  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 [6]. 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 focus on the case where $s_n\rightarrow \infty$ with
$s_n=o(n^{1/3})$. Under mild regularity conditions, we identify the scaling
limit of the weight of the smallest-weight path between two uniform vertices,
and we prove that the number of edges in this path obeys a central limit
theorem with asymptotic mean $s_n\log{(n/s_n^3)}$ and variance
$s_n^2\log{(n/s_n^3)}$. This settles a conjecture of Bhamidi and van der
Hofstad [6]. The proof relies on a decomposition of the smallest-weight tree
into an initial part following invasion percolation dynamics, and a main part
following branching process dynamics. The initial part has been studied in
[14]; the current article focuses on the global branching dynamics.

Probability