The Annals of Applied Probability

Structure of large random hypergraphs

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

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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

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

First available in Project Euclid: 28 January 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Hypergraph component cluster Markov process random graph


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.

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  • Achlioptas, D. (2001). Lower bounds for random 3-SAT via differential equations. Theoret. Comput. Sci. 265 159–185.
  • Achlioptas, D., Kirousis, L. M., Kranakis, E. and Krizanc, D. (2001). Rigorous results for random $(2+p)$-SAT. Theoret. Comput. Sci. 265 109–129.
  • Bollobás, B. (1985). Random Graphs. Academic Press, London.
  • Coppersmith, D., Gamarnik, D., Hajiaghayi, M. and Sorkin, G. B. (2003). Random MAX SAT, random MAX CUT, and their phase transitions. In Proceedings of the Fourteenth Annual ACM–SIAM Symposium on Discrete Algorithms 364–373. ACM Press, New York.
  • Darling, R. W. R., Levin, D. A. and Norris, J. R. (2004). Continuous and discontinuous phase transitions in hypergraph processes. Random Structures Algorithms 24 379–419.
  • Duchet, P. (1995). Hypergraphs. In Handbook of Combinatorics (R. L. Graham, M. Grötschel and L. Lovász, eds.) 1 381–432. North-Holland, Amsterdam.
  • Erdös, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.
  • Ethier, S. N. and Kurtz, T. K. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
  • Karoński, M. and Łuczak, T. (1996). Random hypergraphs. Bolyai Soc. Math. Stud. 2 283–293.
  • Kordecki, W. (1985). On the connectedness of random hypergraphs. Comment. Math. Prace Mat. 25 265–283.
  • Norris, J. R. (2000). Cluster coagulation. Comm. Math. Phys. 209 407–435.
  • Schmidt-Pruzan, J. and Shamir, E. (1985). Component structure in the evolution of random hypergraphs. Combinatorica 5 81–94.
  • Wilson, D. B. (2002). On the critical exponents of random k-SAT. Random Structures Algorithms 21 182–195.
  • Wormald, N. C. (1995). Differential equations for random processes and random graphs. Ann. Appl. Probab. 5 1217–1235.