You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
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 .

Cited Abstract(s):
[1]: The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival @math and (ii) when it is geometric( @math ) we have @math , where the @math are much smaller than previous estimates. We also study the critical value @math for "prolonged persistence" on graphs with @math vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known @math we give estimates on the rate of convergence. Physicists tell us that @math where @math is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.
Less is known for the contact process on general Galton-Watson trees. Recently, Huang and Durrett [1] proved that on Galton-Watson trees, @math if the offspring distribution @math is subexponential. Along with Theorem , we now have the complete characterization of the existence of extinction in the contact process on Galton-Watson trees.