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

TL;DR

This paper investigates the behavior of shortest paths in random graphs with infinite-variance degrees, revealing conditions under which their total weights exhibit tight fluctuations around the average, influenced by branching process properties.

Contribution

It establishes a novel criterion linking the tightness of weight fluctuations to the probability of infinite growth in a related branching process for graphs with infinite-variance degrees.

Findings

Tight fluctuations occur if the associated branching process can reach infinitely many individuals in finite time.

Almost shortest paths can have tight total excess edge weight under certain conditions.

The results connect first-passage percolation behavior to branching process properties in complex networks.

Abstract

We prove results for first-passage percolation on the configuration model with i.i.d. degrees having finite mean, infinite variance and i.i.d. weights with strictly positive support of the form Y=a+X, where a is a positive constant. We prove that the weight of the optimal path has tight fluctuations around the asymptotical mean of the graph-distance if and only if the following condition holds: the random variable X is such that the continuous-time branching process describing first-passage percolation exploration in the same graph with excess edge weight X has a positive probability to reach infinitely many individuals in a finite time. This shows that almost shortest paths in the graph-distance proliferate, in the sense that there are even ones having tight total excess edge weight for various edge-weight distributions.