The Better Angels of Our Nature: Why Vio - By Steven Pinker Page 0,131

municipality to the smallest is 150,000, which is very different from the less-than-fivefold variation in the heights of men.

Also, the distribution of sizes of municipalities isn’t curved like a bell. As the black line in figure 5–10 shows, it is L-shaped, with a tall spine on the left and a long tail on the right. In this graph, city populations are laid out along a conventional linear scale on the black horizontal axis: cities of 100,000, cities of 200,000, and so on. So are the proportions of cities of each population size on the black vertical axis: three-thousandths (3/1000, or 0.003) of a percent of American municipalities have a population of exactly 20,000, two thousandths of a percent have a population of 30,000, one thousandth of a percent have a population of 40,000, and so on, with smaller and smaller proportions having larger and larger populations.59 Now the gray axes at the top and the right of the graph stretch out these same numbers on a logarithmic scale, in which orders of magnitude (the number of zeroes) are evenly spaced, rather than the values themselves. The tick marks for population sizes are at ten thousand, a hundred thousand, a million, and so on. Likewise the proportions of cities at each population size are arranged along equal order-of-magnitude tick marks: one onehundredth (1/100, or 0.01) of a percent, one one-thousandth (1/1,000, or 0.001) of a percent, one ten-thousandth, and so on. When the axes are stretched out like this, something interesting happens to the distribution: the L straightens out into a nice line. And that is the signature of a power-law distribution.

FIGURE 5–10. Populations of cities (a power-law distribution), plotted on linear and log scales

Source: Graph adapted from Newman, 2005, p. 324.

Which brings us back to wars. Since wars fall into a power-law distribution, some of the mathematical properties of these distributions may help us understand the nature of wars and the mechanisms that give rise to them. For starters, power-law distributions with the exponent we see for wars do not even have a finite mean. There is no such thing as a “typical war.” We should not expect, even on average, that a war will proceed until the casualties pile up to an expected level and then will naturally wind itself down.

Also, power-law distributions are scale-free. As you slide up or down the line in the log-log graph, it always looks the same, namely, like a line. The mathematical implication is that as you magnify or shrink the units you are looking at, the distribution looks the same. Suppose that computer files of 2 kilobytes are a quarter as common as files of 1 kilobyte. Then if we stand back and look at files in higher ranges, we find the same thing: files of 2 megabytes are a quarter as common as files of 1 megabyte, and files of 2 terabytes are a quarter as common as files of 1 terabyte. In the case of wars, you can think of it this way. What are the odds of going from a small war, say, with 1,000 deaths, to a medium-size war, with 10,000 deaths? It’s the same as the odds of going from a medium-size war of 10,000 deaths to a large war of 100,000 deaths, or from a large war of 100,000 deaths to a historically large war of 1 million deaths, or from a historic war to a world war.

Finally, power-law distributions have “thick tails,” meaning that they have a nonnegligible number of extreme values. You will never meet a 20-foot man, or see a car driving down the freeway at 500 miles per hour. But you could conceivably come across a city of 14 million, or a book that was on the bestseller list for 10 years, or a moon crater big enough to see from the earth with the naked eye—or a war that killed 55 million people.

The thick tail of a power-law distribution, which declines gradually rather than precipitously as you rocket up the magnitude scale, means that extreme values are extremely unlikely but not astronomically unlikely. It’s an important difference. The chances of meeting a 20-foot-tall man are astronomically unlikely; you can bet your life it will never happen. But the chances that a city will grow to 20 million, or that a book will stay on the bestseller list for 20 years, is merely extremely unlikely—it probably won’t happen, but you could well imagine it happening. I hardly need to point out the