The interplay between network transitivity and community structure in the geometric block model

Abstract

Recent empirical observations suggest that network transitivity is highly correlated with community structure in many real‐world networks. In this article, 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 to and subsequently increases toward an asymptotic upper bound of 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. Our simulation studies confirm these theoretical results.