Skip to main content
×
×
Home

Largest Components in Random Hypergraphs

  • OLIVER COOLEY (a1), MIHYUN KANG (a1) and YURY PERSON (a2)
Abstract

In this paper we consider j-tuple-connected components in random k-uniform hypergraphs (the j-tuple-connectedness relation can be defined by letting two j-sets be connected if they lie in a common edge and considering the transitive closure; the case j = 1 corresponds to the common notion of vertex-connectedness). We show that the existence of a j-tuple-connected component containing Θ(nj) j-sets undergoes a phase transition and show that the threshold occurs at edge probability

$$\frac{(k-j)!}{\binom{k}{j}-1}n^{j-k}.$$
Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph.

Our main original contribution is a bounded degree lemma, which controls the structure of the component grown in the search process.

Copyright
Footnotes
Hide All

The first and third authors were supported by short visit grants 5639 and 5472, respectively, from the European Science Foundation (ESF) within the ‘Random Geometry of Large Interacting Systems and Statistical Physics’ (RGLIS) programme.

The second author is supported by Austrian Science Fund (FWF): P26826, W1230, Doctoral Programme ‘Discrete Mathematics’.

Footnotes
References
Hide All
[1] Aronshtam, L. and Linial, N. (2015) When does the top homology of a random simplicial complex vanish? Random Struct. Alg. 46 2635.
[2] Aronshtam, L. and Linial, N. (2016) The threshold for d-collapsibility in random complexes. Random Struct. Alg. 48 260269.
[3] Aronshtam, L., Linial, N., Łuczak, T. and Meshulam, R. (2013) Collapsibility and vanishing of top homology in random simplicial complexes. Discrete Comput. Geom. 49, no. 2, 317334.
[4] Behrisch, M., Coja-Oghlan, A. and Kang, M. (2010) The order of the giant component of random hypergraphs. Random Struct. Alg. 36 149184.
[5] Behrisch, M., Coja-Oghlan, A. and Kang, M. (2014) Local limit theorems for the giant component of random hypergraphs. Combin. Probab. Comput. 23 331366.
[6] Bohman, T. (2009) The triangle-free process. Adv. Math. 221 16531677.
[7] Bollobás, B. (1984) The evolution of random graphs. Trans. Amer. Math. Soc. 286 257274.
[8] Bollobás, B. (2001) Random Graphs, second edition, Cambridge University Press.
[9] Bollobás, B. and Riordan, O. (2012) Asymptotic normality of the size of the giant component in a random hypergraph. Random Struct. Alg. 41 441450.
[10] Cooley, O., Kang, M. and Koch, K. The size of the giant component in random hypergraphs. Random Struct. Alg., DOI: 10.1002/rsa.20761.
[11] Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 1761.
[12] Janson, S., Łuczak, T. and A. Ruciński, A. (2000) Random Graphs, Wiley.
[13] Karoński, M. and Łuczak, T. (2002) The phase transition in a random hypergraph. J. Comput. Appl. Math. 142 125135.
[14] Krivelevich, M. and Sudakov, B. (2013) The phase transition in random graphs: A simple proof. Random Struct. Alg. 43 131138.
[15] Linial, N. and Meshulam, R. (2006) Homological connectivity of random 2-complexes. Combinatorica 26 475487.
[16] Lu, L. and Peng, X. High-order phase transition in random hypergraphs. arXiv:1409.1174
[17] Łuczak, T. (1990) Component behavior near the critical point of the random graph process. Random Struct. Alg. 1 287310.
[18] Molloy, M. (2005) Cores in random hypergraphs and boolean formulas. Random Struct. Alg. 27 124135.
[19] Nachmias, A. and Peres, Y. (2010) The critical random graph, with martingales. Israel J. Math. 176 2941.
[20] Ravelomanana, and Rijamamy, (2006) Creation and growth of components in a random hypergraph process. In COCOON 2006: Computing and Combinatorics, Springer, pp. 350359.
[21] Schmidt-Pruzan, J. and E. Shamir, E. (1985) Component structure in the evolution of random hypergraphs. Combinatorica 5 8194.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed