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Random networks grown by fusing edges via urns

Published online by Cambridge University Press:  03 November 2022

Kiran R. Bhutani ,
Ravi Kalpathy Open the ORCID record for Ravi Kalpathy [Opens in a new window]  and
Hosam Mahmoud
Kiran R. Bhutani
Affiliation:
Department of Mathematics, The Catholic University of America, Washington, DC, USA
Ravi Kalpathy*
Affiliation:
Department of Mathematics, The Catholic University of America, Washington, DC, USA
Hosam Mahmoud
Affiliation:
Department of Statistics, The George Washington University, Washington, DC, USA
*
*Corresponding author. Email: kalpathy@cua.edu
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Abstract

Many classic networks grow by hooking small components via vertices. We introduce a class of networks that grows by fusing the edges of a small graph to an edge chosen uniformly at random from the network. For this random edge-hooking network, we study the local degree profile, that is, the evolution of the average degree of a vertex over time. For a special subclass, we further determine the exact distribution and an asymptotic gamma-type distribution. We also study the “core,” which consists of the well-anchored edges that experience fusing. A central limit theorem emerges for the size of the core.

At the end, we look at an alternative model of randomness attained by preferential hooking, favoring edges that experience more fusing. Under preferential hooking, the core still follows a Gaussian law but with different parameters. Throughout, Pólya urns are systematically used as a method of proof.

Keywords

networkrandom graphdegree profilePólya urnlimit lawpreferential attachmentphase transitionStirling numberasymptotic analysispolyacenes
Type
Research Article
Information
Network Science , Volume 10 , Issue 4 , December 2022 , pp. 347 - 360
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

Action Editor: Ulrik Brandes

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