Mesoscopic scales in hierarchical configuration models

Remco van der Hofstad; Johan S.H. van Leeuwaarden; Clara Stegehuis

doi:10.1016/j.spa.2018.02.006

Mesoscopic scales in hierarchical configuration models

Remco van der Hofstad, Johan S.H. van Leeuwaarden, Clara Stegehuis

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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 language	English
Pages (from-to)	4246-4276
Number of pages	31
Journal	Stochastic Processes and their Applications
Volume	128
Issue number	12
DOIs	https://doi.org/10.1016/j.spa.2018.02.006
Publication status	Published - 1 Dec 2018

Keywords

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

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10.1016/j.spa.2018.02.006

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@article{d47c07743a3c472fb5f2c7f53652df10,

title = "Mesoscopic scales in hierarchical configuration models",

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.",

keywords = "Bond percolation, Brownian excursions, Community structure, Configuration model, Critical behavior, Random graphs",

author = "{van der Hofstad}, Remco and {van Leeuwaarden}, {Johan S.H.} and Clara Stegehuis",

year = "2018",

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day = "1",

doi = "10.1016/j.spa.2018.02.006",

language = "English",

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T1 - Mesoscopic scales in hierarchical configuration models

AU - van der Hofstad, Remco

AU - van Leeuwaarden, Johan S.H.

AU - Stegehuis, Clara

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Bond percolation

KW - Brownian excursions

KW - Community structure

KW - Configuration model

KW - Critical behavior

KW - Random graphs

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DO - 10.1016/j.spa.2018.02.006

M3 - Article

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VL - 128

SP - 4246

EP - 4276

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 12

ER -

Mesoscopic scales in hierarchical configuration models

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