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Math Trek

Outstanding, superlinear cities

By a new mathematical method, New York City is average and San Francisco exceptional

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New York City seems pretty extraordinary: Its residents make more money, produce more stuff and commit more violent crimes than those of any other U.S. city. And New Yorkers are nearly the most creative, as judged by the total number of patents they produce. But according to mathematician Luís Bettencourt, New York is actually quite average, given its size. For a really exceptional place, swap coasts and look at San Francisco.

The apparently unusual qualities of New York are actually natural and unsurprising products of its size, argues Bettencourt, a researcher at Los Alamos National Laboratory and the Santa Fe Institute, both in New Mexico. Bigger cities are more intense by nature: richer, more productive, more creative and more dangerous. Indeed, Bettencourt and his colleagues have shown, doubling the population of a city gives a 15 percent premium on each of these factors. Since New York City is the most populous city in the United States, New Yorkers should make more money, for example, than other Americans on average. To be exceptional, New Yorkers would need to be raking in even more than the princely sums you’d expect.

Turns out they’re not. In an article this month in PLoS ONE, Bettencourt and his team created a way to measure how exceptional cities are by comparing their characteristics with what mathematics would predict for their size. The team then ranked the exceptionality of 300 U.S. cities based on personal incomes, gross metropolitan product (GMP), number of patents and number of violent crimes. On income, the New York metropolitan area came out a measly 85th place, just 3.8 percent above what one would predict for its size. On GMP, it ranked 167th, and on patents, it ranked 178th. The only exceptional number was for crime, which was surprisingly low:  267th out of 300, a whopping 22 percent below the typical rate for its size.

San Francisco, on the other hand, is rich, productive, creative and moderately safe for its size. By the team’s rankings, the San Francisco metropolitan area comes out 12th for personal income, 19th for patents, 27th for GMP and 131st for violent crime.

“To identify what’s special about a place, you have to separate out the factors that are really just about its size,” Bettencourt says. “Then you can disentangle the general effects of urbanization from the specific character of a town.”

Ideas, the team believes, are the real driver of economic activity and creativity, and when people are in closer contact — as they are in big cities — they tend to share those ideas more. A magazine designer in New York, for example, is much more likely than one in Huntsville, Ala., to bump into someone who knows about new design software or a clever layout trick. As a result, twice as many people are more than twice as productive — a phenomenon known as “superlinear scaling,” since the increase is faster than a linear equation would predict. That’s the origin of the 15 percent premium on per capita income, patents and GMP that Bettencourt and his colleagues have documented in cities around the world. Similarly, crime increases superlinearly as people share bad ideas.

And superlinear scaling applies not just to those phenomena: Knowing just a city’s population, Bettencourt’s team can predict pretty accurately how quickly disease will spread, the number of educational institutions and even how quickly its pedestrians will walk.

But of course, not every city fits the trend perfectly. Some are outliers, lying far off the line showing the average trend. These are the exceptional cities.

It’s easier to be exceptional if you’re small. Corvallis, Ore., for example, is the top-ranked producer of patents, because it’s a small town centered around a big Hewlett-Packard laboratory. Casper, Wyo., ranks second in GMP and sixth in personal income, since its population is just over 50,000 and it’s earned the nickname “Oil City.” But San Francisco breaks the link between smallness and exceptionality. “San Francisco is very exceptional, because it manages to be exceptional while being big,” Bettencourt says. “It’s the most exceptional big city.”

Comparing rankings is more than a fun parlor game: It can help reveal what makes a city prosper. Pittsburgh, for example, has recently become fairly rich and technological, while Buffalo and Cleveland — which, like Pittsburgh, are former industrial cities that went through hard times — are still economically depressed. “These are places that should be compared by people who understand what’s going on on the ground,” Bettencourt says. “These cities might have something to teach each other.” Portland, Ore., and Boulder, Colo., offer another example: They seem to have similar cultures, but Boulder is much more technological and richer while Portland is, by these mathematical measures, rather ho-hum and average.

The team’s measure of exceptionality isn’t ideal for every purpose. If you want to choose the city where you’re least likely to be a victim of violent crime, for example, you’re better off looking at the standard measure of violent crimes per capita — and you’ll probably choose to live in a small town. But Bettencourt’s measure will pick out the cities that have unusually low crime rates given their size, a metric that may lead to a new understanding of the root causes of crime.

Cities are far from the only objects that scale nonlinearly, and the same approach could be revealing in other contexts. Larger animals, for example, tend to live longer, grow up more slowly and have slower heart rates than smaller animals. But the rate of change is sublinear, i.e., slower than a linear equation would predict. Similarly, the productivity of corporations grows sublinearly, with small businesses contributing much more to the economy relative to their size than large corporations. Analyzing exceptional individuals could reveal animals that have evolved unusual strategies or corporations that have found the key to extraordinary success.

But of course, one form of mathematical analysis isn't everything. New Yorkers can still think they're pretty special.

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