, Volume 9, Issue 2, pp 133–143 | Cite as

Survival time of a random graph

  • A. M. Frieze
  • A. M. Frieze


LetV n ={1, 2, ...,n} ande1,e2, ...,e N ,N=\(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) be a random permutation ofVn(2). LetEt={e1,e2, ...,et} andGt=(V n ,E t ). IfΠ is a monotone graph property then the hitting timeτ(Π) forΠ is defined byτ=τ(Π)=min {t:G t ∈Π}. Suppose now thatGτ starts to deteriorate i.e. loses edges in order ofage, e1,e2, .... We introduce the idea of thesurvival time τ =τ′(Π) defined by τt = max {u:(Vn, {eu,eu+1, ...,e T }) ∈Π}. We study in particular the case whereΠ isk-connectivity. We show that
$$\mathop {\lim }\limits_{n \to \infty } \Pr (\tau ' \geqq an) = e^{ - 2a} {\mathbf{ }}for{\mathbf{ }}a \in R^ + $$
$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}E(\tau ') = \frac{1}{n}$$
i.e.τ′/n is asymptotically negative exponentially distributed with mean 1/2.

AMS subject classification (1980)

05 C 80 


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  1. [1]
    B.Bollobás, The evolution of sparse graphs, InGraph Theory and Combinatorics, Proc. Cambridge Combinatorial Conference in houour of Paul Erdős (B. Bollobás, Ed.), Academic Press (1984), 35–57.Google Scholar
  2. [2]
    B.Bollobás,Random Graphs, Academic Press, 1985.Google Scholar
  3. [3]
    B.Bollobás and A.Thomason, Random graphs of small order,Annals of Discrete Mathematics.Google Scholar
  4. [4]
    P. Erdős andA. Rényi, On random Graphs I,Publ. Math. Debrecen. 6 (1959), 290–297.Google Scholar
  5. [5]
    P. Erdős andA. Rényi, On the evolution of random graphs,Publ. Math. Inst. Hungar. Acad. Sci.,7 (1960), 17–61.Google Scholar
  6. [6]
    P. Erdős andA. Rényi, On the strength of connectedness of a random graph,Acta Math. Acad. Sci. Hungar.,12 (1961), 261–267.Google Scholar
  7. [7]
    E.Palmer,Graphical Evolution.Google Scholar

Copyright information

© Akadémiai Kiadó 1989

Authors and Affiliations

  • A. M. Frieze
    • 1
  • A. M. Frieze
    • 1
  1. 1.Department of MathematicsCarnegie Mellon UniversityPittsburgh

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