First passage percolation on random graphs with finite mean degrees

We study first passage percolation on the configuration model. Assuming that each edge has anindependent exponentially distributed edge weight, we derive explicit distributional asymptotics forthe minimum weight between two randomly chosen connected vertices in the network, as well as forthe number of edges on the least weight path, the so-called hopcount.We analyze the configuration model with degree power-law exponent τ > 2, in which the degrees areassumed to be i.i.d. with a tail distribution which is either of power-law form with exponent τ − 1 > 1,or has even thinner tails (τ = ∞). In this model, the degrees have a finite first moment, while thevariance is finite for τ > 3, but infinite for τ ∈ (2, 3).We prove a central limit theorem for the hopcount, with asymptotically equal means and variancesequal to α logn, where α ∈ (0, 1) for τ ∈ (2, 3), while α > 1 for τ > 3. Here n denotes the size of thegraph. For τ ∈ (2, 3), it is known that the graph distance between two randomly chosen connectedvertices is proportional to log log n [25], i.e., distances are ultra small. Thus, the addition of edgeweights causes a marked change in the geometry of the network. We further study the weight of theleast weight path, and prove convergence in distribution of an appropriately centered version.This study continues the program initiated in [5] of showing that logn is the correct scaling for thehopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen verticesis of much smaller order. The case of infinite mean degrees (τ ∈ [1, 2)) is studied in [6], where it isproved that the hopcount remains uniformly bounded and converges in distribution.