  This paper investigates the weighted-averaging dynamic for unconstrained and
constrained consensus problems. Through the use of a suitably defined adjoint
dynamic, quadratic Lyapunov comparison functions are constructed to analyze the
behavior of weighted-averaging dynamic. As a result, new convergence rate
results are obtained that capture the graph structure in a novel way. In
particular, the exponential convergence rate is established for unconstrained
consensus with the exponent of the order of $1-O(1/(m\log_2m))$. Also, the
exponential convergence rate is established for constrained consensus, which
extends the existing results limited to the use of doubly stochastic weight
matrices.
