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The power of the power law

Can we foreshadow wars and other seminal events?

“Where, then, does this order come from, this teaming life I see from my window: urgent spider making her living with her pre-nylon web, coyote crafty across the ridgetop, mighty Rio Grande aswarm with no-see-ems (an invisible insect peculiar to early evenings)?”

- Stuart Kauffman (1939- )

“The wars of the future will be fought by computer technicians and by lawyers and high-altitude specialists, and that may mean war will be increasingly abstract, hard to think about and hard to control.”

- Michael Ignatieff (1947-)

It only takes a little thinking to realize that there are patterns all around us, from the largest things to our fine-grained behaviors. There are regularities in the cosmos (the spiral shape of many galaxies; the way planets revolve constantly and uniformly around their sun, for example), and nature (the consistent features in swirling seashells, and the structure of flowers, trees and rainbows). We can see recurring structures in our behavior (our propensity to congregate together in communities, no matter what part of the world. People cohabit in many family structures, even in the modern world where families are blended, or different from those classic models). Reflect also on the deeply etched patterning of the lifecourse as we age: the constants in the human trajectory through infancy, childhood, adolescence, adulthood, old age, frailty and death. And consider human eating arrangements (basic archetype: three times a day).

It is not too extravagant to claim that sociologically and psychologically, we are primed for pre-existing in patterned structures. It’s even an evocative argument to maintain that the Universe is at some deep level infused with a property – to arrange itself morphologically, into order. That’s what Kauffman has contended: that structure is somehow preordained and we are “at home in the Universe”. If this is so, it’s an underlying compulsion to configure and quite likely a basic of the Universe, life on earth, and your own life cycle.


One model showing how behavior is strongly patterned is the normal distribution. Applied to human affairs, we can explain how people’s height, IQ, job satisfaction, reading ability, weight or blood pressure has few cases at one end of the spectrum, with low levels of the attribute, correspondingly few cases at the other end, with lots of it, and most of us in the bulge in the middle, with medium amounts of the attribute.

This graph shows the range and frequency of cholesterol levels. The clinical measurement lies on the horizontal axis (cholesterol scores), and the frequency of cases is described by the height of the vertical axis. The bell curve emerges when we populate the graph with data, showing the way the cases are distributed. A “normal” distribution is bell-shaped, represented by that hump in the middle, with cases tapering off either side. The “normal” in the normal distribution means that is what we can expect, usually.


The power law is another such model. It also occurs all around us, but is not so common as the normal distribution. It has become a recurring pattern found in things as diverse as how frequently websites are visited, the intensity of earthquakes, or how often words are used in a language.

Take words in English. There are many instances of “the”, “to, and “a”, but few uses of rare words like “lollygag”, “duvetyn” and “didapper”.[i] Listen to people talking, or read any book in a library, and you will get lots of the first three and decreasingly less common words until you get very infrequent occurrences of the last three. When applied to natural languages, that version of the power law is called Zipf's law after the American linguist, George Zipf.

The power law describes a different set of relationships from that of the bell curve. It shows what happens when things aren’t normally spread as depicted in the bell curve. It is when a few extreme, blockbuster events happen, with a long tail of fewer and fewer popular events.

The power law is very effective at providing a picture of the phenomenon it is used to describe. It is more important in many circumstances than the ubiquitous normal distribution. Yet fewer people understand it, and far fewer use it regularly.


But used correctly it can be way more important than the normal distribution. Let me explain. The simplest account of a power law is: “A relationship between two quantities such that one is proportional to a fixed power of the other” ( (note: of the type, y = x∞).

So, if the likelihood of a particular event occurring varies as a consequence of another event, say its intensity, you would have a power law. One quantity changes and there is a relative proportional change in the other quantity.

Power-laws follow a slow distribution curve with a long tail. This curve is also explained another way that many have heard of: the 80-20 rule, or Pareto distribution. The 80-20 rule states that 20% of the causes (of almost anything) will explain 80% of the effect (of almost anything else).

Generally, then, a power law will be found where there are a small number of extreme incidents and a large number of non-extreme incidents. Think about the Internet. There are some hugely popular sites with lots of hits (Google, Wikipedia and Twitter, for instance) and millions of other sites to which only a few people go. Website visits are not randomly or normally distributed: a disproportionate few are much more popular and powerful than the rest.

One key difference between the two types of curve is the mean. There is no meaningful average instance in the power law distribution—no archetypal case—whereas the normal distribution is all about what is a typically recurring case. (You can calculate a mean in a data set expressing the power law. It just doesn’t say anything useful because it doesn’t tell you about the rare or the common cases). It makes sense to talk about people’s average height or IQ. But it makes a lot less sense to describe the average number of web hits, or the average intensity of earthquakes, as you want to know about the powerful websites to go to or the damaging earthquakes to avoid. In power law distributions there are always a few extreme cases and many instances of everything else, and you mostly need knowledge about the hyper examples.

Both biological and human-made phenomena can follow a power law distribution and have been demonstrated in physics, biology and the social sciences. In point of fact, the areas where a power law distribution can be found is quite astounding. Everything from fluctuations in the economy, to the foraging patterns of different species, to wealth. For instance, there are only a few mega-billionaires like Bill Gates, Carlos Slim Helú and family, Warren Buffett, Amancio Ortega Gaona and Larry Ellison, in contrast to the billions of people who eke out an existence in poverty—and whose collective wealth does not go close to equaling the wealth of even one Bill, Carlos or Warren.


An application of the power law a while back caused a stir in academic circles. The results are quite remarkable.

We tend to think of wars as messy, chaotic and unpredictable. They arise from unique circumstances and play out in unpredictable ways. No one can tell if a brief disagreement will escalate, then erupt into full-on human conflict. But is that so? Are they perhaps subject to the power law, and able to be plotted ahead of time?

This is what British scientist Lewis Fry Richardson found after spending seven years gathering data and statistics on all the wars fought between 1850 and 1950. He published his findings in his 1960 work The Statistics of Deadly Quarrels and became known as one of the founders of the science of conflict.

In studying the huge amount of data—there were around 300 wars in that hundred year period alone—Richardson found that the severity and frequency of conflicts were linked. There was an underlying pattern. There were many wars (skirmishes, really) with only a small number of casualties, compared to a few wars with very high fatality rates. Sound familiar? The frequency of wars and the number of casualties were following a power law distribution.

The problem for Richardson was that despite the seeming significance of the find, there was no application. No apparent real-world scenario for which it could be useful. Who could he tell? Which politicians, generals or countries would change course because an academic found a pattern in other people’s mass historical behaviors? And he lacked a really sophisticated model to help with forecasting future events, and persuading others in any case.


More recently, however, everything changed. Neil Johnson and his colleagues from the University of Miami, Florida, found that Richardson’s power law could also be used to predict insurgent attacks – intermittent conflict below the level of full-blown, set-piece wars.

Their formula, Tn = T1n—b, predicted the timing and severity of insurgent attacks against American forces in Afghanistan and Iraq using just the timing of the interval between the first two attacks. And it’s remarkably accurate. See the figure below:

As you can see, the formula follows a progressive curve with an 80-20 type distribution, a power law not a bell curve.

  • Tn represents the number of days between attack number (n).

  • T1n, therefore, is days between the first and second attacks.

  • b is calculated using the attack number, n, and the interval, Tn.

The reason this formula has made some people sit up and pay attention is because it’s surprisingly simple: a knowledge of T—the timing between the first and second attack—is, in most cases, enough to foreshadow (but not yet totally, precisely predict) how and when future attacks will develop. The key things to know are how attacks escalate, and when they will occur.


But we can go further—there’s much more to this than meets the eye. The power law is the gift that just keeps on giving. In a paper published two years later, Simple mathematical law benchmarks human confrontations’, Johnson and his colleagues created a new formula, AB-C which takes the mathematical understanding of conflicts to a whole new level.

The formula describes the distribution and timing of events and encompasses a range of seemingly unrelated conflicts in varied settings including terrorist acts, child-parent disputes and cyber-attacks. They argued that “a simple mathematical law can benchmark them all.”

That law is, of course, the power law. And now it has allowed scholars to find and map patterns in a wider range of conflict scenarios.

Neil Johnson explains: “By picking out a specific baby (and parent), and studying what actions of the parent make the child escalate or de-escalate its cries, we can understand better how to counteract cyber-attacks against a particular sector of U.S. cyber infrastructure, or how an outbreak of civil unrest in a given location (e.g., Syria) will play out, following particular government interventions” (

Consider how this works. A weaker entity (a crying baby, a terrorist, a cyber-hacker, a stockmarket trader) tackles a more powerful one (a parent, an army, a government’s internet defenses, the stock holdings of a large company). The “attacks” accelerate if the attacker does not get the results it first sought. The parent tries harder to pacify the infant; the army doubles its efforts to take out the extremist; the government introduces more stringent cyber-security; the trader buys more stock.

This can be put in a power law model, which can be used to then estimate the frequency and likelihood of future “attacks”. The model can also be used to search and to attempt to predict the severity of the “attacks”, as they, too, follow a power law distribution.

And of course, anyone who has a better idea than just a guess or a belief that such events are merely random, can start to plan counter-measures. Pick the baby up ahead of it starting to cry, identify the likely terrorist beforehand, improve the website’s security, bid up the stock price.


The power law distribution has been found in so many natural and man-made phenomena. It seems to be revealing itself as one of life’s great governing laws. No-one knows where next it will enlighten us.

Further reading:

Gopikrishnan, Parameswaran, Plerou, Vasiliki, Nunes Amaral, Luís A, Meyer, Martin, Stanley, H Eugene (1999). Scaling of the distribution of fluctuations of financial market indices. Physical Review E 60 (5): 5305-5316.

Gutenberg, Beno, Richter, Charles F (1949). Frequency and Energy of Earthquakes. In: Seismicity of the Earth and Associated Phenomena (pp. 16-24). Princeton, NJ: Princeton University Press.

Humphries, Nicolas E, Queiroz, Nuno, Dyer, Jennifer RM, Pade, Nicolas G, Musyl, Michael K, Schaefer, Kurt M, Fuller, Daniel W, Brunnschweiler, Juerg M, Doyle, Thomas K, Houghton, Jonathan DR, Hays, Graeme C, Jones, Catherine S, Noble, Leslie R, Wearmouth, Victoria J, Southall, Emily J, Sims, David W (2010). Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature 465 (7301): 1066-1069.

Johnson, Neil F, Carran, Spencer, Botner, Joel, Fontaine, Kyle, Laxague, Nathan, Nuetzel, Philip, Turnley, Jessica, Tivnan, Brian (2011). Pattern in escalations in insurgent and terrorist activity. Scientific Reports 333 (1): 81-84.

Johnson, Neil F, Medina, Pablo, Zhao, Guannan, Messinger, Daniel S, Horgan, John, Gill, Paul, Bohorquez , Juan C, Mattson, Whitney, Gangi, Devon, Qi, Hong , Manrique, Pedro, Velasquez, Nicolas, Morgenstern, Ana, Restrepo, Elvira, Johnson, Nicholas, Spagat, Michael, Zarama, Roberto (2013). Simple mathematical law benchmarks human confrontations. Scientific Reports 3: 3463.

Kauffman, Stuart (1995). At Home in the Universe: The Search for the Laws of Self-organization and Complexity. New York: Oxford University Press.

Richardson, Lewis F (1960). Statistics of Deadly Quarrels. Pittsburgh: Boxwood Press.


i. Lollygag is to be aimless, idle or to dawdle

Duvetyn is a kind of fabric; smooth and velvet in nature; we have appropriated the word in modern language as a duvet

Didapper is a type of water-bird that constantly dives under the water, disappearing and reappearing

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