A General Dependency Structure for Random Graphs and Its Effect on Monotone Properties

TL;DR

This paper introduces a broad model of dependent random graphs called p-robust graphs, showing they are at least as likely as Erdős–Rényi graphs to possess any monotone property, thus extending classical results to dependent settings.

Contribution

The paper defines p-robust random graphs with dependent edges and proves they preserve monotone properties with at least the same probability as Erdős–Rényi graphs.

Findings

p-robust graphs include dependent edges with probability at least p

Monotone properties are at least as probable in p-robust graphs as in Erdős–Rényi graphs

Results enable extending classical Erdős–Rényi properties to dependent edge models

Abstract

We consider random graphs in which the edges are allowed to be dependent. In our model the edge dependence is quite general, we call it $p$-robust random graph. It means that every edge is present with probability at least $p$, regardless of the presence/absence of other edges. This is more general than independent edges with probability $p$, as we illustrate with examples. Our main result is that for any monotone graph property, the $p$-robust random graph has at least as high probability to have the property as an Erdos-Renyi random graph with edge probability $p$. This is very useful, as it allows the adaptation of many results from classical Erdos-Renyi random graphs to a non-independent setting, as lower bounds.