$p$-th Clustering coefficients $C_{p}$ and Adjacent Matrix for Networks: Formulation based on String

Abstract

The phenomenon of six degrees of separation is an old but interesting problem. The considerations of the clustering coefficient reflecting triangular structures and its extension to square one to six degrees of separation have been made\cite{Newm21}. Recently, Aoyama\cite{Aoyama} has given some considerations to this problem in networks without loops, using a sort of general formalism, "string formalism". In this article, we describe relations between the string formulation proposed by Aoyama and an adjacent matrix. Thus we provided a reformulation of the string formulation proposed by \cite{Aoyama} to analyze networks. According to it, we introduced a series of generalized $q$-$th$ clustering coefficients. The available rules between diagrams of graphs and formulae are also given based on the formulation. Next we apply the formulation to some subjects in order to mainly check consistency with former studies. By evaluating the clustering coefficient for typical networks studied well earlier, we confirm a validity of our formulation. Lastly we applied it to the subject of two degrees of separation.