Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs

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, λ)-graphs (i.e., d-regular graphs on n vertices with non-trivial eigenvalues less than λ in magnitude) our result shows that, with high probability, the graph remains connected under a homogeneous site percolation with survival probability p ≥ 1 − C1n2 with C1 and C2 depending only on λ/d. Index Terms site percolation, algebraic connectivity, matrix concentration inequalities