Large cliques in a power-law random graph

Svante Janson, Tomasz Luczak, Ilkka Norros

Research output: Contribution to journalArticleScientificpeer-review

15 Citations (Scopus)

Abstract

In this paper we study the size of the largest clique ω(G(n, α))in a random graph G(n, α) on n vertices which has power-law degree distribution with exponent α. We show that, for ‘flat’ degree sequences with α > 2, with high probability, the largest clique in G(n, α) is of a constant size, while, for the heavy tail distribution, when 0 <α< 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm with high probability finds in G(n, α) a large clique of size (1 −o(1))ω(G(n, α)) in polynomial time.
Original languageEnglish
Pages (from-to)1124-1135
Number of pages12
JournalJournal of Applied Probability
Volume47
Publication statusPublished - 2010
MoE publication typeA1 Journal article-refereed

Fingerprint

Clique
Random Graphs
Power Law
Heavy Tails
Degree Sequence
Power-law Distribution
Degree Distribution
Polynomial time
Exponent
Power law
Random graphs

Keywords

  • power-law random graph
  • clique
  • greedy algorithm

Cite this

Janson, S., Luczak, T., & Norros, I. (2010). Large cliques in a power-law random graph. Journal of Applied Probability, 47, 1124-1135.
Janson, Svante ; Luczak, Tomasz ; Norros, Ilkka. / Large cliques in a power-law random graph. In: Journal of Applied Probability. 2010 ; Vol. 47. pp. 1124-1135.
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Janson, S, Luczak, T & Norros, I 2010, 'Large cliques in a power-law random graph', Journal of Applied Probability, vol. 47, pp. 1124-1135.

Large cliques in a power-law random graph. / Janson, Svante; Luczak, Tomasz; Norros, Ilkka.

In: Journal of Applied Probability, Vol. 47, 2010, p. 1124-1135.

Research output: Contribution to journalArticleScientificpeer-review

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AB - In this paper we study the size of the largest clique ω(G(n, α))in a random graph G(n, α) on n vertices which has power-law degree distribution with exponent α. We show that, for ‘flat’ degree sequences with α > 2, with high probability, the largest clique in G(n, α) is of a constant size, while, for the heavy tail distribution, when 0 <α< 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm with high probability finds in G(n, α) a large clique of size (1 −o(1))ω(G(n, α)) in polynomial time.

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Janson S, Luczak T, Norros I. Large cliques in a power-law random graph. Journal of Applied Probability. 2010;47:1124-1135.