On connectivity, conductance and bootstrap percolation for a random k-out, age-biased graph

TL;DR

This paper analyzes the properties of a uniform attachment graph, including degree distribution, connectivity, conductance, and bootstrap percolation thresholds, revealing how infection spreads depending on initial infection probability.

Contribution

It provides the first asymptotic analysis of degree, connectivity, conductance, and bootstrap percolation thresholds for the uniform attachment graph model.

Findings

Minimum degree and connectivity follow specific asymptotic distributions.

Conductance of the graph is of order (log n)^{-1}.

Bootstrap percolation exhibits sharp threshold behavior based on initial infection probability.

Abstract

A uniform attachment graph (with parameter $k$), denoted $G_{n,k}$ in the paper, is a random graph on the vertex set $[n]$, where each vertex $v$ makes $k$ selections from $[v-1]$ uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well-studied random graphs: $k$-out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of $G_{n,k}$ to show that the conductance of $G_{n,k}$ is of order $(\log n)^{-1}$. We also study the bootstrap percolation on $G_{n,k}$, where, each vertex is either initially infected with probability $p$, independently of others, or gets infected later as a result of having $r$ infected neighbors at some point. We show that, for $2\le r\le k-1$, if $p\ll (\log n)^{-r/(r-1)}$, then, with probability approaching 1, the process ends before all vertices get infected. On the other hand, if $p\ge \omega(\log n)^{-r/(r-1)}$, where $\omega$ is a certain very slowly growing function, then all the vertices get infected with probability approaching 1.