Relationships Between Characteristic Path Length, Efficiency, Clustering Coefficients, and Graph Density

TL;DR

This paper explores the theoretical relationships between key graph metrics—clustering coefficient, path length, efficiency—and graph density, providing a unified understanding relevant to biological and social networks.

Contribution

It introduces a mathematical framework linking these graph properties through graph density, filling a gap in theoretical comparisons of these metrics.

Findings

Metrics can be expressed in terms of graph density

Unified relationships enhance understanding of network structure

Applicable to neuroscience and social network analysis

Abstract

The graph theoretic properties of the clustering coefficient, characteristic (or average) path length, global and local efficiency, provide valuable information regarding the structure of a graph. These four properties have applications to biological and social networks and have dominated much of the the literature in these fields. While much work has done in applied settings, there has yet to be a mathematical comparison of these metrics from a theoretical standpoint. Motivated by networks appearing in neuroscience, we show in this paper that these properties can be linked together using a single property - graph density.