Mesoscopic scales in hierarchical configuration models

TL;DR

This paper investigates the mesoscopic properties of hierarchical configuration models, revealing how community structures influence critical component sizes and percolation behavior, with results aligning with classical configuration models under certain conditions.

Contribution

It introduces analysis of critical component sizes and percolation in hierarchical configuration models, extending understanding of mesoscopic network scaling behaviors.

Findings

Critical component sizes scale as $O(N^{2/3})$ at criticality.

Rescaled component sizes converge to Brownian excursions.

Conditions on community sizes determine similarity to classical configuration models.

Abstract

To understand mesoscopic scaling in networks, we study the hierarchical configuration model (HCM), a random graph model with community structure. The connections between the communities are formed as in a configuration model. We study the component sizes of the hierarchical configuration model at criticality when the inter-community degrees have a finite third moment. We find the conditions on the community sizes such that the critical component sizes of the HCM behave similarly as in the configuration model. Furthermore, we study critical bond percolation on the HCM. We show that the ordered components of a critical HCM on $N$ vertices are of sizes $O(N^{2/3})$. More specifically, the rescaled component sizes converge to the excursions of a Brownian motion with parabolic drift, as for the scaling limit for the configuration model under a finite third moment condition.