  We study the random geometry of first passage percolation on the complete
graph equipped with independent and identically distributed edge weights,
continuing the program initiated by Bhamidi and van der Hofstad [9]. We
describe our results in terms of a sequence of parameters $(s_n)_{n\geq 1}$
that quantifies the extreme-value behavior of small weights, and that describes
different universality classes for first passage percolation on the complete
graph. We consider both $n$-independent as well as $n$-dependent edge weights.
The simplest example consists of edge weights of the form $E^{s_n}$, where $E$
is an exponential random variable with mean 1.
  In this paper, we investigate the case where $s_n\rightarrow \infty$, and
focus on the local neighborhood of a vertex. We establish that the
smallest-weight tree of a vertex locally converges to the invasion percolation
cluster on the Poisson weighted infinite tree. In addition, we identify the
scaling limit of the weight of the smallest-weight path between two uniform
vertices.
