In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold @math for a Galton-Watson tree is strictly positive if and only if its offspring distribution @math has an exponential tail, i.e., @math , settling a conjecture by Huang and Durrett hd18 . On the random graph with degree distribution @math , we show that if @math has an exponential tail, then for small enough @math the contact process with the all-infected initial condition survives for @math -time w.h.p. (short survival), while for large enough @math it runs over @math -time w.h.p. (long survival). When @math is subexponential, we prove that the contact process w.h.p. displays long survival for any fixed @math .
@cite_4: We consider the contact process with infection rate lambda on a random (d + 1)-regular graph with n vertices, G(n). We study the extinction time tau(Gn) (that is, the random amount of time until the infection disappears) as n is taken to infinity. We establish a phase transition depending on whether lambda is smaller or larger than lambda(1) (T-d), the lower critical value for the contact process on the infinite, (d + 1) -regular tree: if lambda lambda(1) (T-d), it grows exponentially with n. This result differs from the situation where, instead of G(n), the contact process is considered on the d-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on T-d.
@cite_12: We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett (2009), who showed that for arbitrarily small infection parameter @math , the survival time of the process is larger than a stretched exponential function of the number of vertices, @math . We obtain sharp bounds for the typical density of infected sites in the graph, as @math is kept fixed and @math tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.
@cite_23: 
Moreover in @cite_23 , a cutoff phenomenon" of the fraction of infected vertices was established. In @cite_4 , the same result as above is proven for @math with bounded @math . For unbounded degree distribution @math , the only known result was due to Chatterjee and Durrett @cite_12 , where they showed that if @math obeys a power law, then the contact process always displays long survival for any @math , though their survival time was slightly weaker than exponential ( @math for any @math ). Our Theorems and generalize the aforementioned results to any general @math and to an exponentially long survival.