Hostname: page-component-7f64f4797f-tldsr Total loading time: 0.001 Render date: 2025-11-04T15:57:24.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Diameter of the Stochastic Mean-Field Model of Distance

Part of: Graph theory Combinatorial probability Operations research and management science

Published online by Cambridge University Press:  07 August 2017

SHANKAR BHAMIDI  and
REMCO VAN DER HOFSTAD
SHANKAR BHAMIDI
Affiliation:
Department of Statistics, University of North Carolina, Chapel Hill, NC 27599, USA (e-mail: bhamidi@email.unc.edu)
REMCO VAN DER HOFSTAD
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: rhofstad@win.tue.nl)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the complete graph 𝜅n on n vertices with exponential mean n edge lengths. Writing Cij for the weight of the smallest-weight path between vertices i, j ∈ [n], Janson [18] showed that maxi,j∈[n]C ij/logn converges in probability to 3. We extend these results by showing that maxi,j∈[n]Cij − 3 logn converges in distribution to some limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centred graph diameter of the barely supercritical Erdős–Rényi random graph in [22].

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs 90B15: Network models, stochastic

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 6 , November 2017 , pp. 797 - 825
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Addario-Berry, L. Broutin, N. and Lugosi, G. (2010) The longest minimum-weight path in a complete graph. Combin. Probab. Comput. 19 1–19.CrossRefGoogle Scholar
[2] Aldous, D. (1992) Asymptotics in the random assignment problem. Probab. Theory Rel. Fields 93 507–534.Google Scholar
[3] Aldous, D. J. (2001) The ζ (2) limit in the random assignment problem. Random Struct. Alg. 18 381–418.Google Scholar
[4] Aldous, D. J. (2010) More uses of exchangeability: Representations of complex random structures. In Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman (Bingham, N. H. and Goldie, C. M., eds), London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 35–63.Google Scholar
[5] Aldous, D. J. and Bhamidi, S. (2010) Edge flows in the complete random-lengths network. Random Struct. Alg. 37 271–311.Google Scholar
[6] Aldous, D. and Steele, J. M. (2004) The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures (Kesten, H., ed.), Vol. 110 of Encyclopaedia of Mathematical Sciences, Springer, pp. 1–72.Google Scholar
[7] Amini, H. and Lelarge, M. (2015) The diameter of weighted random graphs. Ann. Appl. Probab. 25 1686–1727.Google Scholar
[8] Amini, H. and Peres, Y. (2014) Shortest-weight paths in random regular graphs. SIAM J. Discrete Math. 28 656–672.Google Scholar
[9] Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation, Vol. 2 of Oxford Studies in Probability, The Clarendon Press, Oxford University Press.CrossRefGoogle Scholar
[10] Bhamidi, S. (2008) First passage percolation on locally treelike networks I: Dense random graphs. J. Math. Phys. 49 125218.Google Scholar
[11] 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.Google Scholar
[12] Devroye, L. (1987) Branching processes in the analysis of the heights of trees. Acta Informatica 24 277–298.CrossRefGoogle Scholar
[13] 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.Google Scholar
[14] Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. (2011) Anatomy of a young giant component in the random graph. Random Struct. Alg. 39 139–178.CrossRefGoogle Scholar
[15] Frieze, A. M. (1985) On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47–56.CrossRefGoogle Scholar
[16] Goldstein, L. and Rinott, Y. (2007) Functional BKR inequalities, and their duals, with applications. J. Theoret. Probab. 20 275–293.Google Scholar
[17] Janson, S. (1995) The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph. Random Struct. Alg. 7 337–355.Google Scholar
[18] 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.Google Scholar
[19] Kemperman, J. H. B. (1977) On the FKG-inequality for measures on a partially ordered space. Indag. Math. 39 313–331.Google Scholar
[20] Kingman, J. F. C. (1993) Poisson Processes, Vol. 3 of Oxford Studies in Probability, The Clarendon Press, Oxford University Press.Google Scholar
[21] Pittel, B. (1994) Note on the heights of random recursive trees and random m-ary search trees. Random Struct. Alg. 5 337–347.Google Scholar
[22] Riordan, O. and Wormald, N. (2010) The diameter of sparse random graphs. Combin. Probab. Comput. 19 835–926.CrossRefGoogle Scholar
[23] Salez, J. (2013) Joint distribution of distances in large random regular networks. J. Appl. Probab. 50 861–870.Google Scholar
[24] Smythe, R. T. and Mahmoud, H. M. (1994) A survey of recursive trees. Teor. Ĭmovīr. Mat. Stat. 51 1–29.Google Scholar
[25] Wästlund, J. (2010) The mean field traveling salesman and related problems. Acta Mathematica 204 91–150.CrossRefGoogle Scholar