First Passage Percolation on Inhomogeneous Random Graphs

TL;DR

This paper studies first passage percolation on inhomogeneous random graphs, analyzing shortest path weights and hopcounts, and establishing distributional results including a central limit theorem for the hopcount.

Contribution

It generalizes previous work on Erdős-Rényi graphs to inhomogeneous models, providing new distributional results for shortest paths and hopcounts.

Findings

Distribution of shortest path weights determined

Hopcount follows a central limit theorem

Results hold for finite and infinite average degree cases

Abstract

We investigate first passage percolation on inhomogeneous random graphs. The random graph model G(n,kappa) we study is the model introduced by Bollob\'as, Janson and Riordan, where each vertex has a type from a type space S and edge probabilities are independent, but depending on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal weight path, properly normalized follows a central limit theorem. We handle the cases where lambda(n)->lambda is finite or infinite, under the assumption that the average number of neighbors lambda(n) of a vertex is independent of the type. The paper is a generalization the paper by Bhamidi, van der Hofstad and Hooghiemstra, where FPP is explored on the Erdos-Renyi graphs.