A new approach to the giant component problem

Random Structures and Algorithms

Volumn 34, Issue 2, 2009, Pages 197-216

  Janson, Svante ^a   Luczak, Malwina J ^b  

^a UPPSALA UNIVERSITY   (Sweden)

^b LONDON SCHOOL OF ECONOMICS   (United Kingdom)

Author keywords

Death process; Empirical distribution; Giant component; Random graph

Indexed keywords

EID: 62449296666     PISSN: 10429832     EISSN: 10982418     Source Type: Journal    
DOI: 10.1002/rsa.20231     Document Type: Article

References (27)

1. D. Aldous, Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812-854.
2. R. Arratia, A. D. Barbour, and S. Tavaré, Logarithmic combinatorial structures: A probabilistic approach, EMS, Zurich, 2003.
3. D. Barraez, S. Boucheron, and W. Fernandez de la Vega, On the fluctuations of the giant component, Combin Probab Comput 9 (2000), 287-304.
4. M. Behrisch, A. Coja-Oghlan, and M. Kang, Local limit theorems and the number of connected hypergraphs, Preprint, 2007. arXiv: 0706 . 0497vl.
5. B. Bollobás, Random graphs, 2nd edition, Cambridge University Press, Cambridge, 2001.
6. B. Bollobás, S. Janson, and O. Riordan, The phase transition in inhomogeneous random graphs, Random Struct Alg 31 (2007), 3-122.
7. C. Borgs, J. Chayes, R. van der Hofstad, G. Slade, and J. Spencer, Random subgraphs of finite graphs. III. The phase transition for the n-cube, Combinatorica 26 (2006), 395-410.
8. T. Britton, S. Janson, and A. Martin-Löf, Graphs with specified degree distributions, simple epidemics and local vaccination strategies, Adv Appl Probab 39 (2007), 922-948.
9. T. Britton, M. Deijfen, and A. Martin-Löf, Generating simple random graphs with prescribed degree distribution, J Statist Phys 124 (2006), 1377-1.397.
10. G. Grimmett and S. Janson, Random graphs with forbidden vertex degrees, Preprint, 2007. arXiv: 0712.0270vl.
11. A. Gut, Probability: A graduate course, Springer, New York, 2005.
12. R. van der Hofstad and M. J. Luczak, Random subgraphs of the 2D Hamming graph: the supercritical phase, Probab Theor Relat Fields, (in press) arXiv :0801.1607.
13. S. Janson, The probability that a random multigraph is simple, Preprint, 2006, arXiv:math.CO/0609802.
14. S. Janson, D. E. Knufh, T. Łuczak, and B. Pittel, The birth of the giant component, Random Struct Alg 3 (1993), 233-358.
15. S. Janson and M. Luczak, A simple solution to the k-core problem, Random Struct Alg 30 (2007), 50-62.
16. S. Janson and M. Luczak, Asymptotic normality of the k-core in random graphs, Ann Appl Probab 18 (2008), 1085-1137.
17. S. Janson, T. Łuczak. and A. Ruciński, Random graphs, Wiley, New York, 2000.
18. O. Kallenberg, Foundations of modem probability, 2nd edition, Springer, New York, 2002.
19. M. Kang and T. G. Seierstad, The critical phase for random graphs with a given degree sequence, Combin Probab Comput 17 (2008), 67-86.
20. M. Luczak and T. Łuczak, The phase transition in the cluster-scaled model of a random graph, Random Struct Alg 28 (2006), 215-246.
21. M. Luczak and J. Spencer, The second component in the supercritical 2D Hamming graph. Preprint, 2007. arXiv: 0801.1608.
22. B. D. McKay, Asymptotics for symmetric 0-1 matrices with prescribed row sums, Ars Combin 19A(1985), 15-25.
23. 1/2), Combinatorica 11 (1991), 369-382.
24. M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Struct Alg 6 (1995), 161-179.
25. M. Molloy and B. Reed, The size of the largest component of a random graph on a fixed degree sequence, Combin Probab Comput 7 (1998), 295-306.
26. A. Nachmias and Y. Peres, Critical percolation on random regular graphs. Preprint, 2007. arXiv:0707.2839v2.
27. B. Pittel, On tree census and the giant component in sparse random graphs, Random Struct Alg 1 (1990), 311-342.