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We prove results for first-passage percolation on the configuration modelwith i.i.d. degrees having finite mean, infinite variance and i.i.d. weightswith strictly positive support of the form Y=a+X, where a is a positiveconstant. We prove that the weight of the optimal path has tight fluctuationsaround the asymptotical mean of the graph-distance if and only if the followingcondition holds: the random variable X is such that the continuous-timebranching process describing first-passage percolation exploration in the samegraph with excess edge weight X has a positive probability to reach infinitelymany individuals in a finite time. This shows that almost shortest paths in thegraph-distance proliferate, in the sense that there are even ones having tighttotal excess edge weight for various edge-weight distributions.

Tight fluctuations of weight-distances in random graphs with infinite-variance degrees