SVD or PCA can be used find the largest eigenvalue, but at a cost of $O(n^3)$ complexity. Lanczos algorithm runs much faster on a sparse matrix with complexity $O(dn^2)$ where $d$ is the average number of non-zeros in a row. It is better, but still quadratic.
My question is does anyone know any sub quadratic algorithm to find just the largest eigenvalue. It can be a very approximate algorithm that just capture the magnitude of the largest eigenvalue.