Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs

TL;DR

This paper analyzes how the algebraic connectivity of weighted graphs behaves under site percolation, providing concentration bounds and conditions for maintaining connectivity after random vertex deletions.

Contribution

It introduces a refined concentration inequality for the Laplacian of percolated graphs, establishing bounds on algebraic connectivity in weighted graphs under site percolation.

Findings

The Laplacian of the percolated graph concentrates around its expectation.

A lower bound on algebraic connectivity is derived for percolated graphs.

Graphs remain connected with high probability under certain percolation probabilities.

Abstract

We study the behavior of algebraic connectivity in a weighted graph that is subject to site percolation, random deletion of the vertices. Using a refined concentration inequality for random matrices we show in our main theorem that the (augmented) Laplacian of the percolated graph concentrates around its expectation. This concentration bound then provides a lower bound on the algebraic connectivity of the percolated graph. As a special case for $(n,d,\lambda)$-graphs (i.e., $d$-regular graphs on $n$ vertices with non-trivial eigenvalues less than $\lambda$ in magnitude) our result shows that, with high probability, the graph remains connected under a homogeneous site percolation with survival probability $p\ge 1-C_{1}n^{-C_{2}/d}$ with $C_{1}$ and $C_{2}$ depending only on $\lambda/d$.