Recently I have discovered an online education system Coursera. While browsing through the available courses I noticed one named "Social and Economic Networks: Models and Analysis". Sadly it was already six weeks into the eight week program, so I was not able to formally complete it. Yet the knowledge I have obtained by viewing videos of this course enabled me to prepare couple of posts for this website.
Previously, I have already written about the Erdos-Renyi network formation model. While this time I'll write about the Watts-Strogatz network formation model ("small world" networks). The Barabasi-Albert model ("scale-free" networks) will be left for the next post.
Watts-Strogatz model and the properties of the small world networks
Let us recall that Erdos-Renyi networks were generated completely at random. In this sense the Watts-Strogatz model is almost exact opposite. During the first step we select a known network topology with \( N \) nodes. Note that in the applet below we choose the ring topology, while other topology are also possible (e.g., the node can be connected not to the two nearest nodes, but to four). Next we iterate trough all nodes in the topology and with probability \( p \) create an additional random edge, which does not exists at this time and isn't a loopback.
Note that in the initial deterministic, ring, topology has a large network diameter. This means that the largest distance between any two nodes is relatively large, actually it is proportional to \( N/2 \). While the newly created edges significantly decrease network diameter. Such network has an average path length proportional to \( \lnN \). These newly created edges is a perfectly simple example of how the shortcuts can be created in an actual social networks.
Thus this model is a simplest illustration possible for the "six degrees of separation" theory. This theory states that any two people on Earth are separated by 6 handshakes (i.e., common friends or friends of friends and etc.). Consequently two completely random people, who have almost nothing in common, can easily find each other despite of the huge size of the social networks due to the connections of their friends.
Interactive HTML5 applet
Below you can find a HTML 5 applet, which illustrates how the small world networks can be generated. In the applet you can select a number of nodes in the network, \( N \), and a probability to create a random edge, \( p \). Using mouse wheel you can zoom in and out of the picture. The initial structure is generated automatically each time you press "Restart" button. The random edges can be automatically added via the "Rewire" button (one attempt per second) and also manually via "Add edge" button.