$p$-th Clustering coefficients and $q$-$th$ degrees of separation based on String-Adjacent Formulation

TL;DR

This paper investigates how closed paths and clustering coefficients influence the degrees of separation in networks, using a novel string formalism to analyze scale-free networks and their circle structures.

Contribution

It introduces a formalism based on string adjacency to systematically study the effect of closed paths on degrees of separation, especially in scale-free networks.

Findings

Scale-free network with exponent γ=3 exhibits six degrees of separation.

A phenomenological relation between separation number q and clustering coefficient C_{(p)} is derived.

The formalism reveals crucial information about circle structures in networks.

Abstract

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.