Tag Archives: density gradient

The negative exponential model and the size of cities

Researchers have long noted the tendency for densities to decline as a negative exponential function of distance from the center. They have looked at declines in the density gradient over time as a measure of decentralization in urban areas. They have noted the relationships of the estimated parameters of the model–the density gradient and the density at the center–to a variety of characteristics of urban areas, including, naturally, the size of the area. The consistent finding has been that the gradients tend to be smaller for larger urban areas, while the central densities tend to be larger.

Consider the relationships among the three–the gradient, the central density, and the size of the urban area. If density declines with distance following the negative exponential model, these three values must necessarily be mathematically related. But what affects what? It seems reasonable to believe that the size of the urban area is primarily affected by factors other than the parameters of the negative exponential model.

But what about the model parameters? Housing is long lasting and once established, the patterns in developed areas can remain remarkably stable for many decades. The density of urban development was much higher before widespread use of the automobile. And it turns out that the central densities are very strongly related to the sizes of urban areas in 1910. So it may not be unreasonable to conclude that, at least to some extent the density gradient is determined by the central density and the size of the urban area.

Solving for the mathematical relationship between the gradient, central density, and size yields a somewhat complex expression. However, a simplified approximation can be used. This approximation has the density gradient being directly proportional to the square root of the central density and inversely proportional to the square root of the size of the urban area.

As described in an earlier post and in a paper, I had used my urban patterns data to estimate the parameters of the negative exponential model for large urban areas in the United States from 1950 to 2010. It was straightforward to test for the conformity with the expected relationships among the density gradient, central density, and the size of the urban area. The gradient was indeed approximately inversely proportional to the size of the area, as expected. And the gradient did increase with the central density, though the proportionality was closer to the density itself rather than the square root. It may be possible that this is the result of the fact that the census tract densities in my data (and used by most other researchers) are measures of gross density including nonresidential uses, streets, and vacant land and are therefore lower than the net residential densities within the residential areas alone.

More information on this analysis, including the mathematical derivation of the relationship among the 3 values, is in the paper “Negative Exponential Model Parameters and the Size of Large Urban Areas in the U.S., 1950–2010,” which can be downloaded here.

The negative exponential density gradient and decentralization

Many researchers have used the density gradient from the negative exponential model to study the decentralization of population and housing units in urban areas. The density gradient is the rate of decline of density with distance from the center of the city. A decrease or flattening of the density gradient has been considered to be evidence of the decentralization of population or housing. And the density gradient has been used as a measure of the amount of centralization in an urban area that could be used to compare levels of centralization with other urban areas.

I have estimated the density gradients for 43 large urban areas for each of the census years from 1950 to 2010. And I have developed a separate, “pure” measure of centralization of housing units which I described in the previous post. I am calling this measure the centralization ratio. So this gave me the means of actually looking at the extent to which the density gradient was a good measure of centralization and decentralization.

First, I looked at changes in the density gradient over time and compared it to changes in the centralization ratio. The relationship was reasonably strong. It is appropriate to use the change in the density gradient as a measure of decentralization.

Then I looked at the relationship between the magnitudes of the density gradient and the centralization ratio at single points in time. This time, virtually no relationship. The density gradient does not work as a measure of the level of centralization in an urban area that could be used to make comparisons with other urban areas.

What gives? Why such different findings? The key lay in the fact that the density gradient is strongly inversely related to the size of an urban area. Using the density gradient to predict the centralization ratio resulted in no relationship. But add number of housing units in the urban area to the model, controlling for the size of the area, and a strong relationship emerged. And this is why the change in the density gradient works as a measure of change in centralization over time. The size of the urban area is being subtracted out when you look at the change (with the exception of any change in size over the period).

Someone committed to the idea that the density gradient is a good measure of centralization might object that I have only shown that the centralization ratio and the density gradient are different, not that one is a better measure of centralization. I think I make a good case for the use of the centralization ratio. Also, in developing the measure, I calculated other measures of centralization for a sample of a dozen areas and they were all highly correlated. And an anecdotal point: The three urban areas in my study with the highest centralization ratios were New York, Chicago, and Philadelphia. And all three had density gradients that were below the mean for the 43 large urban areas I looked at.