  We consider first passage percolation on the configuration model. Once the
network has been generated each edge is assigned an i.i.d. weight modeling the
passage time of a message along this edge. Then independently two vertices are
chosen uniformly at random, a sender and a recipient, and all edges along the
geodesic connecting the two vertices are coloured in red (in the case that both
vertices are in the same component). In this article we prove local limit
theorems for the coloured graph around the recipient in the spirit of Benjamini
and Schramm. We consider the explosive regime, in which case the random
distances are of finite order, and the Malthusian regime, in which case the
random distances are of logarithmic order.
