Universality for first passage percolation on sparse uniform and rank-1 random graphs

In [3], we considered first passage percolation on the configuration model equipped with general independent and identically distributed edge weights, where the common distribution function admits a density. Assuming that the degree distribution satisfies a uniform X^2 log X - condition, we analyzed the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well as the asymptotic distribution of the number of edges on this path. Given the interest in understanding such questions for various other random graph models, the aim of this paper is to show how these results extend to uniform random graphs with a given degree sequence and rank-one inhomogeneous random graphs