The contact-process threshold conjecture for random graphs

Let $cmu$ be a degree distribution satisfying the giant component condition, and let $ctilde{\mu}$ be its size-biased distribution. Write $clambda_c^-(\mu)$ and $clambda_c^+(\mu)$ for the short- and long-survival thresholds of the contact process on the corresponding random graphs, and let $clambda_1^{\textsc{gw}}(\widetilde{\mu})$ denote the extinction-survival threshold for the contact process on the Galton–Watson tree with offspring distribution $ctilde{\mu}$. Threshold conjecture. The two random-graph thresholds coincide with the Galton–Watson-tree threshold:

$$\lambda_c^-(\mu)=\lambda_c^+(\mu)=\lambda_1^{\textsc{gw}}(\widetilde{\mu}).$$

This conjectures that the transition from polynomial- to exponential-time survival is sharp and occurs at the extinction-survival threshold of the corresponding Galton–Watson tree. The statement is presented as an expectation in the source and is not established there.