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

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.

11 pages, 5 figures

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• Emergence of scaling in random networks
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• The clustering coefficient and community structure of bipartite networks
• $p$-th Clustering coefficients $C_p$ and Adjacent Matrix for Networks: Formulation based on String
• Some Considerations on Six Degrees of Separation from A Theoretical Point of View
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Cited by in corpus (4)

• Generalized Clustering Coefficients and Milgram Condition for q-th Degrees of Separation
• Effect of Closed Paths in Complex networks on Six Degrees of Separation and Disorder
• The Effect of Network-Topology to Propagation on Networks
• Separation Number and Generalized Clustering Coefficient in Small World Networks based on String Formalism