Small World Networks

by Yaniv Altshuler

As mentioned in the previous post, social trading networks are networks that connect people. As a result, these networks tend to follow the way people make social interactions. One of the most thrilling and important revelations of complex networks research in the past decade is the fact that in a social network – every two strangers can be connected by a usually very short chain of mutual friends. This principle is called “The Small World Phenomenon”, and characterizes many of complex networks we encounter in our daily life.

As an example, let us observe David, a 26 y.o software engineer (and an amateur financial trader in his spare time) from Boston. David has strong social connections with his family and some of his friends from work. He also has social connections to some friends of his wife, Sarah, as well as some of the people who studied with him at MIT. Now let us observe Michael, a 42 y.o architect from South Africa. In theory, Michael and David reside in two very distinct social environments, having no past history or mutual background. However, during his architecture studies in Italy, Michael had become close friends with Natalie – a British designer whose husband is now one the main customers of Sarah’s home based small marketing agency. This way, David and Michael can be “socially connected” through a chain of three “hops”, that comprised of links that were created when David was just a small kid. The same can be shown for Maya – an Israeli chemist that grew up in the Kibbutz David’s mother had volunteered in at the 80es, or even Sourav – the son of an Indian guru that was the spiritual mentor of Greg, Sarah’s ex-boyfriend on his trip to India.

A similar principle can be applied to other types of social networks, and specifically – social trading networks. Based on this phenomenon we can foretell that the “social distance” between any two traders in our network would be fairly small, and hence – a piece of information that is passed along the network through social interactions is expected to proliferate efficiently, reaching most of the network’s members relatively quickly. Notice that this is not the case had we assumed that the network does not follow this pattern (for example, try to imagine a ring-like network, where every trader is interacting with exactly two other members. You can easily see how a knowledge that is generated at some part of the network would have to travel through at least half the number of the network’s members in order to reach the farthest trader). The applications of this notion can be used to predict the portion of network’s members that would be exposed to some information that is introduced to the network by some of its members, the time it would take this information to reach them (the information “freshness”), and many other useful predictive properties.

Formally, we define small-world networks as network in which the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is:

On the specific applications of Small World Networks – next time.