• Bernoulli
  • Volume 19, Number 2 (2013), 363-386.

Weak disorder in the stochastic mean-field model of distance II

Shankar Bhamidi, Remco van der Hofstad, and Gerard Hooghiemstra

Full-text: Open access


In this paper, we study the complete graph $K_{n}$ with $n$ vertices, where we attach an independent and identically distributed (i.i.d.) weight to each of the $n(n-1)/2$ edges. We focus on the weight $W_{n}$ and the number of edges $H_{n}$ of the minimal weight path between vertex $1$ and vertex $n$.

It is shown in (Ann. Appl. Probab. 22 (2012) 29–69) that when the weights on the edges are i.i.d. with distribution equal to that of $E^{s}$, where $s>0$ is some parameter, and $E$ has an exponential distribution with mean $1$, then $H_{n}$ is asymptotically normal with asymptotic mean $s\log n$ and asymptotic variance $s^{2}\log n$. In this paper, we analyze the situation when the weights have distribution $E^{-s}$, $s>0$, in which case the behavior of $H_{n}$ is markedly different as $H_{n}$ is a tight sequence of random variables. More precisely, we use the method of Stein–Chen for Poisson approximations to show that, for almost all $s>0$, the hopcount $H_{n}$ converges in probability to the nearest integer of $s+1$ greater than or equal to $2$, and identify the limiting distribution of the recentered and rescaled minimal weight. For a countable set of special $s$ values denoted by $\mathcal{S}=\{s_{j}\}_{j\geq2}$, the hopcount $H_{n}$ takes on the values $j$ and $j+1$ each with positive probability.

Article information

Bernoulli, Volume 19, Number 2 (2013), 363-386.

First available in Project Euclid: 13 March 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

complete graph extreme value theory first passage percolation hopcount minimal path weight Poisson approximation Stein–Chen method stochastic mean-field model weak disorder


Bhamidi, Shankar; van der Hofstad, Remco; Hooghiemstra, Gerard. Weak disorder in the stochastic mean-field model of distance II. Bernoulli 19 (2013), no. 2, 363--386. doi:10.3150/11-BEJ402.

Export citation


  • [1] Aldous, D. and Steele, J.M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Berlin: Springer.
  • [2] Aldous, D.J. (2001). The $\zeta(2)$ limit in the random assignment problem. Random Structures Algorithms 18 381–418.
  • [3] Aldous, D.J. (2010). More uses of exchangeability: Representations of complex random structures. In Probability and Mathematical Genetics. London Mathematical Society Lecture Note Series 378 35–63. Cambridge: Cambridge Univ. Press.
  • [4] Aldous, D.J., McDiarmid, C. and Scott, A. (2009). Uniform multicommodity flow through the complete graph with random edge-capacities. Oper. Res. Lett. 37 299–302.
  • [5] Barbour, A.D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. New York: The Clarendon Press Oxford Univ. Press. Oxford Science Publications.
  • [6] 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.
  • [7] 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.
  • [8] 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.
  • [9] Braunstein, L.A., Wu, Z., Chen, Y., Buldyrev, S.V., Kalisky, T., Sreenivasan, S., Cohen, R., López, E., Havlin, S. and Stanley, H.E. (2007). Optimal path and minimal spanning trees in random weighted networks. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17 2215–2255.
  • [10] Frieze, A.M. (1985). On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47–56.
  • [11] Grimmett, G. and Kesten, H. (1984). Random electrical networks on complete graphs. J. London Math. Soc. (2) 30 171–192.
  • [12] Havlin, S., Braunstein, L.A., Buldyrev, S.V., Cohen, R., Kalisky, T., Sreenivasan, S. and Stanley, H.E. (2005). Optimal path in random networks with disorder: A mini review. Phys. A 346 82–92.
  • [13] Howard, C.D. (2004). Models of first-passage percolation. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 125–173. Berlin: Springer.
  • [14] 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.
  • [15] 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. Berlin: Springer.
  • [16] Smythe, R.T. and Wierman, J.C. (1978). First-passage Percolation on the Square Lattice. Lecture Notes in Math. 671. Berlin: Springer.
  • [17] Sreenivasan, S., Kalisky, T., Braunstein, L.A., Buldyrev, S.V., Havlin, S. and Stanley, H.E. (2004). Effect of disorder strength on optimal paths in complex networks. Phys. Rev. E 70 46133.