A Physicist Tries to Solve the City

By Reid Ewing

A paper I reviewed recently for a planning journal (that asked to remain unnamed) applies Luis Bettencourt and Geoffrey West's Urban Scaling Theory to an analysis of crime in cities across the U.S., finding that the number of crimes committed follows a superlinear (upward curving) relationship as a function of the population size. That is, crime increases faster than population. If the paper were any good, that conclusion would not exactly be a selling point for big city living. It is not, so don't relocate just yet.

Bettencourt and West, two physicists, argue that virtually any urban phenomenon can be described by a simple formula:

Outcome = a x Population^b where a and b are constants, and b specifically is the power to which population is raised (the exponent).

When b = 1, a relationship is linear.

When b < 1, a relationship is sublinear. This applies, for example, to most urban infrastructure such as road capacity.

When b > 1, a relationship is superlinear. This applies to most economic activity and, superficially, to crime.

Urban Scaling Theory has been topical ever since a popular piece called "A Physicist Solves the City" appeared in the New York Times in 2010. The Times article made the theory sound really profound, and appeared without much of a critique. But one critique in the Times article was itself profound and has tended to be ignored by many who subsequently subscribed to the theory:

"While listening to West talk about cities, it's easy to forget that his confident pronouncements are mere correlations, and that his statistics can only hint at possible explanations. Not surprisingly, many urban theorists disagree with West's conclusions. Some resent the implication that future urban research should revolve around a few abstract mathematical laws."

In urban scaling theory, population size is what is referred to as a confounding variable. As population increases, so does everything else in a city, from number of pigeons, to number of coffee shops, to number of vehicle miles traveled, to number of crimes committed. In a multivariate analysis, one controls for a confounding variable like population either by including it as a control variable or by representing dependent and independent variables on a per unit or per capita basis.

The variables of ultimate interest in such analyses aren't correlates of size but such things as population density and income inequality, controlling for population size, because these variables can be affected by policies and practices.

The use of population exclusively as the independent variable is atheoretical. Because everything in a city increases with size, there will necessarily be a statistical relationship between population and every scaled variable. This relationship may be linear, sublinear, or superlinear. By definition, it has to be one of the three. As every student of statistics knows, any two variables are always correlated to some degree in a bivariate analysis, and any regression analysis will show the relationship between them to be one of the three, sublinear, linear, or superlinear. The important thing isn't the power exponent of the relationship, but the degree of scatter around the regression line.

Original empirical evidence on the number of crimes versus the population of cities comes from Bettencourt and West themselves back in 2007. They found serious crimes to be superlinear vis-à-vis city size, with a b value of 1.16 and with R2 of 0.89. This is significant for two reasons. First, this relationship (crimes versus population) has already been studied extensively by Bettencourt et al. and other researchers cited in the paper I reviewed for that unnamed planning journal. Is one more study groundbreaking enough to warrant publication? The fact that this paper disaggregates crime statistics by type of crime and analyzes relationships over a 16-year period (1995 to 2010) is a modest innovation.

The second significant thing about the Bettencourt et al. study is the R2, 0.89. As statistics students will recall, an R2 measures the proportion of variation in the dependent variable explained by the independent variable or variables. For two variables that are both scale-dependent, 0.89 is not a very high R2. In other words, there is a lot of scatter around the regression line (regression curve). The scatter is what is really interesting. Are some cities above or below the regression line because they have higher or lower population densities than average, or perhaps more interestingly, because they have more or less income inequality or more or less emphasis on community policing?

Our team at the University of Utah developed sprawl measures for 994 metropolitan counties and 221 metropolitan areas in the country. I wonder if this sprawl variable, which is also subject to influence by planners, explains some of the scatter in crime statistics.

The authors of the paper I reviewed proudly trumpet their approach: "The dependent variable is crime counts instead of crime rate ... " and " ... the only independent variable to be examined is the population size of cities." Such bravado. They estimate the constants a and b in the above formula for 176 different years and types of crime using simple regression analysis, and get R2s typically in the neighborhood of 0.7.

Hopefully, in light of the preceding arguments, the shortcomings of their approach are readily apparent to the reader. Interestingly, the authors of the reviewed paper describe several studies that investigate crime in a smarter way, examining "the partial effect of population [on the] crime rate by incorporating a number of socio-economic covariates in their analyses."

Needless to say, I recommended rejection of the reviewed paper in its original form. Its authors failed to grasp an essential point — that a couple of physicists may not have that much to tell planners.

Source: FBI Uniform Crime Statistics; graphic by Torrey Lyons, University of Utah