In this paper, we show how to critically sparsify a given network while improving the convergence rate of the associated average consensus algorithm. Thus, instead of adding new links or reallocating them, we propose novel distributed methods to nd much sparser networks with better convergence results than the original denser ones. We propose two distributed algorithms; a) in the first one, each node solves a local optimization problem using only its two-hop neighborhood, b) the second one is a distributed algorithm based on using, at each node, the power method. As compared with previous work, the reduction in the number of active links is doubled while improving the convergence rate and having a much lower power consumption. Simulation results are presented to verify and show clearly the eciency of our approach.