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The Kruskal-Katona Theorem and a Characterization of System Signatures

Part of: Combinatorial probability Stochastic systems and control

Published online by Cambridge University Press:  30 January 2018

Alessandro D'Andrea  and
Luca De Sanctis
Alessandro D'Andrea*
Affiliation:
Università degli Studi di Roma ‘La Sapienza’
Luca De Sanctis*
Affiliation:
Università degli Studi di Roma ‘La Sapienza’
*
∗ Postal address: Dipartimento di Matematica, Università degli Studi di Roma ‘La Sapienza’, Piazzale Aldo Moro 5, Rome, 00185, Italy.
∗∗ Postal address: Dipartimento di Matematica, Università degli Studi di Roma ‘La Sapienza’, Piazzale Aldo Moro 5, Rome, 00185, Italy. Email address: luca.desanctis@gmail.com
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Abstract

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We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

Keywords

Signaturereliabilitysimplicial complexes

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 93E03: Stochastic systems, general
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 508 - 518
Copyright
© Applied Probability Trust 

References

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