Creating a Small World Graph

Graphs with Intermediate Alpha

We now ask what happens as alpha varies between these two extremes. As Watts did, we fix the size of the society and the average number of connections per agent (at N=200 and k=10) and connect the society according to several values of alpha (1). For each alpha, we measure the characteristic path length and the clustering coefficient for the networks that are formed. Since the connections are probabilistic, for each alpha we repeat the experiment 80 times and average the resulting values.

Below, we have a graph of our results. The measures have been scaled so that both can appear on the same graph for comparison.

For low values of alpha, both path length and clustering are high, and slightly increasing with alpha. For high values of alpha, both measures are low. There is an transitionary period in the middle in which the measures decrease sharply.

What is critical, however, is that the path length makes this transition sooner than the clustering coefficient. Thus, there is a range for which path length is at a low value, while the clustering coefficient remains high. In this range, graphs satisfy the small world property.

¹Watts used values of N=1000, k=10, but our results are qualitatively identical.

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