  We obtain local weak limits in probability for Collapsed Branching Processes
(CBP), which are directed random networks obtained by collapsing random-sized
families of individuals in a general continuous-time branching process. The
local weak limit of a given CBP, as the network grows, is shown to be a related
continuous-time branching process stopped at an independent exponential time.
This is done through an explicit coupling of the in-components of vertices with
the limiting object. We also show that the in-components of a finite collection
of uniformly chosen vertices locally weakly converge (in probability) to i.i.d.
copies of the above limit, reminiscent of propagation of chaos in interacting
particle systems. We obtain as special cases novel descriptions of the local
weak limits of directed preferential and uniform attachment models. We also
outline some applications of our results for analyzing the limiting in-degree
and PageRank distributions.
