# Local Limit Theorems for the Giant Component of Random Hypergraphs

• Michael Behrisch
• Amin Coja-Oghlan
• Mihyun Kang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4627)

## Abstract

Let H d (n,p) signify a random d-uniform hypergraph with n vertices in which each of the $${n}\choose{d}$$ possible edges is present with probability p = p(n) independently, and let H d (n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. We establish a local limit theorem for the number of vertices and edges in the largest component of H d (n,p) in the regime, thereby determining the joint distribution of these parameters precisely. As an application, we derive an asymptotic formula for the probability that H d (n,m) is connected, thus obtaining a formula for the asymptotic number of connected hypergraphs with a given number of vertices and edges. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.

## Keywords

Joint Distribution Connected Graph Random Graph Large Component Random Structure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Andriamampianina, T., Ravelomanana, V.: Enumeration of connected uniform hypergraphs. In: Proceedings of FPSAC (2005)Google Scholar
2. 2.
Barraez, D., Boucheron, S., Fernandez de la Vega, W.: On the fluctuations of the giant component. Combinatorics, Probability and Computing 9, 287–304 (2000)
3. 3.
Barbour, A.D., Karonski, M., Rucinski, A.: A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47, 125–145 (1989)
4. 4.
Behrisch, M., Coja-Oghlan, A., Kang, M.: The order of the giant component of random hypergraphs (preprint, 2007), available at http://arxiv.org/abs/0706.0496
5. 5.
Behrisch, M., Coja-Oghlan, A., Kang, M.: Local limit theorems and the number of connected hypergraphs (preprint, 2007), available at http://arxiv.org/abs/0706.0497
6. 6.
Bender, E.A., Canfield, E.R., McKay, B.D.: The asymptotic number of labeled connected graphs with a given number of vertices and edges. Random Structures and Algorithms 1, 127–169 (1990)
7. 7.
Bender, E.A., Canfield, E.R., McKay, B.D.: Asymptotic properties of labeled connected graphs. Random Structures and Algorithms 3, 183–202 (1992)
8. 8.
Bollobás, B.: Random graphs, 2nd edn. Cambridge University Press, Cambridge (2001)
9. 9.
Coja-Oghlan, A., Moore, C., Sanwalani, V.: Counting connected graphs and hypergraphs via the probabilistic method. In: Random Structures and Algorithms (to appear)Google Scholar
10. 10.
Coppersmith, D., Gamarnik, D., Hajiaghayi, M., Sorkin, G.B.: Random MAX SAT, random MAX CUT, and their phase transitions. Random Structures and Algorithms 24, 502–545 (2004)
11. 11.
Erdös, P., Rënyi, A.: On random graphs I. Publicationes Mathematicae Debrecen 5, 290–297 (1959)
12. 12.
Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960)
13. 13.
van der Hofstad, R., Spencer, J.: Counting connected graphs asymptotically. European Journal on Combinatorics 27, 1294–1320 (2006)
14. 14.
Janson, S.: The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph. Random Structures and Algorithms 7, 337–355 (1995)
15. 15.
Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley, Chichester (2000)
16. 16.
Karoński, M., Łuczak, T.: The number of connected sparsely edged uniform hypergraphs. Discrete Math. 171, 153–168 (1997)
17. 17.
Karoński, M., Łuczak, T.: The phase transition in a random hypergraph. J. Comput. Appl. Math. 142, 125–135 (2002)
18. 18.
Łuczak, T.: On the number of sparse connected graphs. Random Structures and Algorithms 1, 171–173 (1990)
19. 19.
Luczak, M., Łuczak, T.: The phase transition in the cluster-scaled model of a random graph. Random Structures and Algorithms 28, 215–246 (2006)
20. 20.
O’Connell, N.: Some large deviation results for sparse random graphs. Prob. Th. Relat. Fields 110, 277–285 (1998)
21. 21.
Pittel, B.: On tree census and the giant component in sparse random graphs. Random Structures and Algorithms 1, 311–342 (1990)
22. 22.
Pittel, B., Wormald, N.C.: Asymptotic enumeration of sparse graphs with a minimum degree constraint. J. Combinatorial Theory, Series A 101, 249–263 (2003)
23. 23.
Pittel, B., Wormald, N.C.: Counting connected graphs inside out. J. Combin. Theory, Series B 93, 127–172 (2005)
24. 24.
Ravelomanana, V., Rijamamy, A.L.: Creation and growth of components in a random hypergraph process. In: Chen, D.Z., Lee, D.T. (eds.) COCOON 2006. LNCS, vol. 4112, pp. 350–359. Springer, Heidelberg (2006)
25. 25.
Schmidt-Pruzan, J., Shamir, E.: Component structure in the evolution of random hypergraphs. Combinatorica 5, 81–94 (1985)
26. 26.
Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent variables. In: Proc. 6th Berkeley Symposium on Mathematical Statistics and Probability, pp. 583–602 (1970)Google Scholar
27. 27.
Stepanov, V.E.: On the probability of connectedness of a random graph g m(t). Theory Prob. Appl. 15, 55–67 (1970)

## Authors and Affiliations

• Michael Behrisch
• 1
• Amin Coja-Oghlan
• 2
• Mihyun Kang
• 1
1. 1.Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 BerlinGermany
2. 2.Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA 15213USA