Open Access
February 2005 Structure of large random hypergraphs
R. W. R. Darling, J. R. Norris
Ann. Appl. Probab. 15(1A): 125-152 (February 2005). DOI: 10.1214/105051604000000567


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 $|A|=j$.


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R. W. R. Darling. J. R. Norris. "Structure of large random hypergraphs." Ann. Appl. Probab. 15 (1A) 125 - 152, February 2005.


Published: February 2005
First available in Project Euclid: 28 January 2005

zbMATH: 1062.05132
MathSciNet: MR2115039
Digital Object Identifier: 10.1214/105051604000000567

Primary: 05C65
Secondary: 05C80 , 60J75

Keywords: cluster , component , Hypergraph , Markov process , random graph

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.15 • No. 1A • February 2005
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