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Assortativity and bidegree distributions on Bernoulli random graph superpositions

Published online by Cambridge University Press:  19 August 2021

Mindaugas Bloznelis ,
Joona Karjalainen Open the ORCID record for Joona Karjalainen [Opens in a new window]  and
Lasse Leskelä
Mindaugas Bloznelis
Affiliation:
Institute of Informatics, Vilnius University, Vilnius, Lithuania
Joona Karjalainen
Affiliation:
Department of Mathematics and Systems Analysis, School of Science, Aalto University, Espoo, Finland. E-mail: joona.karjalainen@aalto.fi
Lasse Leskelä
Affiliation:
Department of Mathematics and Systems Analysis, School of Science, Aalto University, Espoo, Finland. E-mail: joona.karjalainen@aalto.fi
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Abstract

A probabilistic generative network model with $n$ nodes and $m$ overlapping layers is obtained as a superposition of $m$ mutually independent Bernoulli random graphs of varying size and strength. When $n$ and $m$ are large and of the same order of magnitude, the model admits a sparse limiting regime with a tunable power-law degree distribution and nonvanishing clustering coefficient. In this article, we prove an asymptotic formula for the joint degree distribution of adjacent nodes. This yields a simple analytical formula for the model assortativity and opens up ways to analyze rank correlation coefficients suitable for random graphs with heavy-tailed degree distributions. We also study the effects of power laws on the asymptotic joint degree distributions.

Keywords

Bidegree distributionDegree correlationDegree–degree distributionEmpirical degree distributionErdős–Rényi graphJoint degree distributionPower lawRandom intersection graphStatistical network modelTransitivity
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 4 , October 2022 , pp. 1188 - 1213
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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