Survival time of a random graph


LetV n ={1, 2, ...,n} ande 1,e 2, ...,e N ,N=\(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) be a random permutation ofV n (2). LetE t={e 1,e 2, ...,e t} andG t=(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 thatG τ starts to deteriorate i.e. loses edges in order ofage, e 1,e 2, .... We introduce the idea of thesurvival time τ =τ′(Π) defined by τt = max {u:(V n, {e u,e u+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.

This is a preview of subscription content, access via your institution.


  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.

  2. [2]

    B.Bollobás,Random Graphs, Academic Press, 1985.

  3. [3]

    B.Bollobás and A.Thomason, Random graphs of small order,Annals of Discrete Mathematics.

  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.

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Frieze, A.M., Frieze, A.M. Survival time of a random graph. Combinatorica 9, 133–143 (1989).

Download citation

AMS subject classification (1980)

  • 05 C 80