Before revision:  We shall point out that several random graphs satisfy the sufficient condition and hence, can be sparsified in linear time. For a general spectral sparsification problem, e.g., positive subset selection problem, a nonnegative UGA algorithm is proposed which needs O(mn^2+ n^3/\epsilon^2) time and the convergence is established .
After revision:  Finally, extensive experimental results, including applications to graph clustering and least squares regression, show the effectiveness of proposed approaches .