Spread of Infection over P.A. random graphs with edge insertion

TL;DR

This paper studies how infections spread rapidly in a special class of evolving random graphs that combine preferential attachment and edge insertion, showing that a small number of steps can infect a large portion of the network.

Contribution

It introduces a new model of random graphs with time-dependent edge insertion and proves rapid infection spread under certain conditions.

Findings

Infection spreads within 3 steps to a positive fraction of the graph.

Graphs are highly susceptible to infection under integrability conditions.

Provides a lower bound for maximum degree in the graph.

Abstract

In this work we investigate a bootstrap percolation process on random graphs generated by a random graph model which combines preferential attachment and edge insertion between previously existing vertices. The probabilities of adding either a new vertex or a new connection between previously added vertices are time dependent and given by a function $f$ called the edge-step function. We show that under integrability conditions over the edge-step function the graphs are highly susceptible to the spread of infections, which requires only $3$ steps to infect a positive fraction of the whole graph. To prove this result, we rely on a quantitative lower bound for the maximum degree that might be of independent interest.