Mathematics in everyday life

The power of cities


In the 1930s the Swiss born biologist Max Kleiber studied how much energy different animals expend at rest, and noticed something curious.  A human weighs about 10 times more than a cat.  But rather than expending 10 times the energy of a resting tabby, we only spend 6 times as much.  This number is not arbitrary — A cow is about 10 times heavier, and also expends about 6 times the energy of a human.

Kleiber was the first to notice this regularity: He showed that energy expenditure follows a 3/4 power law.  What this means is that if you double the size of an animal, it will use about 2^3/4, or about 1.7 times as much energy.  If you increase the size tenfold, it will use about 10^3/4 times as much energy. Amazingly organisms from bacteria to whales follow this law.

Power laws are all around us: If you add twice the amount of salt, a dish will not taste twice as salty.  Rather, it will appear about 2^1.4 or 2.6 times as salty. A star twice the mass of our Sun will be 10 times as bright. There are many other example, and surprisingly even human constructs behave similarly. Cities are particularly fascinating: If the size of a city doubles, we see more than a doubling of the number of patents, inventors and artists, and the amount of time we spend in traffic. All of these quantities follow power laws.

On the other hand some quantities grow more slowly – if a city doubles in size, we spend less than twice the gasoline or electricity – bigger cities are more efficient. These may be reasons why cities grow in size.  But all is not rosy – unfortunately, the number of crimes and cases of a disease more than double in a city twice as large.

We do not yet know exactly why such different quantities as a city’s crime rate, road density and the number of inventors behave so predictably.  But scientists have plausible theories:  It is possible that ideas, information and inspiration behave like diseases, and spread more easily when populations are larger and denser. The larger the city, the more contacts we have, the higher the chance that we will hear the latest important news and insight, or hear about a good job opportunity.

More than half the world’s population now lives in urban areas. Can we reap the benefits of living in a city – the higher wages, and higher energy efficiency – without the disproportionate increase in crime, pollution and disease? The city of Zürich in Kleibner’s native Switzerland suggests that this may be possible.   Zürich has grown tremendously in the last 20 years. But proper planning has kept traffic reasonable, and crime low. The underlying laws that govern how cities behave gives me hope that we will also be able to understand the mechanisms behind these laws. This will allow urban planners and administrators to avoid the mistakes of the past. They will be able to work with physicists and mathematicians to help cities reach their full potential.

References and notes:

Here is a nice article about Kleiber’s Law (there are many other good ones easy to find with Google).  A reason for why it may hold has been proposed given in the 1990s. However, it relies on the fractal geometry of the circulatory system, while Kleiber’s Law seems to extend to organisms that do not have one. The mechanisms behind the law are therefore still under debate.

For discussion of power laws and perception you can see this Wikipedia entry on Steven’s power law.  The laws here are a bit controversial because quantifying subjective experiences is difficult. More information about how luminosity scales with the mass of stars is here.

Here are some references (not complete) on how to explain power law scaling in cities. Arbesman, Kleinberg and Strogatz assume that the network of human contacts is assumed to have a hierarchical, self-similar (fractal) structure. Under certain conditions, with an increasing city, the increased number of contacts can lead to power law growth of the the overall benefit.  However the assumption that interactions are hierarchical may be too strong. It could be simply the increase in density that facilitates the interchange of ideas and information, as explained hereand reviewed hereLuís Bettencourt’s explanation develops this idea, but is also more complete.

A couple more interesting short references: A call for a science of cities (polisology?)  and an overview of why innovation thrives in cites.

Luís Bettencourt and Geoffrey West also gives a nice review of the statistical findings and how they could be used.  Unfortunately, behind a paywall in Nature

Here is another take on city complexity,  and here.