The Annals of Probability

Universality for first passage percolation on sparse random graphs

Shankar Bhamidi, Remco van der Hofstad, and Gerard Hooghiemstra

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider first passage percolation on the configuration model with $n$ vertices, and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform $X^{2}\log{X}$-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path namely the hopcount.

Writing $L_{n}$ for the weight of the optimal path, we show that $L_{n}-(\log{n})/\alpha_{n}$ converges to a limiting random variable, for some sequence $\alpha_{n}$. Furthermore, the hopcount satisfies a central limit theorem (CLT) with asymptotic mean and variance of order $\log{n}$. The sequence $\alpha_{n}$ and the norming constants for the CLT are expressible in terms of the parameters of an associated continuous-time branching process that describes the growth of neighborhoods around a uniformly chosen vertex in the random graph. The limit of $L_{n}-(\log{n})/\alpha_{n}$ equals the sum of the logarithm of the product of two independent martingale limits, and a Gumbel random variable. So far, for sparse random graph models, such results have only been shown for the special case where the edge weights have an exponential distribution, wherein the Markov property of this distribution plays a crucial role in the technical analysis of the problem.

The proofs in the paper rely on a refined coupling between shortest path trees and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination.

Article information

Ann. Probab., Volume 45, Number 4 (2017), 2568-2630.

Received: May 2014
Revised: September 2015
First available in Project Euclid: 11 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Central limit theorem continuous-time branching processes extreme value theory first passage percolation hopcount Malthusian rate of growth point process convergence Poisson point process stable-age distribution random graphs


Bhamidi, Shankar; van der Hofstad, Remco; Hooghiemstra, Gerard. Universality for first passage percolation on sparse random graphs. Ann. Probab. 45 (2017), no. 4, 2568--2630. doi:10.1214/16-AOP1120.

Export citation


  • [1] Aldous, D. and Lanoue, D. (2012). A lecture on the averaging process. Probab. Surv. 9 90–102.
  • [2] Aldous, D. J. (2013). When knowing early matters: Gossip, percolation and Nash equilibria. In Prokhorov and Contemporary Probability Theory. Springer Proc. Math. Stat. 33 3–27. Springer, Heidelberg.
  • [3] Amini, H., Draief, M. and Lelarge, M. (2013). Flooding in weighted sparse random graphs. SIAM J. Discrete Math. 27 1–26.
  • [4] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • [5] Baroni, E., van der Hofstad, R. and Komjáthy, J. (2015). Nonuniversality of weighted random graphs with infinite variance degree. J. Appl. Probab. 54 146–164.
  • [6] Bhamidi, S. (2008). First passage percolation on locally treelike networks. I. Dense random graphs. J. Math. Phys. 49 125218, 27.
  • [7] Bhamidi, S. and van der Hofstad, R. (2012). Weak disorder asymptotics in the stochastic mean-field model of distance. Ann. Appl. Probab. 22 29–69.
  • [8] 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.
  • [9] 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.
  • [10] 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.
  • [11] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2014). Universality for first passage percolation on sparse uniform and rank-1 random graphs. Preprint.
  • [12] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [13] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
  • [14] 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.
  • [15] Chatterjee, S. (2014). Superconcentration and Related Topics. Springer, Cham.
  • [16] Draief, M. and Massoulié, L. (2010). Epidemics and Rumours in Complex Networks. London Mathematical Society Lecture Note Series 369. Cambridge Univ. Press, Cambridge.
  • [17] Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth&Brooks/ Cole Advanced Books&Software, Pacific Grove, CA.
  • [18] van den Esker, H., van der Hofstad, R. and Hooghiemstra, G. (2008). Universality for the distance in finite variance random graphs. J. Stat. Phys. 133 169–202.
  • [19] Flaxman, A., Gamarnik, D. and Sorkin, G. (2011). First-passage percolation on a ladder graph, and the path cost in a VCG auction. Random Structures Algorithms 38 350–364.
  • [20] Hammersley, J. M. (1966). First-passage percolation. J. Roy. Statist. Soc. Ser. B 28 491–496.
  • [21] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
  • [22] 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.
  • [23] Howard, C. D. (2004). Models of first-passage percolation. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 125–173. Springer, Berlin.
  • [24] Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, New York.
  • [25] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16 221–259.
  • [26] Janson, S. (2009). The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.
  • [27] Janson, S. (2010). Susceptibility of random graphs with given vertex degrees. J. Comb. 1 357–387.
  • [28] Janson, S. and Luczak, M. J. (2009). A new approach to the giant component problem. Random Structures Algorithms 34 197–216.
  • [29] Kallenberg, O. (1976). Random Measures. Akademie-Verlag, Berlin.
  • [30] 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.
  • [31] Leskovec, J., Backstrom, L. and Kleinberg, J. (2009). Meme-tracking and the dynamics of the news cycle. In Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 497–506. ACM, New York.
  • [32] Leskovec, J., McGlohon, M., Faloutsos, C., Glance, N.and Hurst, M. (2007). Patterns of cascading behavior in large blog graphs. In Proceedings of the 2007 SIAM International Conference on Data Mining 551–556. Minneapolis, MN. 551–556.
  • [33] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
  • [34] Mihail, M., Papadimitriou, C. and Saberi, A. (2006). On certain connectivity properties of the Internet topology. J. Comput. System Sci. 72 239–251.
  • [35] Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6 161–179.
  • [36] 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.
  • [37] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • [38] Samuels, M. L. (1971). Distribution of the branching-process population among generations. J. Appl. Probab. 8 655–667.
  • [39] Smythe, R. T. and Wierman, J. C. (1978). First-Passage Percolation on the Square Lattice. Lecture Notes in Math. 671. Springer, Berlin.
  • [40] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.