The phase transition and connectedness in uniformly grown random graphs

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 3243, Issue , 2004, Pages 1-18

  Bollobás, Béla ^a,b   Riordan, Oliver ^b,c  

^a University of Memphis   (United States)

^b TRINITY COLLEGE   (United Kingdom)

^c UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

CRITICAL VALUE; DENSITY PARAMETERS; GIANT COMPONENTS; INFINITE GRAPH; RANDOM GRAPHS; REAL-WORLD NETWORKS;

GRAPH THEORY;

EID: 35048828306     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-540-30216-2_1     Document Type: Article

References (21)

1. A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science 286 (1999), 509-512.
2. B. Bollobás, The evolution of random graphs. Trans. Amer. Math. Soc. 286 (1984), 257-274.
3. B. Bollobás, S. Janson and O. Riordan, The phase transition in the uniformly grown random graph has infinite order, to appear in Random Structures and Algorithms.
4. B. Bollobás and O. Riordan, Mathematical results on scale-free random graphs, in Handbook of Graphs and Networks, Stefan Bornholdt and Heinz Georg Schuster (eds.), Wiley-VCH, Weinheim (2002), 1-34.
5. B. Bollobás and O. Riordan, Robustness and vulnerability of scale-free random graphs, Internet Mathematics 1 (2003), 1-35.
6. B. Bollobás and O. Riordan, The diameter of a scale-free random graph, Combinatorica 24 (2004), 5-34.
7. B. Bollobás and O. Riordan, Slow emergence of the giant component in the growing m-out graph, submitted. (Preprint available from http://www.dpmms.cam.ac.uk/~omr10/.)
8. B. Bollobás, O. Riordan, J. Spencer and G. Tusnády, The degree sequence of a scale-free random graph process, Random Structures and Algorithms 18 (2001), 279-290.
9. D.S. Callaway, J.E. Hopcroft, J.M. Kleinberg, M.E.J. Newman and S.H. Strogatz, Are randomly grown graphs really random?, Phys. Rev. E 64 (2001), 041902.
10. S.N. Dorogovtsev, J.F.F. Mendes and A.N. Samukhin, Anomalous percolation properties of growing networks, Phys. Rev. E 64 (2001), 066110.
11. R. Durrett, Rigorous result for the CHKNS random graph model, Proceedings, Discrete Random Walks 2003, Cyril Banderier and Christian Krattenthaler, Eds. Discrete Mathematics and Theoretical Computer Science AC (2003), 95-104. http://dmtcs.loria.fr/proceedings/
12. R. Durrett and H. Kesten, The critical parameter for connectedness of some random graphs, in A Tribute to Paul Erdös (A. Baker, B. Bollobás and A. Hajnal, eds), Cambridge Univ. Press, Cambridge, 1990, pp 161-176,
13. P. Erdös, and A. Rényi, On random graphs I., Publications Mathematicae Debrecen 5 (1959), 290-297.
14. P. Erdös and A. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17-61.
15. E.N. Gilbert, Random graphs, Annals of Mathematical Statistics 30 (1959), 1141-1144.
16. S. Janson, D.E. Knuth, T. Luczak and B. Pittel, The birth of the giant component, Random Structures and Algorithms 3 (1993), 233-358.
17. S. Kalikow and B. Weiss, When are random graphs connected?, Israel J. Math. 62 (1988), 257-268.
18. T. Luczak, Component behavior near the critical point of the random graph process, Random Structures and Algorithms 1 (1990), 287-310.
19. O. Riordan, The small giant component in scale-free random graphs, to appear in Combinatorics, Probability and Computing.
20. L.A. Shepp, Connectedness of certain random graphs. Israel J. Math. 67 (1989), 23-33.
21. Y. Zhang, A power law for connectedness of some random graphs at the critical point, Random Structures and Algorithms 2 (1991), 101-119.