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 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 $\lambda /d$.