Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-07T11:26:03.692Z Has data issue: false hasContentIssue false
  • English
  • Français

The Diameter of Sparse Random Graphs

Published online by Cambridge University Press:  05 October 2010

OLIVER RIORDAN  and
NICHOLAS WORMALD
OLIVER RIORDAN
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK (e-mail: riordan@maths.ox.ac.uk)
NICHOLAS WORMALD
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada (e-mail: nwormald@uwaterloo.ca)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper we study the diameter of the random graph G(n, p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ > 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish two-point concentration in the case that np → ∞. For p =(1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n, p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.

References

[1]Addario-Berry, L., Broutin, N. and Goldschmidt, C. The continuum limit of critical random graphs. Preprint available from arXiv:0903.4730.Google Scholar
[2]Albert, R., Jeong, H. and Barabási, A.-L. (1999) Diameter of the world-wide web. Nature 401 130–131.CrossRefGoogle Scholar
[3]Athreya, K. B. and Ney, P. E. (1972) Branching Processes, Springer.Google Scholar
[4]Barabási, A.-L., Albert, R. and Jeong, H. (2000) Scale-free characteristics of random networks: The topology of the world-wide web. Physica A 281 69–77.CrossRefGoogle Scholar
[5]van den Berg, J. and Kesten, H. (1985) Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556–569.Google Scholar
[6]Bollobás, B. (1981) The diameter of random graphs. Trans. Amer. Math. Soc. 267 41–52.Google Scholar
[7]Bollobás, B. (1984) The evolution of random graphs. Trans. Amer. Math. Soc. 286 257–274.CrossRefGoogle Scholar
[8]Bollobás, B. (1985) Random Graphs, Academic Press.Google Scholar
[9]Bollobás, B. and Chung, F. R. K. (1988) The diameter of a cycle plus a random matching. SIAM J. Discrete Math. 1 328–333.Google Scholar
[10]Bollobás, B. and Fernandez de la Vega, W. (1982) The diameter of random regular graphs. Combinatorica 2 125–134.CrossRefGoogle Scholar
[11]Bollobás, B., Janson, S. and Riordan, O. (2007) The phase transition in inhomogeneous random graphs. Random Struct. Alg. 31 3–122.Google Scholar
[12]Bollobás, B. and Riordan, O. (2004) The diameter of a scale-free random graph. Combinatorica 24 5–34.Google Scholar
[13]Bollobás, B. and Riordan, O. (2009) Random graphs and branching processes. In Handbook of Large-Scale Random Networks, Vol. 18 of Bolyai Society Mathematical Studies (Bollobás, B., Kozma, R. and Miklós, D., eds), pp. 15–115.Google Scholar
[14]Burtin, J. D. (1973) Asymptotic estimates of the diameter and the independence and domination numbers of a random graph. Dokl. Akad. Nauk SSSR 209 765–768. Translated in Soviet Math. Dokl. 14 (1973) 497–501.Google Scholar
[15]Burtin, J. D. (1974) Extremal metric characteristics of a random graph I. Teor. Verojatnost. i Primenen. 19 740–754.Google Scholar
[16]Chung, F. and Lu, L. (2001) The diameter of sparse random graphs. Adv. Appl. Math. 26 257–279.Google Scholar
[17]Chung, F. and Lu, L. (2002) The average distances in random graphs with given expected degrees. Proc. Nat. Acad. Sci. USA 99 15879–15882 (electronic).Google Scholar
[18]Chung, F. and Lu, L. (2003) The average distance in a random graph with given expected degrees. Internet Math. 1 91–113.Google Scholar
[19]Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. Anatomy of a young giant component in the random graph. Random Struct. Alg., to appear. Available at: http://arxiv.org/abs/0906.1839).Google Scholar
[20]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
[21]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.Google Scholar
[22]van den Esker, H., van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2006) Distances in random graphs with infinite mean degrees, Extremes 8 111–141.CrossRefGoogle Scholar
[23]Fernholz, D. and Ramachandran, V. (2007) The diameter of sparse random graphs. Random Struct. Alg. 31 482–516.CrossRefGoogle Scholar
[24]Harris, T. E. (1948) Branching processes. Ann. Math. Statist. 19 474–494.Google Scholar
[25]van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2005) Distances in random graphs with finite variance degrees. Random Struct. Alg. 27 76–123.Google Scholar
[26]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).Google Scholar
[27]Janson, S. (2002) On concentration of probability. In Contemporary Combinatorics, Vol. 10 of Bolyai Society Mathematical Studies, János Bolyai Mathematical Society, pp. 289–301.Google Scholar
[28]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.Google Scholar
[29]Kolmogorov, A. N. (1938) Zur Lösung einer biologischen Aufgabe. Comm. Math. Mech. Chebyshev Univ. Tomsk 2 1–6.Google Scholar
[30]Łuczak, T. (1990) Component behavior near the critical point of the random graph process. Random Struct. Alg. 1 287–310.Google Scholar
[31]Łuczak, T. (1991) Cycles in a random graph near the critical point. Random Struct. Alg. 2 421–440.Google Scholar
[32]Łuczak, T. (1998) Random trees and random graphs. Random Struct. Alg. 13 485–500.Google Scholar
[33]Łuczak, T. and Seierstad, T. G. (2008/2009) The diameter behavior in the random graph process. Mittag–Leffler Preprint Report no. 5.Google Scholar
[34]Nachmias, A. and Peres, Y. (2008) Critical random graphs: Diameter and mixing time. Ann. Probab. 36 1267–1286.CrossRefGoogle Scholar
[35]Newman, M. E. J., Strogatz, S. H. and Watts, D. J. (2001) Random graphs with arbitrary degree distribution and their applications. Phys. Review E 64 026118.Google Scholar
[36]Pittel, B. and Wormald, C. (2005) Counting connected graphs inside-out. J. Combin. Theory Ser. B 93 127–172.Google Scholar
[37]Reimer, D. (2000) Proof of the van den Berg–Kesten conjecture. Combin. Probab. Comput. 9 27–32.Google Scholar
[38]Sanwalani, V. and Wormald, N. The diameter of random regular graphs. In preparation.Google Scholar
[39]Wormald, N. C. (1999) Models of random regular graphs. In Surveys in Combinatorics 1999, Vol. 267 of London Mathematical Society Lecture Notes (Lamb, J. D. and Preece, D. A., eds), Cambridge University Press, pp. 239–298.CrossRefGoogle Scholar