This chart explains why cities are at the center of innovation

Density + connectivity = productivity.
Density + connectivity = productivity.
Image: AP Photo/HuYuanjia
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It’s obvious from human history that people have long found unique value in living and working in cities, even if for reasons they couldn’t quite articulate. Put people together, and opportunities and ideas and wealth seem to grow at a more powerful rate than a simple sum of all our numbers. This has been intuitively true for centuries of city-dwellers.

“What people didn’t know,” says MIT researcher Wei Pan, “is why.”

There have been plenty of theories. Adam Smith famously figured that people become more productive when we’re able to specialize, each of us honing a separate area of expertise. And when lots of us elbow into cities, we’re able to specialize in ways that we can’t when a rural farmer must also double as his own butcher, accountant and milkmaid. Other economists have suggested that cities become great agglomerators of industry when factories cluster together around economies of scale and communal access to transportation.

“We think there’s an underlying completely different way of thinking here, which is very different from the economist’s way of thinking,” says Pan, a doctoral candidate in computational social science in the MIT Media Laboratory’s Human Dynamics LabPrevious work by researchers at the Santa Fe Institute has proven the math behind the power of cities: As they grow in population, all kinds of positive outcomes like patents and GDP and innovation (and negative ones like STDs and crime) grow at an exponential factor of 1.1 to 1.3.

This means that all the benefits (and downsides) that come from cities don’t just grow linearly; they grow super-linearly, and at roughly the exact same scale, with a growth rate that looks on a graph something like this blue line (a linear relationship is shown in red):

Via Desmos

As for why this happens, though, Pan pushes aside theories about the location of manufacturing or the specialty of trade. “It’s more fundamental than that,” he says. “Cities are about people. It’s just that simple.”

In a new paper published in Nature CommunicationsPan and several colleagues argue that the underlying force that drives super-linear productivity in cities is the density with which we’re able to form social ties. The larger your city, in other words, the more people (using this same super-linear scale) you’re likely to come into contact with.

“If you think about productivity, it’s all about ideas, information flows, how easily you can access ideas and opportunities,” Pan says. “We believe that the interaction mechanism is what drives the productivity of the city.”

It’s not possible for scientists to measure your social ties in the same way they can measure GPD or crime incidents or STD infections (despite their best wishes, they can’t put sensors on all of us). But this study examined a proxy for the same idea: The researchers looked at phone logs between anonymized telephone numbers all over the country, in search of the number of people who we communicate with inside our own metropolitan statistical area.

“If you look at the interaction patterns of cities,” Pan says, “You will see that they grow super-linearly with population with the same growth rate as productivity, as innovation, as crime, as HIV, as STDs.”

All of those facets of urban life have appeared until now to share a somewhat mysterious mathematical relationship. But this research suggests that this particular super-linear growth rate is directly tied to how dense cities enable us to connect to each other. As cities grow, our connections to each other grow by an exponential factor. And those connections are the root of productivity.

“What really happens when you move to a big city is you get to know a lot of different people, although they are not necessarily your ‘friends,'” Pan says. “These are the people who bring different ideas, bring different opportunities, and meetings with other great people that may help you.”

Bad infrastructure that divides us has the potential to wipe out the most fundamental benefit of cities.

Maybe this doesn’t sound like a novel discovery—that the inherent power of cities lies in our individual connections to each other. Other researchers have nipped at a similar idea, calculating, for example, the “social interaction potential” of place. But until recently, most economic thinking sidestepped the sheer value of human interaction in favor of explanations about the proximity of manufacturing, or the processes of production.

This explanation for the power of cities also raises some curious questions about those places that remain an exception to the model. In some African cities and Eastern mega-cities, innovation and productivity don’t grow super-linearly. Populations grow, but the benefits don’t accrue with them as we would expect. This is likely because transportation infrastructure in those places is so poor that people aren’t able to connect across town to each other. “To live on the west side of Beijing,” Pan says, you never go to the east side.”

And this tells us that bad infrastructure that divides us has the potential to wipe out the most fundamental benefit of cities.

The model also raises another question for future planning: Is it possible for metropolitan regions to grow so big that the super-linear relationship between population and productivity breaks down? Researchers don’t know, Pan says.

“It’s really hard to build a mega-city with good traffic infrastructure. We don’t have flying cars,” he jokes. “Once we have that, we’ll know better. But so far, it’s hard.”

The models suggest that when cities reach around 40 million residents, the benefits will start to tail off and eventually decline. But there isn’t a city in the world currently approaching that theoretical limit. And so we have a lot of time to think about what will happen when we get there.

Emily Badger is a staff writer at The Atlantic Cities.

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