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. 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.
Original languageEnglish
Article number1612.02668
Number of pages27
JournalarXiv
Issue number1612.02668
Publication statusPublished - 8 Dec 2016

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Configuration
Model
Moment Conditions
Community Structure
Excursion
Scaling Limit
Graph Model
Criticality
Random Graphs
Brownian motion
Scaling
Moment
Converge
Community

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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. 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.",
keywords = "math.PR",
author = "{van der Hofstad}, R.W. and {van Leeuwaarden}, J.S.H. and C. Stegehuis",
year = "2016",
month = "12",
day = "8",
language = "English",
journal = "arXiv",
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}

Mesoscopic scales in hierarchical configuration models. / van der Hofstad, R.W.; van Leeuwaarden, J.S.H.; Stegehuis, C.

In: arXiv, No. 1612.02668, 1612.02668, 08.12.2016.

Research output: Contribution to journalArticleAcademic

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AU - van der Hofstad, R.W.

AU - van Leeuwaarden, J.S.H.

AU - Stegehuis, C.

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N2 - 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.

AB - 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.

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