  We consider first passage percolation on sparse random graphs with prescribed
degree distributions and general independent and identically distributed edge
weights assumed to have a density. Assuming that the degree distribution
satisfies a uniform X^2\log{X}-condition, we analyze the asymptotic
distribution for the minimal weight path between a pair of typical vertices, as
well the number of edges on this path or hopcount.
  The hopcount satisfies a central limit theorem where the norming constants
are expressible in terms of the parameters of an associated continuous-time
branching process. Centered by a multiple of \log{n}, where the constant is the
inverse of the Malthusian rate of growth of the associated branching process,
the minimal weight converges in distribution. The limiting random variable
equals the sum of the logarithms of the martingale limits of the branching
processes that measure the relative growth of neighborhoods about the two
vertices, and a Gumbel random variable, and thus shows a remarkably universal
behavior.
  The proofs rely on a refined coupling between the shortest path problems on
these graphs and continuous-time branching processes, and on a Poisson point
process limit for the potential closing edges of shortest-weight paths between
the source and destination.
  The results extend to a host of related random graph models, ranging from
random r-regular graphs, inhomogeneous random graphs and uniform random graphs
with a prescribed degree sequence.
