  The phenomenon of six degrees of separation is an old but attractive subject.
The deep understanding has been uncovered yet, especially how closed paths
included in a network affect six degrees of separation are an important subject
left yet. For it, some researches have been made\cite{Newm21}, \cite{Aoyama}.
  Recently we have develop a formalism \cite{Toyota3},\cite{Toyota4} to explore
the subject based on the string formalism developed by Aoyama\cite{Aoyama}. The
formalism can systematically investigate the effect of closed paths, especially
generalized clustering coefficient $C_{(p)}$ introduced in \cite{Toyota4}, on
six degrees of separation. In this article, we analyze general $q$-th degrees
of separation by using the formalism developed by us. So we find that the scale
free network with exponent $\gamma=3$ just display six degrees of separation.
Furthermore we drive a phenomenological relation between the separation number
$q$ and $C_{(p)}$ that has crucial information on circle structures in
networks.
