Speeding up non-Markovian First Passage Percolation with a few extra edges

TL;DR

This paper investigates how adding a single edge can significantly accelerate the spread of processes with heavy-tailed passage times on graphs, especially in critical random graph models, by analyzing infection times and growth limits.

Contribution

It provides bounds on infection times in non-Markovian SI models with heavy-tailed distributions and demonstrates how minimal edge additions drastically speed up spreading in critical random graphs.

Findings

Adding one edge greatly increases infected vertices in finite expected time.

Expected infection time scales as O(k^{1/α}) for k infected vertices.

Edge addition accelerates spreading in critical and near-critical random graph models.

Abstract

One model of real-life spreading processes is First Passage Percolation (also called SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with i.i.d.~heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow, because of bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power law distribution $\mathbb{P}(\xi>t)\sim t^{-\alpha}$, with infinite mean. For any finite connected graph $G$ with a root $s$, we find the largest number of vertices $\kappa(G,s)$ that are infected in finite expected time, and prove that for every $k \leq \kappa(G,s)$, the expected time to infect $k$ vertices is at most $O(k^{1/\alpha})$. Then, we show that adding a single edge from $s$ to a random vertex in a random tree $\mathcal{T}$ typically increases $\kappa(\mathcal{T},s)$ from a bounded variable to a fraction of the size of $\mathcal{T}$, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton-Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erd\H{o}s-R\'enyi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.