A small-world network is a generalisation of the small-world phenomenon, as in the phenomenon where we suddenly burst out "It's a small world" when we meet a stranger who we find is linked by a mutual acquaintance.

The small-world phenomenon applies to social networks. Watts and Strogatz (1998) argue that a similar phenomenon can apply to any network.

They propose that we can measure whether a network is a small world or not according to two graph measurements of the network: clustering coefficient and mean-shortest path length.

If the clustering coefficient is significantly higher than would be expected for a random network, and the mean shortest-path length is lower, then the network is a small world. We can see how this works for the small-world phenomenon: most people have a relatively small circle of friends who generally all know each other (highly clustered), but the shortest-path length from one person to any other is possibly very short.

References

• Watts, D. J. (1999). Small Worlds: The Dynamics of Networks Between Order and Randomness. Princeton University Press. ISBN 0691005419.

• Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of 'small-world' networks. Nature 393, 440--442 (4 June 1998).