Hierarchical Configuration Model

TL;DR

This paper introduces the hierarchical configuration model, a new class of random graphs with community structures, analyzing their connectivity, degree distribution, clustering, and percolation properties.

Contribution

It presents a novel hierarchical graph model combining configuration models and community graphs, with analytical results on structural properties and phase transitions.

Findings

Largest component size characterized

Degree distribution and clustering coefficient derived

Conditions for giant percolation cluster identified

Abstract

We introduce a class of random graphs with a community structure, which we call the hierarchical configuration model. On the inter-community level, the graph is a configuration model, and on the intra-community level, every vertex in the configuration model is replaced by a community: i.e., a small graph. These communities may have any shape, as long as they are connected. For these hierarchical graphs, we find the size of the largest component, the degree distribution and the clustering coefficient. Furthermore, we determine the conditions under which a giant percolation cluster exists, and find its size.