Power Laws and Cities

Bettencourt et al. in PNAS looked a variety of cities of various sizes. They wanted to determine what the effect of population size of the city has on their properties including physical properties like roads, but also economic properties like consumption.

What they found was very interesting. What they found was that 1) these properties all appear to follow a power law with respect to population and 2) these properties fall into specific categories depending on the power law.

The first thing we should talk about before going to this paper is what is a power law. Power laws are ubiquitous in both physical and life sciences. An example of a power law would be Coulomb's law or Newton's law of gravitation. A power law has two important traits. The first is that it is in the following form:

The critical part of the equation above is that exponent z. The exponent says something about what happens when the value of x gets larger. In essence, the exponent says something about the effects of scaling.

The second trait of power laws that is important is they are scale invariant. Scale invariance is the property of maintaining the same characteristics regardless of scale and lacking a characteristic scale. For example, if you looked at a picture of rocks from space or a picture of rocks up close, the rocks would look pretty much the same -- assuming there wasn't clouds in the way. They would be sort of rough looking. Another example is coastlines. Whether you are close up or far away, the jagged nature of coastlines is the same regardless of scale.

The equation above satisfies this quality of scale invariance. Imagine the graph of y = sqrt(x). No matter where you set the axes -- how near or far away you are -- the left part of the line is starting at the bottom near zero and the right part of the line is rising to infinity at a slower and slower rate.

Basically, the key part about power laws is that regardless of scale an A is an A is an A. A rock is a rock is a rock. They have no particular scale where A follow particular laws and a different scale where A follows different laws. Applying this to cities, a city is a city is a city. No matter how big or small, if the city follows a power law the properties are the same at any scale. One city is just bigger than another. (This is not particularly intuitive if you think of cities. Why would a small town follow the same laws as NYC? This is why this paper is cool.)

(Power laws have a bunch of other interesting traits. If you want a good review for the lay-person, I recommend Gisiger et al. 2001.)

Getting back to Bettencourt et al., they looked at an extensive set of data from cities all over the world. What they wanted to determine was whether certain factors in cities like crime related to the cities population. For them, all these power laws would be in the following form:

N(t) is the population of a particular city at time t. Y is a property of the city, and Beta is the exponent of the power law.

They fit the data to the power law above and calculated the exponent Beta for each property tested. Here is there data for world cities, displaying the exponent values for each property:

Looking at these data, they found two important things.

• 1) A variety of traits of cities obey power law functions of population.
• 2) The Beta exponents for each of these properties fall into three general classes.

Class 1 (at the top of the chart) includes properties that increase exponentially with population. These are the properties with Beta ~ 1.2. These include aggregate measures of production like total wages and new patents as well as aggregate measures of consumption like total electrical consumption. This class also includes negative factors like AIDS cases and crime. They call these sociological properties.

Class 2 (in the middle of the chart) includes properties that increase linearly with population. These are the properties with Beta ~ 1. These include measures of consumption and production per capita like individual power consumption and individual employment. They call these individual properties.

Class 3 (at the bottom of the chart) includes properties where the rate of increase slows as population increases. These are the properties with Beta ~ .8. These include measures where economies of scale can be applied like road surface and length of electrical cable. They call these biological properties. (These are like biological systems because if you look at say the metabolism of an animal as it increases in size it slows down.)

Why is this it interesting that traits of cities would vary with population in such defined ways?

Well, first if you are a city planner you can use this information to...well...plan. In the transition of your city from small to large you can expect an exponential increase in production but also in crime. Also, you can expect that the cost of things like electricity -- to which economies of scale can be applied -- will slowly become less of your budget per capita while AIDS treatment might become more.

Second, it would appear that the power laws for cities show a trait that physicists call universality. Universality means that the exponents defining the power laws of particular systems fall within a set of narrowly defined values, ans the particular value of the exponent is determined by the arrangement of the system.

Put more simply, why should roads and electrical cables scale at the same rate? You could argue that roads and electrical cables tend to occur together -- cables are often on the sides of roads. (We might expect similar effects for say total number of internet routers.) It is the geometry of the system -- communication and transportation routes -- that makes the two factors scale together. Similarly, why do new patents and total wages scale together? You could make the argument that since individual innovation produces wage growth, we would expect the two to covary.

Universality is in my opinion one of the most fascinating aspects of systems analysis. It allows you to ignore the particular traits of the system you are dealing with, and get into the fundamental traits that determine how that system works. With respect to cities, universality implies that organization of cities implies that certain traits will move together. It might be possible to encourage growth in one by encouraging growth in another of the same class.

Anyway, I recommend reading the whole paper. This is just a taste. The authors come up with a mathematical model showing the relative contributions of exponential and non-exponential growth properties and how these would lead to boom and bust cycles in cities. It is interesting stuff.

Hat-tip: Faculty of 1000