On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F19%3A00507737" target="_blank" >RIV/67985840:_____/19:00507737 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.4418/2019.74.1.8" target="_blank" >http://dx.doi.org/10.4418/2019.74.1.8</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4418/2019.74.1.8" target="_blank" >10.4418/2019.74.1.8</a>

Result language
angličtina
Original language name
On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs
Original language description
Suppose that you have n colours and m mutually independent dice, each of which has r sides. Each dice lands on any of its sides with equal probability. You may colour the sides of each die in any way you wish, but there is one restriction: you are not allowed to use the same colour more than once on the sides of a die. Any other colouring is allowed. Let X be the number of different colours that you see after rolling the dice. How should you colour the sides of the dice in order to maximize the Shannon entropy of X? In this article we investigate this question. It is shown that the entropy of X is at most 1/2 log(n/2 + 16 + 1/2) log(πe) and that the bound is tight, up to a constant additive factor, in the case of there being equally many coins and colours. Our proof employs the differential entropy bound on discrete entropy, along with a lower bound on the entropy of binomial random variables whose outcome is conditioned to be an even integer. We conjecture that the entropy is maximized when the colours are distributed over the sides of the dice as evenly as possible.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GJ18-01472Y" target="_blank" >GJ18-01472Y: Graph limits and inhomogeneous random graphs</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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