Mesoscopic scales in hierarchical configuration models

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Abstract

To understand mesoscopic scaling in networks, we study the hierarchical configuration model (HCM), a random graph model with community structure. Connections between communities are formed as in a configuration model. We study the component sizes of HCM at criticality, and we study critical bond percolation. We find the conditions on the community sizes such that the critical component sizes of HCM behave similarly as in the configuration model. We show that the ordered components of a critical HCM on [Formula presented] vertices are [Formula presented]. More specifically, the rescaled component sizes converge to the excursions of a Brownian motion with parabolic drift.

Original languageEnglish
Pages (from-to)4246-4276
Number of pages31
JournalStochastic Processes and their Applications
Volume128
Issue number12
DOIs
Publication statusPublished - 1 Dec 2018

Keywords

  • Bond percolation
  • Brownian excursions
  • Community structure
  • Configuration model
  • Critical behavior
  • Random graphs

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