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Local Limit Theorems for the Giant Component of Random Hypergraphs†

Published online by Cambridge University Press:  13 February 2014

MICHAEL BEHRISCH ,
AMIN COJA-OGHLAN  and
MIHYUN KANG
MICHAEL BEHRISCH
Affiliation:
Institute of Transportation Systems, German Aerospace Center, Rutherfordstrasse 2, 12489 Berlin, Germany (e-mail: michael.behrisch@dlr.de)
AMIN COJA-OGHLAN
Affiliation:
Goethe University, Mathematics Institute, 60054 Frankfurt am Main, Germany (e-mail: acoghlan@math.uni-frankfurt.de)
MIHYUN KANG
Affiliation:
TU Graz, Institut für Optimierung und Diskrete Mathematik (Math B), Steyrergasse 30, 8010 Graz, Austria (e-mail: kang@math.tugraz.at)
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Abstract

Let Hd(n,p) signify a random d-uniform hypergraph with n vertices in which each of the $\binom{n}{d}$ possible edges is present with probability p=p(n) independently, and let Hd(n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of Hd(n,p) and Hd(n,m) in the regime $(d-1)\binom{n-1}{d-1}p>1+\varepsilon$, resp. d(d−1)m/n>1+ϵ, where ϵ>0 is arbitrarily small but fixed as n → ∞. The proofs are based on a purely probabilistic approach.

Keywords

Primary 05C80Secondary 05C65
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 3 , May 2014 , pp. 331 - 366
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

†

An extended abstract version of this work appeared in the proceedings of RANDOM 2007, Vol. 4627 of Lecture Notes in Computer Science, Springer, pp. 341–352.

References

[1]Alon, N. and Spencer, J. (2000) The Probabilistic Method, second edition, Wiley.Google Scholar
[2]Andriamampianina, T. and Ravelomanana, V. (2005) Enumeration of connected uniform hypergraphs. In Proc. FPSAC 2005.Google Scholar
[3]Barbour, A. D., Karoński, M. and Ruciński, A. (1989) A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47 125–145.Google Scholar
[4]Barraez, D., Boucheron, S. and Fernandez de la Vega, W. (2000) On the fluctuations of the giant component. Combin. Probab. Comput. 9 287–304.Google Scholar
[5]Behrisch, M. (2007) Stochastical models for networks in the life sciences. PhD thesis, Humboldt Universität zu Berlin.Google Scholar
[6]Behrisch, M., Coja-Oghlan, A. and Kang, M. (2010) The order of the giant component of random hypergraphs. Random Struct. Alg. 36 149–184.Google Scholar
[7]Behrisch, M., Coja-Oghlan, A. and Kang, M. (2014) The asymptotic probability that a random d-uniform hypergraph is connected. Combin. Probab. Comput. doi:10.1017/S0963548314000029.Google Scholar
[8]Bender, E. A., Canfield, E. R. and McKay, B. D. (1990) The asymptotic number of labeled connected graphs with a given number of vertices and edges. Random Struct. Alg. 1 127–169.Google Scholar
[9]Bender, E. A., Canfield, E. R. and McKay, B. D. (1992) Asymptotic properties of labeled connected graphs. Random Struct. Alg. 3 183–202.Google Scholar
[10]Bollobás, B. (2001) Random Graphs, second edition, Cambridge University Press.CrossRefGoogle Scholar
[11]Bollobás, B. and Riordan, O. (2012) Asymptotic normality of the size of the giant component via a random walk. J. Combin. Theory Ser. B 102 53–61.Google Scholar
[12]Bollobás, B. and Riordan, O. (2012) Asymptotic normality of the size of the giant component in a random hypergraph. Random Struct. Alg. 41 441–450.Google Scholar
[13]Coja-Oghlan, A., Moore, C. and Sanwalani, V. (2007) Counting connected graphs and hypergraphs via the probabilistic method. Random Struct. Alg. 31 288–329.Google Scholar
[14]Coppersmith, D., Gamarnik, D., Hajiaghayi, M. and Sorkin, G. B. (2004) Random MAX SAT, random MAX CUT, and their phase transitions. Random Struct. Alg. 24 502–545.Google Scholar
[15]Erdős, P. and Rényi, A. (1959) On random graphs I. Publicationes Math. Debrecen 5 290–297.Google Scholar
[16]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5 17–61.Google Scholar
[17]van der Hofstad, R. and Spencer, J. (2006) Counting connected graphs asymptotically. European J. Combin. 27 1294–1320.Google Scholar
[18]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.Google Scholar
[19]Karoński, M. and Łuczak, T. (1997) The number of connected sparsely edged uniform hypergraphs. Discrete Math. 171 153–168.Google Scholar
[20]Karoński, M. and Łuczak, T. (2002) The phase transition in a random hypergraph. J. Comput. Appl. Math. 142 125–135.Google Scholar
[21]Łuczak, T. (1990) On the number of sparse connected graphs. Random Struct. Alg. 1 171–173.Google Scholar
[22]O'Connell, N. (1998) Some large deviation results for sparse random graphs. Probab. Theory Rel. Fields 110 277–285.Google Scholar
[23]Pittel, B. (1990) On tree census and the giant component in sparse random graphs. Random Struct. Alg. 1 311–342.Google Scholar
[24]Pittel, B. and Wormald, N. C. (2003) Asymptotic enumeration of sparse graphs with a minimum degree constraint. J. Combin. Theory Ser. A 101 249–263.Google Scholar
[25]Pittel, B. and Wormald, N. C. (2005) Counting connected graphs inside out. J. Combin. Theory Ser. B 93 127–172.Google Scholar
[26]Ravelomanana, V. and Rijamamy, A. L. (2006) Creation and growth of components in a random hypergraph process. In Computing and Combinatorics, Vol. 4112 of Lecture Notes in Computer Science, Springer, pp. 350–359.CrossRefGoogle Scholar
[27]Rudin, W. (1987) Real and Complex Analysis, third edition, McGraw-Hill.Google Scholar
[28]Schmidt-Pruzan, J. and Shamir, E. (1985) Component structure in the evolution of random hypergraphs. Combinatorica 5 81–94.Google Scholar
[29]Stepanov, V. E. (1970) On the probability of connectedness of a random graph gm(t). Theory Probab. Appl. 15 55–67.Google Scholar