First passage percolation on random graphs with finite mean degrees

Annals of Applied Probability

Volumn 20, Issue 5, 2010, Pages 1907-1965

  Bhamidi, Shankar ^a   Van Der Hofstad, Remco ^b   Hooghiemstra, Gerard ^c  

^a The University of North Carolina at Chapel Hill   (United States)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

^c DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Central limit theorem; Coupling to continuous time branching processes; First passage percolation; Flows; Hopcount; Random graph; Universality

Indexed keywords

EID: 77957744298     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/09-AAP666     Document Type: Article

References (37)

1. ADDARIO-BERRY, L., BROUTIN, N. and LUGOSI, G. (2010). The longest minimum-weight path in a complete graph. Preprint. Combin. Probab. Comput. 19 1-19.
2. ATHREYA, K. B. and KARLIN, S. (1967). Limit theorems for the split times of branching processes. J. Math. Mech. 17 257-277. MR0216592
3. ATHREYA, K. B. and NEY, P. E. (2004). Branching Processes. Dover, Mineola, NY. MR2047480
4. BERTOIN, J. and SIDORAVICIUS, V. (2009). The structure of typical clusters in large sparse random configurations. J. Stat. Phys. 135 87-105. MR2505727
5. BHAMIDI, S. (2008). First passage percolation on locally treelike networks. I. Dense random graphs. J. Math. Phys. 49 125218. MR2484349
6. BHAMIDI, S., VAN DER HOFSTAD, R. and HOOGHIEMSTRA, G. (2009). Extreme value theory, Poisson-Dirichlet distributions and first passage percolation on random networks. Preprint.
7. BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge. MR1015093
8. BOLLOBÁS, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge. MR1864966
9. BRAUNSTEIN, L. A., BULDYREV, S. V., COHEN, R., HAVLIN, S. and STANLEY, H. E. (2003). Optimal paths in disordered complex networks. Phys. Rev. Lett. 91 168701.
10. BÜHLER, W. J. (1971). Generations and degree of relationship in supercritical Markov branching processes. Z. Wahrsch. Verw. Gebiete 18 141-152. MR0295444
11. CHUNG, F. and LU, L. (2003). The average distance in a random graph with given expected degrees. Internet Math. 1 91-113. MR2076728
12. CHUNG, F. and LU, L. (2006). Complex Graphs and Networks. CBMS Regional Conference Series in Mathematics 107. Amer. Math. Soc., Providence, RI. MR2248695
13. COHEN, R. and HAVLIN, S. (2003). Scale-free networks are ultrasmall. Phys. Rev. Lett. 90 058701.
14. DEIJFEN, M., VAN DEN ESKER, H., VAN DER HOFSTAD, R. and HOOGHIEMSTRA, G. (2009). A preferential attachment model with random initial degrees. Ark. Mat. 47 41-72. MR2480915
15. DING, J., KIM, J. H., LUBETZKY, E. and PERES, Y. (2009). Diameters in supercritical random graphs via first passage percolation. Preprint. Available at http://arxiv.org/abs/0906.1840.
16. DOMMERS, S., VAN DER HOFSTAD, R. and HOOGHIEMSTRA, G. (2010). Diameters in preferential attachment graphs. J. Stat. Phys. 139 72-107.
17. DURRETT, R. (2007). Random Graph Dynamics. Cambridge Univ. Press, Cambridge. MR2271734
18. FERNHOLZ, D. and RAMACHANDRAN, V. (2007). The diameter of sparse random graphs. Random Structures Algorithms 31 482-516. MR2362640
19. GUT, A. (2005). Probability: A Graduate Course. Springer, New York. MR2125120
20. HAMMERSLEY, J. M. and WELSH, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Bernouilli-Bayes-Laplace: Anniversary Volume 61-110. Springer, New York. MR0198576
21. HOWARD, C. D. (2004). Models of first-passage percolation. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 125-173. Springer, Berlin. MR2023652
22. JANSON, S. (1999). One, two and three times log n/n for paths in a complete graph with random weights. Combin. Probab. Comput. 8 347-361. MR1723648
23. JANSON, S. (2009). The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205-225. MR2497380
24. JANSON, S. and LUCZAK, M. J. (2009). A new approach to the giant component problem. Random Structures Algorithms 34 197-216. MR2490288
25. JANSON, S., LUCZAK, T. and RUCINSKI, A. (2000). Random Graphs. Wiley, New York. MR1782847
26. MOLLOY, M. and REED, B. (1995). A critical point for random graphs with a given degree sequence. In Proceedings of the Sixth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science, "Random Graphs'93" (Poznań, 1993) 6 161-179. Wiley, New York. MR1370952
27. MOLLOY, M. and REED, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput. 7 295-305. MR1664335
28. NORROS, I. and REITTU, H. (2006). On a conditionally Poissonian graph process. Adv. in Appl. Probab. 38 59-75. MR2213964
29. REITTU, H. and NORROS, I. (2004). On the power law random graph model of massive data networks. Performance Evaluation 55 3-23.
30. SMYTHE, R. T. and WIERMAN, J. C. (1978). First-Passage Percolation on the Square Lattice. Lecture Notes in Math. 671. Springer, Berlin.
31. VAN DEN ESKER, H., VAN DER HOFSTAD, R., HOOGHIEMSTRA, G. and ZNAMENSKI, D. (2005). Distances in random graphs with infinite mean degrees. Extremes 8 111-141. MR2275914 (Pubitemid 44325340)
32. VAN DER HOFSTAD, R., HOOGHIEMSTRA, G. and VAN MIEGHEM, P. (2001). First-passage percolation on the random graph. Probab. Engrg. Inform. Sci. 15 225-237. MR1828576 (Pubitemid 33677294)
33. VAN DER HOFSTAD, R., HOOGHIEMSTRA, G. and VAN MIEGHEM, P. (2002). The flooding time in random graphs. Extremes 5 111-129. MR1965974
34. VAN DER HOFSTAD, R., HOOGHIEMSTRA, G. and VAN MIEGHEM, P. (2005). Distances in random graphs with finite variance degrees. Random Structures Algorithms 27 76-123. MR2150017 (Pubitemid 41229042)
35. VAN DER HOFSTAD, R., HOOGHIEMSTRA, G. and ZNAMENSKI, D. (2007). Distances in random graphs with finite mean and infinite variance degrees. Electron. J. Probab. 12 703-766 (electronic). MR2318408
36. VAN MIEGHEM, P., HOOGHIEMSTRA, G. and VAN DER HOFSTAD, R. (2000). A scaling law for the hopcount. Technical report 2000125, Delft Univ. Technology. Available at http://www.nas.ewi.tudelft.nl/people/Piet/.
37. WÄSTLUND, J. (2006). Random assignment and shortest path problems. In Proceedings of the Fourth Colloquium on Mathematics and Computer Science, Institut. Élie Cartan, Nancy France. DMTCS Proc. AG 31-38.