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The Entropy of Random-Free Graphons and Properties

Published online by Cambridge University Press:  16 May 2013

HAMED HATAMI  and
SERGUEI NORINE
HAMED HATAMI
Affiliation:
School of Computer Science, McGill University, Montreal, Canada (e-mail: hatami@cs.mcgill.ca)
SERGUEI NORINE
Affiliation:
Department of Mathematics & Statistics, McGill University, Montreal, Canada (e-mail: snorin@math.mcgill.ca)
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Abstract

Every graphon defines a random graph on any given number n of vertices. It was known that the graphon is random-free if and only if the entropy of this random graph is subquadratic. We prove that for random-free graphons, this entropy can grow as fast as any subquadratic function. However, if the graphon belongs to the closure of a random-free hereditary graph property, then the entropy is O(n log n). We also give a simple construction of a non-step-function random-free graphon for which this entropy is linear, refuting a conjecture of Janson.

Keywords

05C99
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 4 , July 2013 , pp. 517 - 526
Copyright
Copyright © Cambridge University Press 2013 

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