I was teaching a course on graphs and networks a few years ago, soon after Facebook became popular, and I mentioned that the graph of connections between the students in the room, defined by their Facebook "friends" would be an interesting one. Within 24 hours, several students had asked to be friends; I said that I felt it would compromise me to be linked to some but not all the class. However, we were able to discuss aspects of graph properties that related to Facebook.
I have a love of practical uses of graphs and networks, so was delighted to find a new one. It is the "Map of the World Drawn Entirely Using Facebook Connections" (found here among other places). Based on a large number of connections in Facebook, lines are drawn between them, and the colour of the line relates to the number of connections. Many people have links within their home city, still more are linked within their home country, and then there are international ones.
There are several fascinating aspects to the map. National coasts are very learly defined. Look at Florida, for instance. There are numerous links within the state, and these so outnumber the links to neighbouring states, that the coast of southern USA is clear. The same is true of the west of Britain. There are not many links between Wales and the west of England, so that the Bristol Channel is clearly marked. It is hard to see the boundaries between most countries, though Spain is clearly separated from France and Portugal, and (hardly surprisingly) the boundaries of Israel are well marked.
The author comments on the emptiness of China, Brazil and Russia. There are empty spaces in the world's deserts as well -- in Australia, the Sahara, and Central Asia.
The more I look at the map, the more I find of interest. A wonderful illustration of the power of mathematics and Operational Research.