Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs

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 all non-trivial eigenvalues of the adjacency matrix less than $\lambda$ in magnitude) our result shows that, with high probability, the graph remains connected under a homogeneous site percolation with survival probability $p\geq1-C_{1}n^{-C_{2}/d}$ with $C_{1}$ and $C_{2}$ depending only on $\lambda /d$ .