Inspired by these questions, the following paper not only presents a unique and playful way of mapping American commercialism, but also uses the analysis as an opportunity to conduct a comprehensive investigation of density-equalizing maps, or "cartograms." First, I discuss the development and methodology of cartograms, including a relatively recent method proposed by Gastner and Newman in 2004. Then, I use Gastner and Newman's method to create elegant cartograms depicting the distribution of Walmarts, McDonalds, and Starbucks in the fifty United States. I examine some of the interesting trends evidenced by these cartograms, and finally I conclude with a discussion of the advantages and disadvantages of cartograms in comparison with other common methods of graphical data presentation.

However, there are challenges associated with using cartography to analyze or present statistical data. Apart from needing additional geographic information and the increased complexity of the data presentation, a significant hurdle is that population density is extremely variable. Smaller populations tend to be found in larger geographic regions (such as rural areas), which can lead to misleading visual impressions. Perhaps the best way to visualize data that is affected by spatial characteristics is to actually use spatial characteristics to distinguish the trends on display. This is the idea that inspired cartograms.

Cartograms are maps in which the sizes of geographic regions appear in proportion to their population or some other analogous property. These density-equalizing maps are useful for the representation of census results, election returns, disease incidence, and vital statistics such as the distribution of Walmarts, McDonalds, and Starbucks across the country. Several methods for making cartograms have been proposed over the years, but most of these ideas are inordinately complex or suffer from a lack of readability due to the distortion needed to scale regions and have them still fit together.

In 2004, Gastner and Newman at the University of Michigan proposed a new method of making cartograms. Their method is not only faster and more conceptually simple than previous methods, but also produces useful and easily readable cartograms. Using the relatively straightforward principles of linear diffusion, a cartogram is created from a given population density (or some other analogous property) by allowing the population to "flow away" from high-density areas into low-density ones, until the density is equalized everywhere. Areas shrink and grow and distort to stay connected, producing a cartogram that is in fact unique for a given dataset. (For the formulas and Fourier transforms, Gastner and Newman's 2004 paper is a good reference.) The degree to which the data is binned is important to achieving the desired balance of distortion and recognizably: a very fine level of data binning will cause substantial local distortions, while a coarser level of binning will result in a cartogram with features that are easier to recognize, but gives a less accurate impression of the true population distribution.

The rest of this paper focuses on my use of Gastner and Newman's method to develop cartograms displaying the distribution of Walmarts, McDonalds, and Starbucks in the fifty United States. The Appendix provides annotated references to the datasets and software I used in this study, as well as provides some examples of code I wrote to analyze and display the density-equalized cartograms in R.

My paper. Link to pdf of report.