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First passage percolation on Erdős-Rényi graphs with general weights

Fraser Daly, Matthias Schulte, Seva Shneer

TL;DR

This work analyzes first passage percolation on the supercritical Erdős–Rényi graph with general nonnegative edge weights. Using the method of moments, the authors show that, after centering and scaling, the point process of hopcounts and total weights for near-minimal paths converges to a Cox process whose random intensity is governed by a fixed-point random variable $W$ connected to a branching-process structure. They obtain the limiting distribution for the minimal total weight and its hopcount(s), with distinct behaviors for non-arithmetic and arithmetic weight distributions, thereby generalizing earlier results that required densities for edge weights. The approach yields a unified framework for the joint asymptotics of hopcounts and weights, and provides precise asymptotics for the minimal-weight path in terms of a random intensity that encapsulates the graph's local growth and the weight distribution.

Abstract

We consider first passage percolation on the Erdős--Rényi graph with $n$ vertices in which each pair of distinct vertices is connected independently by an edge with probability $λ/n$ for some $λ>1$. The edges of the graph are given non-negative i.i.d. weights with a non-degenerate distribution such that the probability of zero is not too large. We consider the paths with small total weight between two distinct typical vertices and analyse the joint behaviour of the numbers of edges on such paths, the so-called hopcounts, and the total weights of these paths. For $n\to\infty$, we show that, after a suitable transformation, the pairs of hopcounts and total weights of these paths converge in distribution to a Cox process, i.e., a Poisson process with a random intensity measure. The random intensity measure is controlled by two independent random variables, whose distribution is the solution of a distributional fixed point equation and is related to branching processes. For non-arithmetic and arithmetic edge-weight distributions we observe different behaviour. In particular, we derive the limiting distribution for the minimal total weight and the corresponding hopcount(s). Our results generalise earlier work of Bhamidi, van der Hofstad and Hooghiemstra, who assume that edge weights have an absolutely continuous distribution. The main tool we employ is the method of moments.

First passage percolation on Erdős-Rényi graphs with general weights

TL;DR

This work analyzes first passage percolation on the supercritical Erdős–Rényi graph with general nonnegative edge weights. Using the method of moments, the authors show that, after centering and scaling, the point process of hopcounts and total weights for near-minimal paths converges to a Cox process whose random intensity is governed by a fixed-point random variable connected to a branching-process structure. They obtain the limiting distribution for the minimal total weight and its hopcount(s), with distinct behaviors for non-arithmetic and arithmetic weight distributions, thereby generalizing earlier results that required densities for edge weights. The approach yields a unified framework for the joint asymptotics of hopcounts and weights, and provides precise asymptotics for the minimal-weight path in terms of a random intensity that encapsulates the graph's local growth and the weight distribution.

Abstract

We consider first passage percolation on the Erdős--Rényi graph with vertices in which each pair of distinct vertices is connected independently by an edge with probability for some . The edges of the graph are given non-negative i.i.d. weights with a non-degenerate distribution such that the probability of zero is not too large. We consider the paths with small total weight between two distinct typical vertices and analyse the joint behaviour of the numbers of edges on such paths, the so-called hopcounts, and the total weights of these paths. For , we show that, after a suitable transformation, the pairs of hopcounts and total weights of these paths converge in distribution to a Cox process, i.e., a Poisson process with a random intensity measure. The random intensity measure is controlled by two independent random variables, whose distribution is the solution of a distributional fixed point equation and is related to branching processes. For non-arithmetic and arithmetic edge-weight distributions we observe different behaviour. In particular, we derive the limiting distribution for the minimal total weight and the corresponding hopcount(s). Our results generalise earlier work of Bhamidi, van der Hofstad and Hooghiemstra, who assume that edge weights have an absolutely continuous distribution. The main tool we employ is the method of moments.
Paper Structure (9 sections, 14 theorems, 138 equations, 3 figures)

This paper contains 9 sections, 14 theorems, 138 equations, 3 figures.

Table of Contents

  1. Introduction, notation and main results
  2. Proofs of Theorems \ref{['thm:main']} and \ref{['thm:shortest_path']}
  3. Factorial moments of the limiting point processes
  4. Truncated exponential renewal functions
  5. Counting short paths: first moments
  6. Some random variables associated with trees
  7. Counting short paths: factorial moments
  8. Proof of Theorem \ref{['thm:main']}
  9. Proof of Theorem \ref{['thm:shortest_path']}

Key Result

Theorem 1.1

Assume that eqn:probability_weight_zero is satisfied. Let $\mathbb{P}_{N}$ be the distribution of a standard Gaussian random variable $N$, and let $W_1$ and $W_2$ be independent copies of $W$.

Figures (3)

  • Figure 1: In the left panel, no paths cross each other. In contrast, the two lower paths (in green and red) cross in the panel on the right.
  • Figure 2: Two trees from $\mathcal{T}_4$ describing the behaviour of the non-crossing paths from the left panel of Figure \ref{['fig:crossings']} in the neighbourhoods of $1$ and $n$.
  • Figure 3: In this example for $q=1$ the highest path (in blue) is crossed by the lowest path leaving $1$ (in red).

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 18 more