The Annals of Applied Probability

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

Shankar Bhamidi and Remco van der Hofstad

Full-text: Open access

Abstract

In the recent past, there has been a concerted effort to develop mathematical models for real-world networks and to analyze various dynamics on these models. One particular problem of significant importance is to understand the effect of random edge lengths or costs on the geometry and flow transporting properties of the network. Two different regimes are of great interest, the weak disorder regime where optimality of a path is determined by the sum of edge weights on the path and the strong disorder regime where optimality of a path is determined by the maximal edge weight on the path. In the context of the stochastic mean-field model of distance, we provide the first mathematically tractable model of weak disorder and show that no transition occurs at finite temperature. Indeed, we show that for every finite temperature, the number of edges on the minimal weight path (i.e., the hopcount) is Θ(log n) and satisfies a central limit theorem with asymptotic means and variances of order Θ(log n), with limiting constants expressible in terms of the Malthusian rate of growth and the mean of the stable-age distribution of an associated continuous-time branching process. More precisely, we take independent and identically distributed edge weights with distribution Es for some parameter s > 0, where E is an exponential random variable with mean 1. Then the asymptotic mean and variance of the central limit theorem for the hopcount are s log n and s2 log n, respectively. We also find limiting distributional asymptotics for the value of the minimal weight path in terms of extreme value distributions and martingale limits of branching processes.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 1 (2012), 29-69.

Dates
First available in Project Euclid: 7 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1328623695

Digital Object Identifier
doi:10.1214/10-AAP753

Mathematical Reviews number (MathSciNet)
MR2932542

Zentralblatt MATH identifier
1248.60012

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

Keywords
Flows random graphs first passage percolation hopcount central limit theorem weak disorder continuous-time branching process stable-age distribution theory mean-field model of distance Cox point processes

Citation

Bhamidi, Shankar; van der Hofstad, Remco. Weak disorder asymptotics in the stochastic mean-field model of distance. Ann. Appl. Probab. 22 (2012), no. 1, 29--69. doi:10.1214/10-AAP753. https://projecteuclid.org/euclid.aoap/1328623695


Export citation

References

  • [1] Addario-Berry, L., Broutin, N. and Reed, B. (2006). The diameter of the minimum spanning tree of a complete graph. In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities. Discrete Math. Theor. Comput. Sci. Proc., AG 237–248. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
  • [2] Aldous, D. (1992). Asymptotics in the random assignment problem. Probab. Theory Related Fields 93 507–534.
  • [3] 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. Springer, Berlin.
  • [4] Bhamidi, S. (2008). First passage percolation on locally treelike networks. I. Dense random graphs. J. Math. Phys. 49 125218, 27.
  • [5] 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.
  • [6] 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.
  • [7] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). Weak disorder in the stochastic mean-field model of distance II. Bernoulli. To appear.
  • [8] Bollobás, B., Gamarnik, D., Riordan, O. and Sudakov, B. (2004). On the value of a random minimum weight Steiner tree. Combinatorica 24 187–207.
  • [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] Chen, Y., López, E., Havlin, S. and Stanley, H. E. (2006). Universal behavior of optimal paths in weighted networks with general disorder. Phys. Rev. Lett. 96 68702.
  • [11] Frieze, A. M. (1985). On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47–56.
  • [12] Gradshteyn, I. S. and Ryzhik, I. M. (1965). Table of Integrals, Series, and Products, 4th ed. Academic Press, New York.
  • [13] 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.
  • [14] Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, London.
  • [15] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16 221–259.
  • [16] 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.
  • [17] Riordan, O. and Wormald, N. (2008). The diameter of sparse random graphs. Preprint.
  • [18] Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw-Hill, New York.
  • [19] 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.
  • [20] 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.
  • [21] van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2002). The flooding time in random graphs. Extremes 5 111–129.
  • [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] 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).