Survival time of a random graph

Abstract

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

$$\mathop {\lim }\limits_{n \to \infty } \Pr (\tau ' \geqq an) = e^{ - 2a} {\mathbf{ }}for{\mathbf{ }}a \in R^ + $$

(1))

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}E(\tau ') = \frac{1}{n}$$

(2))

i.e.τ′/n is asymptotically negative exponentially distributed with mean 1/2.