The Annals of Applied Probability

The diameter of weighted random graphs

Hamed Amini and Marc Lelarge

Full-text: Open access

Abstract

In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 3 (2015), 1686-1727.

Dates
First available in Project Euclid: 23 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1427124140

Digital Object Identifier
doi:10.1214/14-AAP1034

Mathematical Reviews number (MathSciNet)
MR3325285

Zentralblatt MATH identifier
1351.60009

Subjects
Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]
Secondary: 90B15: Network models, stochastic

Keywords
First-passage percolation weighted diameter random graphs

Citation

Amini, Hamed; Lelarge, Marc. The diameter of weighted random graphs. Ann. Appl. Probab. 25 (2015), no. 3, 1686--1727. doi:10.1214/14-AAP1034. https://projecteuclid.org/euclid.aoap/1427124140


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References

  • [1] Amini, H. (2011). Epidemics and percolation in random networks. Ph.D. thesis, ENS-INRIA. Available at http://www.di.ens.fr/~amini/Publication/Thesis.pdf.
  • [2] Amini, H., Draief, M. and Lelarge, M. (2013). Flooding in weighted sparse random graphs. SIAM J. Discrete Math. 27 1–26.
  • [3] Amini, H. and Lelarge, M. (2012). Upper deviations for split times of branching processes. J. Appl. Probab. 49 1134–1143.
  • [4] Amini, H. and Peres, Y. (2014). Shortest-weight paths in random regular graphs. SIAM J. Discrete Math. 28 656–672.
  • [5] Bender, E. A. and Canfield, E. R. (1978). The asymptotic number of labeled graphs with given degree sequences. J. Combin. Theory Ser. A 24 296–307.
  • [6] Bhamidi, S. (2008). First passage percolation on locally treelike networks. I. Dense random graphs. J. Math. Phys. 49 125218, 27.
  • [7] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). Extreme value theory, Poisson–Dirichlet distributions, and first passage percolation on random networks. Adv. in Appl. Probab. 42 706–738.
  • [8] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Probab. 20 1907–1965.
  • [9] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). First passage percolation on the Erdős–Rényi random graph. Combin. Probab. Comput. 20 683–707.
  • [10] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2012). Universality for first passage percolation on sparse random graphs. Preprint. Available at arXiv:1210.6839.
  • [11] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.
  • [12] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [13] Bollobás, B. and Fernandez de la Vega, W. (1982). The diameter of random regular graphs. Combinatorica 2 125–134.
  • [14] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
  • [15] Chung, F. and Lu, L. (2003). The average distance in a random graph with given expected degrees. Internet Math. 1 91–113.
  • [16] Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. (2010). Diameters in supercritical random graphs via first passage percolation. Combin. Probab. Comput. 19 729–751.
  • [17] Fernholz, D. and Ramachandran, V. (2007). The diameter of sparse random graphs. Random Structures Algorithms 31 482–516.
  • [18] Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete 66 335–366.
  • [19] Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35 683–692.
  • [20] 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.
  • [21] Janson, S. and Luczak, M. J. (2007). A simple solution to the $k$-core problem. Random Structures Algorithms 30 50–62.
  • [22] Janson, S. and Luczak, M. J. (2008). Asymptotic normality of the $k$-core in random graphs. Ann. Appl. Probab. 18 1085–1137.
  • [23] Janson, S. and Luczak, M. J. (2009). A new approach to the giant component problem. Random Structures Algorithms 34 197–216.
  • [24] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
  • [25] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [26] Kesten, H. (1986). Aspects of first passage percolation. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 125–264. Springer, Berlin.
  • [27] Klenke, A. and Mattner, L. (2010). Stochastic ordering of classical discrete distributions. Adv. in Appl. Probab. 42 392–410.
  • [28] Lelarge, M. (2012). Diffusion and cascading behavior in random networks. Games Econom. Behav. 75 752–775.
  • [29] 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.
  • [30] Riordan, O. and Wormald, N. (2010). The diameter of sparse random graphs. Combin. Probab. Comput. 19 835–926.
  • [31] 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.