The evolution of subcritical Achlioptas processes

Random Structures and Algorithms

Volumn 47, Issue 1, 2015, Pages 174-203

  Riordan, Oliver ^a   Warnke, Lutz ^b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Achlioptas processes; Branching processes; Random graphs; Size rules; Susceptibility

Indexed keywords

EID: 84932195662     PISSN: 10429832     EISSN: 10982418     Source Type: Journal    
DOI: 10.1002/rsa.20530     Document Type: Article

References (22)

1. D. Achlioptas, R.M. D'Souza, and J. Spencer, Explosive percolation in random networks, Science 323 (2009), 1453-1455.
2. M. Aizenman, D.J. Barsky, Sharpness of the phase transition in percolation models, Comm Math Phys 108 (1987), 489-526.
3. S. Bhamidi, A. Budhiraja, and X. Wang, Aggregation models with limited choice and the multiplicative coalescent, Random Struct Algorithms, to appear.
4. T. Bohman, A. Frieze, Avoiding a giant component Random Struct Algorithms 19 (2001), 75-85.
5. T. Bohman, A. Frieze, M. Krivelevich, P.-S. Loh, and B. Sudakov, Ramsey games with giants, Random Struct Algorithms 38 (2011), 1-32.
6. T. Bohman and D. Kravitz, Creating a giant component Combin Probab Comput 15 (2006), 489-511.
7. B. Bollobas, S. Janson, and O. Riordan, The phase transition in inhomogeneous random graphs, Random Struct Algorithms 31 (2007) 3-122.
8. B. Bollobás, Random graphs, 2nd edition, Cambridge University Press, Cambridge, 2001.
9. B. Bollobás, O. Riordan, Random graphs and branching processes, In Handbook of large-scale random networks, Bolyai Soc Math Stud 18 (2009), 15-115.
10. S. Janson, Networking-Smoothly does it, Science 333 (2011), 298-299.
11. S. Janson, D.E. Knuth, T. Łuczak, and B. Pittel, The birth of the giant component, Random Struct Algorithms 4 (1993), 231-358.
12. S. Janson and J. Spencer, Phase transitions for modified Erdős-Rényi processes, Ark Math 50 (2012) 305-329.
13. M. Kang, W. Perkins and, J. Spencer, The Bohman-Frieze process near criticality, Random Struct Algorithms, 43 (2013), 221-250.
14. C. McDiarmid, On the method of bounded differences, In Surveys in Combinatorics (Norwich, 1989), J. Siemons (Editor), London Math. Soc. Lecture Note Ser., vol. 141, Cambridge University Press, Cambridge, 1989, 148-188.
15. M.V. Men'shikov, Coincidence of critical points in percolation problems, Dokl Akad Nauk SSSR 288 (1986), 1308-1311.
16. O. Riordan and L. Warnke, Explosive percolation is continuous, Science 333 (2011), 322-324.
17. O. Riordan and L. Warnke, Achlioptas processes are not always self-averaging, Physical Review E 86 (2012), 011129.
18. O. Riordan and L. Warnke, Achlioptas process phase transitions are continuous, Ann Appl Probab 22 (2012), 1450-1464.
19. O. Riordan and L. Warnke, Convergence of Achlioptas processes via differential equations with unique solutions, Preprint (2013), arXiv:1111.6179.
20. J. Spencer and N.C. Wormald, Birth control for giants, Combinatorica 27 (2007), 587-628.
21. N.C. Wormald, Differential equations for random processes and random graphs, Ann Appl Probab 5 (1995), 1217-1235.
22. N.C. Wormald, The differential equation method for random graph processes and greedy algorithms, In Lectures on approximation and randomized algorithms, M. Karoński and H.J. Prömel (Editors), PWN, Warsaw, 1999, 73-155.