Separation Number and Generalized Clustering Coefficient in Small World Networks based on String Formalism

TL;DR

This paper reformulates string formalism for small world networks using adjacency matrices, introduces generalized clustering coefficients, and explores the relation between separation number, cycle structures, and clustering in small world networks.

Contribution

It presents a new reformulation of string formalism with adjacency matrices and applies it to analyze six degrees of separation in small world networks.

Findings

Effective analysis of separation number and cycle structures in small world networks.

Power law relation between Milgram key quantity and clustering coefficients.

Contrast in properties between small world and scale free networks.

Abstract

We reformulated the string formalism given by Aoyama, using an adjacent matrix of a network and introduced a series of generalized clustering coefficients based on it. Furthermore we numerically evaluated Milgram condition proposed by their article in order to explore $q$-$th$ degrees of separation in scale free networks. In this article, we apply the reformulation to small world networks and numerically evaluate Milgram condition, especially the separation number of small world networks and its relation to cycle structures are discussed. Considering the number of non-zero elements of an adjacent matrix, the average path length and Milgram condition, we show that the formalism proposed by us is effective to analyze the six degrees of separation, especially effective for analyzing the relation between the separation number and cycle structures in a network. By this analysis of small world networks, it proves that a sort of power low holds between $M_n$, which is a key quantity in Milgram condition, and the generalized clustering coefficients. This property in small world networks stands in contrast to that of scale free networks.