  We prove non-universality results for first-passage percolation on the
configuration model with i.i.d. degrees having infinite variance. We focus on
the weight of the optimal path between two uniform vertices. Depending on the
properties of the weight distribution, we use an example-based approach and
show that rather different behaviors are possible. When the weights are a.s.
larger than a constant, the weight and number of edges in the graph grow
proportionally to loglog(n), as for the graph distances. On the other hand,
when the continuous-time branching process describing the first passage
percolation exploration through the graph reaches infinitely many vertices in
finite time, the weight converges to the sum of two i.i.d. random variables
representing the explosion times of the processes started from the two sources.
This non-universality is in sharp contrast to the setting where the degree
sequence has a finite variance (see Bhamidi, Hofstad and Hooghiemstra arXiv:
1210.6839).
