  We propose a weight design method to increase the convergence rate of
distributed consensus. Prior work has focused on symmetric weight design due to
computational tractability. We show that with proper choice of asymmetric
weights, the convergence rate can be improved significantly over even the
symmetric optimal design. In particular, we prove that the convergence rate in
a lattice graph can be made independent of the size of the graph with
asymmetric weights. We then use a Sturm-Liouville operator to approximate the
graph Laplacian of more general graphs. A general weight design method is
proposed based on this continuum approximation. Numerical computations show
that the resulting convergence rate with asymmetric weight design is improved
considerably over that with symmetric optimal weights and Metropolis-Hastings
weights.
