A modified bootstrap percolation on a random graph coupled with a lattice

TL;DR

This paper introduces a novel random graph model combining a lattice with long-range edges, analyzes its diameter, and studies phase transitions in non-monotonous bootstrap percolation, motivated by brain research.

Contribution

It presents a new graph model with long-range edges, analyzes its diameter, and investigates phase transitions in bootstrap percolation, providing sharp bounds and mean-field approximations.

Findings

Graph diameter is Θ(log N) with high probability.

Phase transitions occur in bootstrap percolation on the model.

Sharp bounds on critical parameters are established.

Abstract

In this paper a random graph model $G_{\mathbb{Z}^2_N,p_d}$ is introduced, which is a combination of fixed torus grid edges in $(\mathbb{Z}/N \mathbb{Z})^2$ and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices $u,v\in(\mathbb{Z}/N \mathbb{Z})^2$ with graph distance $d$ on the torus grid is $p_d=c/Nd$, where $c$ is some constant. We show that, {\em whp}, the diameter $D(G_{\mathbb{Z}^2_N,p_d})=\Theta (\log N)$. Moreover, we consider non-monotonous bootstrap percolation on $G_{\mathbb{Z}^2_N,p_d}$. We prove the presence of phase transitions in mean-field approximation and provide fairly sharp bounds on the error of the critical parameters. Our model addresses interesting mathematical questions of non-monotonous bootstrap percolation, and it is motivated by recent results of brain research.