We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is a Galton-Watson tree then, in certain circumstances, the branching random walks and contact processes will have weak survival phases. We also provide bounds on critical values.
"Branching random walks and contact processes on Galton-Watson trees." Electron. J. Probab. 19 1 - 12, 2014. https://doi.org/10.1214/EJP.v19-3118