Modelling City Growth using Kleiber's Law
David Thompson Secondary
Floor Location : M 116 F
Cities play such a crucial role in all of our lives, whether we like it or not. Wouldn’t it be nice to easily know important information about any city simply given that city’s population? By applying a biological law to a social scenario, that of cities, one is able to do just that. First, a bit of background… Kleibers’s Law is a biological law, which states that an organism’s mass to the three-fourths power is proportional to its basal metabolic rate (energy used while at rest). Kleiber’s law follows a power law equation. This equation is in the form y=ax^b. In this project I compared characteristics of Kleiber’s law (the exponent b in the model y=ax^b) with how growth aspects of a city (such as GDP, housing units, etc.) scale with that city’s population. This was done using computer code written by me to find the best line fit (of the form y=ax^b) for the data set, through nonlinear regression. I tested this law with different data sets (accessed through a programming application, and from public sources), and found that as I had predicted, similarities are apparent when applied to model aspects of city growth. These similarities are found in the exponent b, which in most cases was close to Kleiber’s Law predicted value of ¾. The similarities can and were quantified by finding the R^2 coefficient of determination for each data set. This coefficient tells us how well the actual data fits the predicted best line fit. For city data sets whose predicted b value (from their best line fit) was greater than one, superlinearity was present, meaning that for an increase in population, per capita, values of the observed city growth aspect increase. This differs from city data sets whose predicted b value was less than one, where sublinearity was present, and for an increase in population, per capita, values of the observed city growth aspect decrease. Throughout my analysis, I encountered both cases. It is easy to tell which type of case was present for each data set since, when graphing the data, I added a table showing the best line fit exponent (b), along with some statistical analysis such as the standard error for the data set, and p and t values (which help in confirming the results). When graphing, I also linearized the data points and the predicted best line fit by taking the log of both variables. This allows us to easily see if the data points are close to the predicted line, or not. I then added sigma bands, which show us what percentage of the data points lie between two lines. With the results of this project, experiments reconfirm the claim that a variation of Kleiber’s law is applicable to model aspects of a city’s growth. This creates multiple applications such as a smart method for governments to distribute funds to certain aspects of their city.