Small-world network

In mathematics and related fields, a small-world network is a specific kind of network (to be more precise a special kind of a complex network) in which the distribution of connectivity is not confined to a certain scale, and where every node can be reached from every other by a small number of hops or steps. It is a generalisation of the small-world phenomenon, as in the phenomenon of meeting a stranger who we find is linked by a mutual acquaintance.

The small-world phenomenon applies to social networks. Duncan J. Watts and Steven Strogatz (1998) have identified it as a general feature of certain networks and propose that a similar phenomenon can occur in any network.

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

They state that if the clustering coefficient is significantly higher than would be expected for a random network, and the mean shortest-path length is lower than would be expected for a regular network, then the network is a small world. The small-world phenomenon can be used as an example: 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 in the whole world is possibly very short.

Properties of small-world networks
By virtue of the above definition, small-world networks will inevitably have high representation of cliques, and subgraphs that are a few edges shy of being cliques, i.e. small-world networks will have sub-networks that are characterized by the presence of connections between almost any two nodes within them. This follows from the requirement of a high cluster coefficient. Secondly, most pairs of nodes will be connected by at least one short path. This follows from the requirement that the mean-shortest path length be small.

Additionally, there are several properties that are commonly associated with small-world networks even though they are not required for that classification. Typically there is an over abundance of hubs - nodes in the network with a high number of connections (known as high degree). These hubs serve as the common connections mediating the short path lengths between other edges. By analogy, the small-world network of airline flights has a small mean-path length (i.e. between any two cities you are likely to have to take three or fewer flights) because many flights are routed through hub cities.

This property is often analyzed by considering the fraction of nodes in the network that have a particular number of connections going into them (the degree distribution of the network). Networks with a greater than expected number of hubs will have a greater fraction of nodes with high degree, and consequently the degree distribution will be enriched at high degree values. This is known colloquially as a fat-tailed distribution. Specifically, if a small-world network has a degree-distribution which can be fit with a power law distribution, it is taken as a sign that the network is small-world. Power law distributions have fat tails when compared to exponential distributions characteristic of random networks. These networks are known as scale-free networks.