An Empirical Study of the Relation Between Community Structure and Transitivity

TL;DR

This study empirically explores the relationship between community structure strength and network transitivity, revealing that they influence each other and are affected by community size distribution, with some surprising findings about their correlation.

Contribution

It provides the first empirical analysis of how community structure strength and transitivity are related in real-world networks using random models.

Findings

Transitivity increases with community structure strength.

Community size distribution affects the relationship.

Weak community structures can still exhibit low transitivity.

Abstract

One of the most prominent properties in real-world networks is the presence of a community structure, i.e. dense and loosely interconnected groups of nodes called communities. In an attempt to better understand this concept, we study the relationship between the strength of the community structure and the network transitivity (or clustering coefficient). Although intuitively appealing, this analysis was not performed before. We adopt an approach based on random models to empirically study how one property varies depending on the other. It turns out the transitivity increases with the community structure strength, and is also affected by the distribution of the community sizes. Furthermore, increasing the transitivity also results in a stronger community structure. More surprisingly, if a very weak community structure causes almost zero transitivity, the opposite is not true and a network with a close to zero transitivity can still have a clearly defined community structure. Further analytical work is necessary to characterize the exact nature of the identified relationship.