Small World Networks and the Power Law

A Small World Network is model of interconnected nodes which consists of small clusters of densely connected nodes - with each node only having a small number of connections - which are linked by a much smaller number of nodes with vastly more connections. The result is a very robust network that has an optimum number of connections between any one point and another. Small world networks occur naturally in many different contexts from social networks to neural networks showing the innate strengths of their loose, organic structure and the efficiency for transfer of information. The six degrees of separation theory derives from the powerful bridging capability of small world networks.

The Power Law is distribution curve  - closely related to the Long Tail and the Pareto Principle/80-20 rule - that is manifest in many different contexts from economic distribution to the range of activity on forums between users. One unique characteristic of the power law distribution is that the mean average is unrepresentative of the group as a whole because of the huge variation exponential difference between the values at one end of the curve and the other. This creates a dynamic  where the items at the top end of the curve are fundamentally different in nature to those at the mode, this is because the amount of power they possess transforms them into a fundamentally different entity and therefore they occupy a distinct role.

The best way to illustrate how the dynamic functions is to show how the power law enables small world networks to form in social networks. The power law distribution applies to the amount of contacts each person has in a vast social network with the top 20% having vastly more than the lower 80%. Most people, those who have the mode number of contacts, would be formed in tightly connected clusters - because the selection of contacts isn’t large enough to be sustain looser contacts - with perhaps a couple of connections with more highly connected individuals. Those in the top 20% however have a vast number or contacts, most of which are those in the mode and therefore part of dense clusters. The resultant effect is that the clusters are linked together by the highly connected individuals who act as hubs through which the vast social network is binded together.