Before revision:  We provide an O(m)-time randomized algorithm for the \ell_2-norm flow diffusion problem, obtaining the approximation factor of 1+1/\mathrm{poly local cut structure, we give an alternative approach for finding locally-biased low conductance cuts .
After revision:  We provide an O(m)-time randomized algorithm for the \ell_2-norm flow diffusion problem, a recently proposed diffusion model based on network flow with demonstrated graph clustering related applications both in theory and in practice. Examples include finding locally-biased low conductance cuts. Using a known connection between the optimal dual solution of the flow diffusion problem and the local cut structure, we give an alternative approach for finding locally-biased low conductance cuts .