Chapter 3. Small World Graphs

Many networks in the real world, including social networks, have the “small world property”, which is that the average distance between nodes, measured in number of edges on the shortest path, is much smaller than expected.

In this chapter, I present Stanley Milgram’s famous Small World Experiment, which was the first demonstration of the small world property in a real social network. Then we’ll consider Watts-Strogatz graphs, which are intended as a model of small world graphs. I’ll replicate the experiment Watts and Strogatz performed and explain what it is intended to show.

Along the way, we’ll see two new graph algorithms: breadth-first search (BFS) and Dijkstra’s algorithm for computing the shortest path between nodes in a graph.

Stanley Milgram

Stanley Milgram was an American social psychologist who conducted two of the most famous experiments in social science, the Milgram experiment, which studied people’s obedience to authority (https://thinkcomplex.com/milgram) and the Small World Experiment, which studied the structure of social networks (https://thinkcomplex.com/small).

In the Small World Experiment, Milgram sent a package to several randomly-chosen people in Wichita, Kansas, with instructions asking them to forward an enclosed letter to a target person, identified by name and occupation, in Sharon, Massachusetts (which happens to be the town near Boston where I grew up). The subjects were told that they could mail the letter directly to ...

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