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Large Cliques in a Power-Law Random Graph

Part of: Combinatorial probability Graph theory

Published online by Cambridge University Press:  14 July 2016

Svante Janson ,
Tomasz Łuczak  and
Ilkka Norros
Svante Janson*
Affiliation:
Uppsala University
Tomasz Łuczak*
Affiliation:
Adam Mickiewicz University
Ilkka Norros*
Affiliation:
VTT Technical Research Centre of Finland
*
∗Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. Email address: svante.janson@math.uu.se
∗∗Postal address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland. Email address: tomasz@amu.edu.pl
∗∗∗Postal address: VTT Technical Research Centre of Finland, PO Box 1000, 02044 VTT, Finland. Email address: ilkka.norros@vtt.fi
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Abstract

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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.

Keywords

Power-law random graphcliquegreedy algorithm

MSC classification

Secondary: 05C80: Random graphs 05C69: Dominating sets, independent sets, cliques 60C05: Combinatorial probability
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1124 - 1135
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Bianconi, G. and Marsili, M. (2006). Emergence of large cliques in random scale-free networks. Europhys. Lett. 74}, 740–746.Google Scholar
[2] Bianconi, G. and Marsili, M. (2006). Number of cliques in random scale-free network ensembles. Physica D 224}, 1–6.Google Scholar
[3] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
[4] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation}. Cambridge University Press.CrossRefGoogle Scholar
[5] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31}, 3–122.Google Scholar
[6] Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Statist. Phys.} 124, 1377–1397.CrossRefGoogle Scholar
[7] Chung, F. and Lu, L. (2002). Connected components in random graphs with given expected degree sequences. Ann. Combinatorics 6, 125–145.Google Scholar
[8] Durrett, R. (2007). Random Graphs Dynamics}. Cambridge University Press.Google Scholar
[9] Feller, W. (1971). An Introduction to Probability Theory and Its Applications}, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
[10] Frieze, A. and McDiarmid, C. (1997). Algorithmic theory of random graphs. Random Structures Algorithms 10}, 5–42.3.0.CO;2-Z>CrossRefGoogle Scholar
[11] Janson, S. (2008). Asymptotic equivalence and contiguity of some random graphs. Preprint. Available at http://arxiv.org/abs/0802.1637v1.Google Scholar
[12] Janson, S., Łuczak, T. and Ruciński, A. (2000). Random Graphs}. Wiley-Interscience, New York.CrossRefGoogle Scholar
[13] Norros, I. (2009). A mean-field approach to some Internet-like random networks. In Proc. 21st Internat. Teletraffic Congress (Paris, September 2009), North-Holland, Amsterdam, pp. 1–8.Google Scholar
[14] Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. Appl. Prob. 38}, 59–75.Google Scholar
[15] Riordan, O. (2005). The small giant component in scale-free random graphs. Combinatorics Prob. Comput. 14}, 897–938.Google Scholar
[16] Van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2006). Distances in random graphs with infinite mean degrees. Extremes 8, 111–141.Google Scholar