Generalized relationships between characteristic path length, efficiency, clustering coefficients, and density

Abstract

Graph theoretic properties such as 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 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 both real-world data and computer simulations, we present asymptotic linear relationships between the characteristic path length, global efficiency, and graph density, and also between the clustering coefficient and local efficiency. In the current literature, these properties are often presented as independent metrics; however, we show in this paper that they are inextricably linked.