The interplay between network transitivity and community structure

TL;DR

This paper explores the theoretical relationship between network transitivity and community structure in the geometric block model, revealing phase transitions and non-monotonic behaviors of clustering coefficients.

Contribution

It derives the limits of clustering coefficients in the GBM, showing phase transitions and differences between balanced and unbalanced communities.

Findings

Limits of clustering coefficients exhibit phase transitions.

Clustering coefficients do not have a monotonic relationship with community strength.

In balanced GBM, the limit decreases then increases as community strength varies.

Abstract

Recent empirical observations suggest that network transitivity is highly correlated with community structure in many real-world networks. In this paper, we theoretically investigate this relationship by deriving the limits of the global and average clustering coefficients for the geometric block model (GBM). Both limits exhibit a phase transition; specifically, the functional forms of the limit functions differ between the weak and strong community structure strength regimes. For a GBM with balanced communities, the limits of the global and average clustering coefficients are identical, whereas these limits differ for unbalanced communities. In general, the clustering coefficients do not exhibit a monotonic relationship with community structure strength. Particularly, for a balanced GBM where the within-community edge probability is a constant multiple of the between-community edge probability, the limit decreases from $3/4$ to $3/5$ and subsequently increases toward an asymptotic upper bound of $3/4$ as the multiple grows from one. A similar pattern is observed for the global clustering coefficient in unbalanced settings, where both limits exhibit an explicit dependence on community size.