growing surface as time goes by follows a power law distribution with an exponent ranging between 0 and 1 where an exponent of 0 refers to a smooth growth and 1 refers to a stiff growth. The surface roughness is measured by the variance of heights of surface locations. [8]

Does it really exist? Why does it exist?

Verifying existence of a power law distribution is not easy and requires enough number of samples to show the “tail” of the distribution. The tail of the distribution refers to the samples with large (or rare) values. For example, for the power law distributions in Figure 2, the portion of the distribution when x is greater than 10 (i.e. x>10) roughly corresponds to the “tail.” The tail corresponds to the rare samples. Though statistical theory calls those rare samples “outliers,” the distribution will not be a power law distribution without them. They are strictly parts of pieces that constitute a power law relationship, and observing them typically requires long periods or large numbers of measurements. Due to this difficulty, the existence of the power law is questioned for many real systems. Most of the time, claims of the existence of the power law typically come with an error factor indicating the confidence of the claim. The bottom-line is to observe trends in the samples and thus establish sufficient confidence (e.g., more than 95%) that the power law distribution does exist in the samples.

For those systems with clear exhibition of power law, the root causes of it have been of high interest. The “rich get richer” rule is intuitively one of the root causes, and it is intuitively a natural dynamic to get attracted by a rich member rather than a poor one. Growth certainly naturally follows the “rich get richer” rule, but we have system components slowing their growth, flattening, and then deteriorating. So, not everything is growing, and actually, we have as many things deteriorating as growing. For instance, participants join or leave the Internet or the social networks, and likewise, people join (i.e. birth) or leave (i.e. death) society. How does the power law stay in such systems then?

Due to the “rich get richer” intuition, the power law is considered to be the signature of “self-organization.” The fact that so many natural or synthetic systems are exhibiting this signature deserves the question: “Is it really self-organization?” Maintaining a global power law distribution for a system requires either (i) every member joining or leaving the system according to the “rich get richer” rule and having global knowledge of the whole system or (ii) somebody who knows everything about the system and gives explicit direct orders to each member when they are joining or leaving. Which one is more likely?

Murat Yuksel is an Assistant Professor at the CSE Department of The University of Nevada – Reno (UNR), Reno, NV.

References

[1] Wikipedia, “Inverse-square law,” http://en.wikipedia.org/wiki/Inverse-square_law

[2] M. A. Santos, R. Coelho, G. Hegyi, Z. Néda, and J. Ramasco. 2007. “Wealth distribution in modern and medieval societies,” The European Physical Journal, Volume 143, Number 1, pages 81-85.

[3] J. C. Bohorquez, S. Gourley, A. R. Dixon, M. Spagat, and N. F. Johnson. 2009. “Common ecology quantifies human insurgency,” Nature, Volume 462, December, pages 911-914.

[4] T. Karagiannis, M. Molle, and M. Faloutsos. 2004. “Long-Range Dependence: Ten Years of Internet Traffic Modeling,” IEEE Internet Computing, September/October, pages 57-64.

[5] T. Lay and T. Wallace. 1995. Modern Global Seismology, Academic Press, San Diego, CA.

[6] M. Faloutsos, P. Faloutsos, and C. Faloutsos. 1999. “On power-law relationships of the Internet topology,” ACM Computer Communication Review, Volume 29, Issue 4.

[7] R. Kumar, J. Novak, and A. Tomkins. 2006. “Structure and evolution of online social networks,” Proceedings of ACM SIGKDD, pages 611-617.

[8] A. L. Barabasi and H. E. Stanley. 1995. Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, England.

Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. He argued that an object in motion can never pass from one position to another, because between the two there is always an “infinity” of other positions, however close, that must be successively traversed in the course of the motion, and this “infinity” can never be exhausted. David Hilbert is a German mathematician who postulated a hypothetical hotel with “countably infinitely” many rooms, all of which are occupied. Since the hotel has “infinitely” many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, and so on, and fit a newcomer into room 1. By repeating this procedure, it may be argued that it is possible to make room for any finite number of new guests, although every room of this hotel initially contains a guest.

The following two quotations from two contemporary authors may provide more substance about the nature of the problem:

On the other hand, involvement with the infinite brings with it a huge range of difficulties. In particular, there are the many puzzles and paradoxes that have been outlined in the pages of this book. Moreover, there are the many quite fundamental problems that arise for such apparently simple notions as counting, adding, maximizing, and so forth. Because we are so firmly wedded to limit notions-“best,” “first,” “greatest,” “maximum,” and so forth-that do not sit easily with the infinite, it is very hard to see how we can make our peace with the infinite.

The infinite has always been a slippery concept. Even the commonly accepted mathematical view, developed by Georg Cantor, may not have truly placed infinity on a rigorous foundation.

In the present article, we attempt to summarize Rene Guenon’s (1886-1951) alternative way of thinking on the idea of infinite from the perspective of the traditional metaphysical science. Much more detailed presentation of his perspective can be found in his (1886-1951) valuable study The Metaphysical Principles of the Infinitesimal Calculus from which we will extensively quote here.

Guenon considered mathematics as providing a particularly proper symbolism for the expression of metaphysical truths to the extent that they are expressible. However, he states that “in order for this to be so it is above all necessary that these sciences be rid of the various errors and confusions that have been introduced by the false views of the moderns.” To follow Guenon’s paradigm it is necessary to start with the metaphysical notion of the universal-All which comprehends all possibilities, the non-manifested as well as the manifested Universe, that is, the cosmos. The universal All leaves outside itself only the impossible that is a pure nothing. A determination is to define a certain domain of possibilities in relation to all the rest which is expressed by Spinoza (1632-1677) as omnis determinatio negatio est (all determination is a negation). The first of all determinations is Being itself. “Number is only a mode of quantity, and quantity itself only a category or special mode of being, not coextensive with it, or more precisely still, quantity is only a condition proper to one certain state of existence in the totality of universal existence.” Number, space, and time are all determined conditions.

The Infinite, understood in its true, metaphysical sense, has no limits since its opposite, finite is synonymous with limited. Therefore, according to Guenon,

… one cannot correctly apply this term to anything other than that which has absolutely no limits, that is to say the universal All. Furthermore, there can obviously be only one Infinite, for two supposedly distinct infinities would limit and therefore inevitably exclude one another.

He further states, “The Infinite, in its true sense, can have neither opposite nor complementarity.” The scholastic distinction between “the infinite in a certain respect” and “the absolute infinite” cannot be accepted. If a thing is not limited in a certain sense or in a certain respect than one can legitimately conclude that it is limited in no way at all, and since a determined thing does not include every possibility, as such it can only be finite.

Given any number, one can form the next by adding a unit gives the sequence of numbers to us. Therefore, we cannot actually reach its limits. However, the impossibility of reaching the limits of certain things in the manifested Universe should not cause the illusion that these determined things have no limits at all. In order to replace the false notion of “determined infinite,” Guenon introduces, the idea of the indefinite, which is precisely the idea of a development of possibilities the limits of which we cannot actually reach; and this is why we (Guenon) regard this distinction between the Infinite and the indefinite as fundamental to all questions in which the so-called mathematical infinite appears.

According to Descartes (1596-1650), the indefinite is that of which we do not perceive the limits, and which in reality could be infinite. On the contrary, Guenon affirms that

(T)he indefinite cannot be infinite because it always implies a certain determination, whether it is a question of extension, duration, divisibility, or some other possibility; in a word, whatever the indefinite may be, and according to whatever aspect it is considered, it is still of the finite and can only be of the finite.

The idea of an “infinite number” understood as “the greatest of all numbers,” or “the number of all numbers” is contradictory. The impossibility of an “infinite number” can be established by various arguments:

(T)o every whole number (integer) there corresponds another number equal to its double, such that one can make the two sequences correspond term by term, with the result that the number of terms must be the same in both; but there are obviously twice as many whole numbers as there are even, since even numbers alternate by twos in the sequence of whole numbers; one thus ends up with a manifested contradiction.

Guenon insists that number, despite its indefinitude, is by no means applicable to all that exists and the multitude of all numbers cannot constitute a number, which, moreover, is finally only an application of the incontestable truth that what limits a certain order of possibilities must necessarily be beyond and outside that which it limits.

On the other hand, the idea of multitude, contrary to that of number, is applicable to all that exists which allows one to speak of the multitude of divine attributes for example, or again of the multitude of angels, that is, of beings belonging to states that are not subject to quantity, where, consequently, there can be no question of number.

Number itself can also be regarded as a species of multitude, but on the added condition that it be a “multitude measured by the unit” according to the expression of Saint Thomas Aquinas (1225-1274).

The term “indefinite” consists of something unfinished. The “non-measured” is that which has not yet been defined, which is only incompletely realized within manifestation. The multitude of all numbers is “innumerable” or “non-measured,” which is not to say they are infinite, but merely that they are indefinite.

Guenon calls whole number as true number or pure number. He accepts that the numbers other than whole numbers can be considered as the extensions or generalizations of the idea of number. However, he adds that these extensions are also distortions. According to Guenon, numerical quantity has a discontinuous character, whereas spatial or temporal magnitudes, for example, are continuous quantities. “Between these two modes of quantity is a difference of nature such that a correspondence between the two cannot be perfectly established.” He distinguishes the arithmetical unit from the “units of measurement,” which are magnitudes of another sort than number, notably geometric magnitudes. He defines a continuous quantity as an extension-however small it might be that will always remain indefinitely divisible.

Guenon is against atomism, which necessarily implies the discontinuity of all things. He argues extension cannot be composed of indivisible elements, for these elements would have to be extensionless to be truly indivisible, and a sum of elements with no extension can no more constitute an extension than a sum of zeros can constitute a number, that is why points are not the elements or parts of a line; the true linear elements are always distances between points, which latter are only their extremities. Points multiplied by any quantity at all can never produce length, since, rigorously speaking, they are null with respect to length; the true elements of a magnitude must always be of the same nature as the magnitude, although incomparably less: this leaves no room for indivisibles.

Further,

The point, which, being indivisible, is by that very fact without extension, that is, spatially null, but which, as we (Guenon) have explained elsewhere, is nonetheless the very principle of all extension.

For Guenon, Zeno of Elea’s arguments are against atomism and indeed, they prove that without continuity there would be no possible motion.

It is this very conception of motion that is in error, for it amounts in short to regarding the continuous as if it were composed of points, or of final, indivisible elements, like the notion according to which bodies are composed of atoms; and this would amount to saying that in reality there is no continuity, for whether it is a question of points or atoms, these final elements can only be discontinuous.

And, “The possibility of motion presupposes the union, or rather the combination, of both temporal and spatial continuity.”

We consider Guenon as an important and prominent example of thinkers who tried to remind people of the traditional metaphysical ideas. This metaphysical perspective does not share the modern tendency to attribute more importance to the practical applications of science than to science itself. This perspective attempts to link science back to principles of a higher order so that a particular science can be used as a support for elevating oneself to a higher knowledge.

Let me give you some everyday examples. For instance, most customer complaints (eighty percent) stem from a very few reasons (twenty percent). Eighty percent of problems in a company are caused by twenty percent of the employees. This idea is often applied to data such as sales figures: twenty percent of clients are responsible for eighty percent of sales volume. Eighty percent of our expenditure, stems from twenty percent of the items we buy. If we can differentiate cheap items from expensive items, then, we can develop ways to save on our shopping. We need to realize that we make eighty percent of our phone calls to the same twenty percent of our acquaintances. If we look at the clothes we wear, we realize that eighty percent of the time we wear the same twenty percent of them. We can also use this principle to plan our daily schedule and to order things according to their importance. For example, we may calculate the length of the time to do something and determine the importance of the thing we are to do. We calculate the percentages according to all the things waiting to be done. Later, with twenty percent of our time, we put our attention and energy into the things which carry eighty percent of the importance. We leave the things with twenty percent importance to later because those things require eighty percent of our time. What needs to be observed here is that quality is preferred over quantity. Some of us postpone tasks which can be done in a short time, due to the belief that we can somehow do it later. According to the Pareto Principle, it is not a good idea to delay things which are important in terms of quality even if they take a bit of our time to carry out.

Bediüzzaman used this rule very intelligently in his works in terms ofat explaining the wisdom behind seeming realities. manay-Ii harfi. For example, in the answer to the second question under the Twelfth Letter, Bediüzzaman uses the Pareto Principle persuasively as an answer to the following question: “As sending Prophets has caused many or even most people to become unbelievers because of Satan’s seduction, how can you say that creating evil things and acts is good, that raising Prophets is a mercy for humanity?”

In this answer, Bediüzzaman emphasizes the importance of quality over quantity. According to him, quantity has no importance in relation to quality. Moreover, Bediüzzaman shows that it is not an evil to lose atheists and hypocrites, who are many but less important in terms of quality, when you compare them with prophets, saints, and the righteous who are few, in terms of quantity. It is interesting that he uses a twenty percent to eighty percent ratio in each of his two examples:

“As quality is always far more important than quantity, we should consider only qualitative values in making our judgment. To cite an example: 100 date-stones are worth only 100 cents until they are planted and grow into palm trees. But if only 20 grow into trees and the remaining 80 rot because of over-watering, how can you say it is an evil to plant and water them? Everyone would agree that it is wholly good to have 20 trees at the expense of 80 date-stones, since 20 trees will give 20,000 date-stones.

Again, 100 peacock eggs are worth maybe 500 cents. But if she sits on the eggs and only 20 hatch, who can say it is an evil that 80 eggs were spoiled in return for 20 peacocks? On the contrary, it is wholly good to have 20 peacocks at the expense of 80 eggs, because the 20 peacocks will be worth far more than the eggs and will lay more eggs.”

In another example, Bediüzzaman explains why he remained distant from politics in the Thirteenth Letter as a response to the third question:

“…We are travelers in this world. Basing myself on the Qur’an’s light, I say that humanity has reached a marsh in this century. Whole caravans of humanity are trying, with great difficulty, to advance in this putrid marsh. A small minority follow a safe way and some have extricated themselves, but the majority continues to flail around in the dark. Although 20 percent of this majority seems quite happy with this struggle, mistaking its dirt and filth for musk and ambergris, whereas the other 80 percent knows that it is in a filthy marsh but cannot see the safe path (leading them out). We must bring that majority out of the marsh. To do so, we must use a mace to knock the 20 percent back to its senses or provide the 80 percent with a light to see a way to safety. I see that most people hold maces, but almost no one gives light to the helpless 80 percent. If some still have light, they are not trusted because they also carry maces. People are afraid of being beaten after being drawn to the light. Besides, the light may be extinguished

if the mace is broken.”

In this example, Bediüzzaman states that the first thing that needs to be done for those who have deviated from the right path is to show them the Qur’anic truths instead of helping them through politics which is associated with hitting someone on the head. Again it is interesting to see that Bediüzzaman used the Pareto Principle to explain the example. Bediüzzaman considers human beings to be walking in a dark swamp. Moreover, he believes that the priority is to enlighten most of the people’s road (eighty percent) with a small (twenty percent) effort, rather than to help a small number of people (twenty percent) with a large amount of political power (eighty percent of effort).

In the Twenty-Eighth Letter’s seventh matter, Bediüzzaman emphasizes that twenty percent of scholars surpasses the other eighty percent in terms of quality while he is examining how strong truths seem weak in the hands of weak people: “Eighty percent of mankind are not investigative scholars who can penetrate to reality, recognize reality as reality and accept it as such. They rather accept matters by way of imitation, which they hear from acceptable and reliable people, in consequence of their good opinions of them.”

Therefore, it is possible to say that Bediüzzaman, an investigative observer, found the Pareto Principle without using any contemporary methods such as surveys. Besides being known as an eminent Islamic Scholar and Saint of Islam admired by most of the people, Bediüzzaman use of mathematics, physics, astronomy, and sociology in the Risale-i Nur show us how to contemplate the universe using modern science.

Persuasion through Probability in the Risale-i Nur

Bediüzzaman often uses mathematical logic and probability in his works which were written to save the faith from evil (thought). For instance, under the section on the stratagems of Satan in the Twenty-Ninth Letter, Bediüzzaman states how harmful it can be if you use the feeling of fear against its purpose of creation. He gives an example of how he persuaded a person, who had to fear of drowning, to embark on a boat willingly without any fear through giving an excellent example of probability:

An important man (may God’s mercy be upon him) was afraid to travel by boat. One evening, we went to Galata bridge to take the ferry to Eyup.

He did not want to get on, saying that he feared he would drown. When I asked him how many boats were in the Golden Horn, he replied that there might be as many as one thousand. When I asked him how many boats sank each year, he replied usually one or two, and sometimes none.

I made this analogy: “Since a year has 365 days, your chance of drowning is 1:365,000. Why does such a small chance scare you?” I asked: “How much longer do you expect to live?” He answered: “Maybe 10 years; I am old already.” I contunied: “As there are 3,650 days in 10 years, your chance of dying today is 1:3,650. But since we do not know when we will die, you could die at any time. So repent and weep! Write your last will and testament!”

Seeing the truth in my words, he got on the boat even though trembling. On the boat, I told him: “God Almighty placed fear in our nature so that we might preserve our life, not ruin it. He did not give us fear to make life an unbearable burden full of pain and torment. If there is a risk of 1:2 or 1:3 or 1:4, or at most 1:5 or 6, it may be permissible and tolerable to fear and avoid the risk. But to fear a chance of 1:20, 1:30, or 1:40 is groundless suspicion, a sort of paranoia that changes life into a torment.”