Space and time are favorite subjects of mine, since they are the root concepts for two of the most fundamental types of questions we can ask, where and when questions. I discussed three dimensions in detail in a previous post, so I am going to dive into the subject of cartograms and show why you should be careful about your two-dimensional thinking as well. I’ll give you a question to stick behind your ear before I begin: how do tiny island nations like Britain and Japan manage to dramatically influence the world, while huge continents like Africa and South America often don’t even register on the radar? Let me warn you right now, that’s a trick question.
We live on a sphere. Since we don’t really think effectively in 3d, we invented maps. You probably know that the common Mercator projection distorts the areas of countries, while roughly preserving their shapes. You might also know that there exists a projection called the Peter projection, or “equal area” projection, that corrects the area distortion at the expense of introducing a shape distortion. The latter is a favorite of super-liberals, since a side-by-side comparison of the two projections will usually show startling West-East and North-South biases, lending credence to claims that the Mercator projection is racist (simple example, Mexico is vastly bigger than Alaska, but looks the same size on the Mercator projection, India is almost the size of all Europe, but looks no bigger than Scandinavia). Personally, I found the Peter projection an intriguing curiosity when I first encountered it on the wall of the bathroom of a vegan co-op I was living in at the time, in Ann Arbor (yes, the place was a hippie cliche of sorts), but I somehow couldn’t see how the Mercator distortions really mattered that much.
I understood why I was under-whelmed by the Peter map when I encountered cartograms. The inevitable distortions introduced by flattening a sphere pale in comparison to the distortions introduced by the ways in which we use maps to represent other geospatial information. It turns out that when you are talking about things like population with the aid of maps, it is important that your maps be wildly inaccurate. Cartograms were invented by Michael Gastner and Mark Newman of the University of Michigan as a way to visualize the results of the 2004 US Presidential elections. In particular, they wanted to show why the race was so close even though maps of red vs. blue states were overwhelmingly red. To do this, they invented a clever diffusion algorithm that distorted geography in proportion to population density, so that the voting pattern in the election looked like this, by state (these graphics below are from Mark’s election 2004 page, used per his permissions on the page).
And if you apply the diffusion algorithm at the county level, with some color scaling, you get this much richer, fractal-like look at the political leanings of America.
These maps became so hugely popular that a nonprofit organization, Worldmapper (where you can find dozens of provocative maps and posters to annoy friends with), was born, and went slightly nuts with the method. Since the diffusion algorithm can be applied to any dataset that has geographic coordinates, it turns out to be a pretty powerful tool. Here are two maps that should help you answer the question you have stuck behind your ear. These are from Mark’s page of global cartograms constructed with various important geospatial datasets (again used per permissions as described on his page). Here is a cartogram of GDP by country:
And here is one by population
Check out the rest of the thought provoking maps, including the ones on AIDS and childhood mortality. Then ask yourself again, is the UK/Japan question possibly a trivial one, once you have the right visualization in your head? Like sphere squashing, cartogram generation is a mathematically under-constrained problem, so it probably needs taste and skill to use for honest visualizations, but done right, cartograms are quite eye-opening.
Challenge: Visualize Connections
I first learned about Mark’s work when I attended a talk by him about an earlier body of work for which he is famous, on complex network visualization (all that good stuff about scale-free and small world networks, Erdos numbers and the like), and started exchanging occasional emails with him on the subject. One of his contributions was coming up with really good ways to visualize abstract graphs, like social networks, which have no natural spatial coordinates associated with them. I am sort of surprised his two contributions to visualization haven’t yet come together (unless I missed some recent developments), so here is a challenge for you.
Putting abstract network visualization together with geographic distortion brings up an interesting question: how do you visualize geospatial graphs?
For example, consider the volume of phone, Internet and air traffic among all medium or larger sized cities in the world (appropriately defined). Assume that’s a good metric of economic connectedness and globalization — Bangalore is actually closer to San Francisco by this metric than Mysore or Monterey are to the former and latter respectively. Can you visualize this?
Such a visualization would be really useful for understanding the economic structure of the world, and the rate at which ‘globalization’ is progressing (a much better visual metaphor than that awful flat world metaphor — I’ll do a Friedman polemic some other time). For example, literal geographic proximity is important for some things — an earthquake in San Francisco would impact Monterey more than it would impact Bangalore. But a major financial crisis at, say, Google, might propagate faster to Bangalore than to Monterey. What is the right map for visualizing this sort of relational-geographic information? Is there a method analogous to the diffusion method that takes as input a connection-strength data set (2 locations per data point) rather than a point density data set (1 location per data point)?