Mean clustering coefficients: the role of isolated nodes and leafs on clustering measures for small-world networks

TL;DR

This paper proposes a new way to calculate clustering coefficients in small-world networks that accounts for isolated nodes and leaves, revealing that traditional methods often underestimate clustering and can misclassify network properties.

Contribution

It introduces a formula incorporating leaf and isolated node proportions to improve clustering coefficient estimates, significantly affecting network classification.

Findings

Traditional clustering measures underestimate neighborhood connectivity in sparse networks.

Applying the new formula can change network classification from non-small-world to small-world.

The alternative disconnectedness measure D is less affected by leafs and isolated nodes.

Abstract

Many networks exhibit the small-world property of the neighborhood connectivity being higher than in comparable random networks. However, the standard measure of local neighborhood clustering is typically not defined if a node has one or no neighbors. In such cases, local clustering has traditionally been set to zero and this value influenced the global clustering coefficient. Such a procedure leads to underestimation of the neighborhood clustering in sparse networks. We propose to include $\theta$ as the proportion of leafs and isolated nodes to estimate the contribution of these cases and provide a formula for estimating a clustering coefficient excluding these cases from the Watts and Strogatz (1998 Nature 393 440-2) definition of the clustering coefficient. Excluding leafs and isolated nodes leads to values which are up to 140% higher than the traditional values for the observed networks indicating that neighborhood connectivity is normally underestimated. We find that the definition of the clustering coefficient has a major effect when comparing different networks. For metabolic networks of 43 organisms, relations changed for 58% of the comparisons when a different definition was applied. We also show that the definition influences small-world features and that the classification can change from non-small-world to small-world network. We discuss the use of an alternative measure, disconnectedness D, which is less influenced by leafs and isolated nodes.