Diffusion-based method for producing density-equalizing maps

Abstract

Map makers have for many years searched for a way to construct cartograms, maps in which the sizes of geographic regions such as countries or provinces appear in proportion to their population or some other analogous property. Such maps are invaluable for the representation of census results, election returns, disease incidence, and many other kinds of human data. Unfortunately, to scale regions and still have them fit together, one is normally forced to distort the regions' shapes, potentially resulting in maps that are difficult to read. Many methods for making cartograms have been proposed, some of them are extremely complex, but all suffer either from this lack of readability or from other pathologies, like overlapping regions or strong dependence on the choice of coordinate axes. Here, we present a technique based on ideas borrowed from elementary physics that suffers none of these drawbacks. Our method is conceptually simple and produces useful, elegant, and easily readable maps. We illustrate the method with applications to the results of the 2000 U.S. presidential election, lung cancer cases in the State of New York, and the geographical distribution of stories appearing in the news. Suppose we wish to represent on a map some data concerning, to take the most common example, the human population. For instance, we might wish to show votes in an election, incidence of a disease, number of cars, televisions, or phones in use, numbers of people falling in one group or another of the population, by age or income, or any of very many other variables of statistical, medical, or demographic interest. The typical course under such circumstances would be to choose one of the standard projections for the area of interest and plot the data on it with some color code or similar representation. Such maps, however, can be misleading. A plot of disease incidence, for example, will inevitably show high incidence in cities and low incidence in rural areas, solely because more people live in cities.The obvious cure for this problem is to plot a fractional measure rather than raw incidence data; we plot some measure of the number of cases per capita, binned in segments small enough to give good spatial resolution but large enough to give reliable sampling. This method has its own problems, however, because it discards all information about where most of the cases are occurring. One case per thousand people means something entirely different in Sydney from what it means in Siberia.What we would like is some representation of the data that factors out variations in the population density but, at the same time, shows how many cases are occurring in each region. It appears at first that these two goals are irreconcilable, but this is not the case. On a normal area-preserving or approximately area-preserving projection, such as a Robinson projection or an equal-area conic projection, they are indeed irreconcilable. However, if we can construct a projection in which areas on the map are proportional not to areas on the ground but...

Gridded cartograms as a method for visualising earthquake risk at the global scale

“…The diffusion-based method for producing density-equalising maps published by Gastner and Newman (2004) describes one of the most substantial recent advances in the computer-generation of contiguous cartograms. Now over a decade old, this invention was dubbed as 'one small step for two men, one giant leap for mapping' by Dorling (2006, p. 35).…”

“…Design concepts for contiguous area cartograms are often particularly close to the wider field of other geographical map projections, because the transformation of the real geographical shape of an area stands in the centre of interest, and the relative geographical location remains preserved in some form. The implementation of algorithms that facilitate the digital creation of computer-generated contiguous cartograms has been demonstrated by a number of approaches since the first concepts were created at the GIS-pioneering institution of Harvard (Dougenik, Chrisman, & Niemeyer, 1985;Edelsbrunner & Waupotitsch, 1997;Gastner & Newman, 2004;Gusein-Zade & Tikunov, 1993;Kocmoud & House, 1998;Tobler, 1986;Wolf, 2005).…”

Mapping election results in proportional electoral systems

“…If you believe that every individual's circumstance is equally important in the spatial makeup of a society, then you should use area cartograms in mapping census data (pp. 171-172). they are coupled with elections from the early algorithmic beginnings -for example, the physical accretion models by Hunter and Young (1968) were inspired by cartograms of British general election results drawn in the Times Newspaper in 1964 and 1966. Visualisation of election results is regarded as a suitable use case for applying cartograms and several authors have done so to demonstrate their algorithms, including Dougenik, Chrisman, and Niemeyer (1985) and Gastner and Newman (2004). The visual parameters of contiguous cartograms were criticised by Roth, Woodruff, and Johnson (2010), who also proposed an alternative method based on using transparency as a visual variable, again using elections data for demonstration purposes.…”

“…A subtype called a contiguous aerial cartogram (sometimes also a density-equalizing map) transforms the base map so that the size of each enumeration unit corresponds to the value of selected attributes. For the purpose of this study, I applied the algorithm developed by Gastner and Newman (2004) to transform the base map in order to reflect the distribution of registered voters.…”

User-Centric Time-Distance Representation of Road Networks

“…While effective for geographic information professionals, these representations do not appear to be sufficiently intuitive to aid the average user in everyday decision making, as suggested by the fact that their use in maps for lay-people is uncommon. Conversely, cartograms [7,8,9] have been successful at representing areal data, giving promise to a similar representation of time-distance. To date, the attempts to integrate time with a geographical view of space either were aimed at professionals, or required considerable manual design intervention.…”