Tight Fluctuations of Weight-Distances in Random Graphs with Infinite-Variance Degrees

Enrico Baroni,Remco Van Der Hofstad,Júlia Komjáthy

doi:10.1007/s10955-018-2213-8

Enrico Baroni, Remco Van Der Hofstad + Show 1 more

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https://doi.org/10.1007/s10955-018-2213-8

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Abstract

We prove results for first-passage percolation on the configuration model with degrees having asymptotic finite mean, infinite variance and i.i.d. edge-weights with strictly positive support of the form Y=a+X, where a is a positive constant and the excess edge-weight X is a non-negative random variable with zero as the infimum of its support. We prove that the weight of the optimal path between two uniformly chosen vertices has tight fluctuations around the asymptotic mean of the graph-distance if and only if the following condition holds: the random variable X is such that the age-dependent branching process describing first-passage percolation exploration in the same graph with edge-weights from distribution 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 even exist paths having tight total excess edge-weight for appropriate edge-weight distributions. Our proof makes use of a degree-dependent percolation model that we believe is interesting in its own right, as well as tightness results for distances in scale-free configuration models that we prove to hold under rather weak conditions on the degrees.

Highlights

First Passage Percolation (FPP) has been introduced as a model for the spread of a material in a random medium
We study first-passage percolation in the setting of the configuration model random graph, with degrees having finite mean and infinite variance
If we assume that the edges have a passage-time represented by a collections of i.i.d. random variables, a second problem is to determine the geometry of the time-minimizing paths between two points and the way in which they differ from graphdistance paths

Summary

Introduction

First Passage Percolation (FPP) has been introduced as a model for the spread of a material in a random medium (see [10]). In more recent times, motivated by the boost in interest in complex networks and the related random graph models for them, it has appeared as a mathematical tool for studying dynamics in complex networks. A typical setting in this sense is a transportation network in which roads, corresponding to edges, have certain transport times, corresponding to weights on the edges (see [13]). The corresponding mathematical model is a simple and connected graph G, such that every edge e has a random variable Ye assigned to it, representing the passage time through the edge e, where the edge-weights (Ye) are a collection of independent and identically distributed (i.i.d.) random variables. For an extensive introduction to FPP on random graphs, we refer to [18, Chapter 3]

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