  We study first passage percolation on the configuration model (CM) having
power-law degrees with exponent $\tau\in [1,2)$. To this end we equip the edges
with exponential weights. We derive the distributional limit of the minimal
weight of a path between typical vertices in the network and the number of
edges on the minimal weight path, which can be computed in terms of the
Poisson-Dirichlet distribution. We explicitly describe these limits via the
construction of an infinite limiting object describing the FPP problem in the
densely connected core of the network. We consider two separate cases, namely,
the {\it original CM}, in which each edge, regardless of its multiplicity,
receives an independent exponential weight, as well as the {\it erased CM}, for
which there is an independent exponential weight between any pair of direct
neighbors. While the results are qualitatively similar, surprisingly the
limiting random variables are quite different.
  Our results imply that the flow carrying properties of the network are
markedly different from either the mean-field setting or the locally tree-like
setting, which occurs as $\tau>2$, and for which the hopcount between typical
vertices scales as $\log{n}$. In our setting the hopcount is tight and has an
explicit limiting distribution, showing that one can transfer information
remarkably quickly between different vertices in the network. This efficiency
has a down side in that such networks are remarkably fragile to directed
attacks. These results continue a general program by the authors to obtain a
complete picture of how random disorder changes the inherent geometry of
various random network models.
