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Isn't that actually an example for graph centrality (betweenness)? https://en.wikipedia.org/wiki/Centrality
Yes - one of the classical ways of computing betweeness centrality is to calculate the number of shortest paths passing through each edge. In this case however, he's only computing the shortest paths to/from a single vertex.
This is actually really useful - you have a map of 'what is the fastest way home from anywhere in the city?' On moving to a new city, getting a map like this and memorising a few of the arteries would help loads with familiarization.
This isn't exactly what the OP has, what you are looking for is a transit time heat map.

http://www.transitheatmap.com/index.html

http://project.wnyc.org/transit-time/#40.72280,-73.95464,12,...

If you are busing... Sure the argument can be made that this doesn't take into account light-timers for walking, but if you're greedily selecting for shortest distance, this is exactly what you want when walking.
If you are busing...

Transit heat maps are generally used for autos and buses, but could just as easily be made for walking. You'd just code the average wait time for timers into the heat map algorithm.

Nitpick: I don't think the world flights to Paris map is correct. It shows flights from Noumea, New Caledonia going through Australia. The shorter route is through Seoul.
Very interesting. How are the roads weighted in this? Meaning is there any travel time involved? Or is it just length? It would be interesting to play with different weighting and see if it changes.
Should be time in all cases (if default settings of the routing engines are used)
This is awesome. I wanted to something like this for a long time but never got around to it.
The roads to rome is indeed not open source although the tools are: http://roadstorome.moovellab.com/about

I was also playing around with this and GraphHopper as well: https://graphhopper.com/blog/2016/01/19/alternative-roads-to...

Using some simple modification of the MiniUIGraph tool which you can find here: https://gist.github.com/karussell/768e828a01f71ac7f46c

BTW: If you have questions related to GraphHopper feel free to ask them :)

BTW2: Funny how this visualization side-project got more stars in a few hours then one of the used routing engines in 4 years. Bad marketing ;) ?