Some Considerations on Six Degrees of Separation from A Theoretical Point of View

In this article we discuss six degrees of separation, which has been proposed by Milgram, from a theoretical point of view. Simply if one has $k$ friends, the number $N$ of indirect friends goes up to $\sim k^d$ in $d$ degrees of separation. So it would easily come up to population of whole world. That, however, is unacceptable. Mainly because of nonzero clustering coefficient $C$, $N$ does not become $\sim k^d$. In this article, we first discuss relations between six degrees of separation and the clustering coefficient in the small world network proposed by Watt and Strogatz\cite{Watt1},\cite{Watt2}. Especially, conditions that $(N)>$ (population of U.S.A or of the whole world) arises in the WS model is explored by theoretical and numerical points of view. Secondly we introduce an index that represents velocity of propagation to the number of friends and obtain an analytical formula for it as a function of $C$, $K$, which is an average degree over all nodes, and some parameter $P$ concerned with network topology. Finally the index is calculated numerically to study the relation between $C$, $K$ and $P$ and $N$.