  We study the complete graph equipped with a topology induced by independent
and identically distributed edge weights. The focus of our analysis is on the
weight W_n and the number of edges H_n of the minimal weight path between two
distinct vertices in the weak disorder regime. We establish novel and simple
first and second moment methods using path counting to derive first order
asymptotics for the considered quantities. Our results are stated in terms of a
sequence of parameters (s_n) that quantifies the extreme-value behaviour of the
edge weights, and that describes different universality classes for first
passage percolation on the complete graph. These classes contain both
n-independent and n-dependent edge weight distributions. The method is most
effective for the universality class containing the edge weights E^{s_n}, where
E is an exponential(1) random variable and s_n log n -> infty, s_n^2 log n ->
0. We discuss two types of examples from this class in detail. In addition, the
class where s_n log n stays finite is studied. This article is a contribution
to the program initiated in \cite{BhaHof12}.
