In this talk I will investigate the question of explosion of branching processes, i.e., when is it possible that a BP produces infinitely many offspring in finite time. Two important cases in terms of application are age-dependent BPs and BPs arising from epidemic models where individuals are only contagious in a possibly random interval after being infected. This imposes dependencies between the birth-time of the children of an individual.

The motivation for studying the explosiveness question is to understand weighted distances in locally tree-like random graphs, such as the configuration model, in the regime where the degree distribution is a power-law with exponent between (2,3). Here, the local neighborhood of a vertex and thus the initial stages of the spreading can be the approximated by an infinite mean offspring BP. I will explain the recent results on this area. This part is joint work with Enrico Baroni and Remco van der Hofstad.