The Annals of Applied Probability

Structure of large random hypergraphs

R. W. R. Darling and J. R. Norris

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Abstract

The theme of this paper is the derivation of analytic formulae for certain large combinatorial structures. The formulae are obtained via fluid limits of pure jump-type Markov processes, established under simple conditions on the Laplace transforms of their Lévy kernels. Furthermore, a related Gaussian approximation allows us to describe the randomness which may persist in the limit when certain parameters take critical values. Our method is quite general, but is applied here to vertex identifiability in random hypergraphs. A vertex $v$ is identifiable in $n$ steps if there is a hyperedge containing $v$ all of whose other vertices are identifiable in fewer steps. We say that a hyperedge is identifiable if every one of its vertices is identifiable. Our analytic formulae describe the asymptotics of the number of identifiable vertices and the number of identifiable hyperedges for a Poisson $(β)$ random hypergraph $Λ$ on a set $V$ of $N$ vertices, in the limit as $N→∞$. Here $β$ is a formal power series with nonnegative coefficients $β_0,β_1,…,$ and $(Λ(A))_{A⊆V}$ are independent Poisson random variables such that $Λ(A)$, the number of hyperedges on $A$, has mean $Nβ_j/\pmatrix{{N}\cr{j}}$ whenever $|A|=j$.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 1A (2005), 125-152.

Dates
First available in Project Euclid: 28 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1106922324

Digital Object Identifier
doi:10.1214/105051604000000567

Mathematical Reviews number (MathSciNet)
MR2115039

Zentralblatt MATH identifier
1062.05132

Subjects
Primary: 05C65: Hypergraphs
Secondary: 60J75: Jump processes 05C80: Random graphs [See also 60B20]

Keywords
Hypergraph component cluster Markov process random graph

Citation

Darling, R. W. R.; Norris, J. R. Structure of large random hypergraphs. Ann. Appl. Probab. 15 (2005), no. 1A, 125--152. doi:10.1214/105051604000000567. https://projecteuclid.org/euclid.aoap/1106922324


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The Annals of Applied Probability

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