  Reaching consensus among states of a multi-agent system is a key requirement
for many distributed control/optimization problems. Such a consensus is often
achieved using the standard Laplacian matrix (for continuous system) or Perron
matrix (for discrete-time system). Recent interest in speeding up consensus
sees the development of finite-time consensus algorithms. This work proposes an
approach to speed up finite-time consensus algorithm using the weights of a
weighted Laplacian matrix. The approach is an iterative procedure that finds a
low-order minimal polynomial that is consistent with the topology of the
underlying graph. In general, the lowest-order minimal polynomial achievable
for a network system is an open research problem. This work proposes a
numerical approach that searches for the lowest order minimal polynomial via a
rank minimization problem using a two-step approach: the first being an
optimization problem involving the nuclear norm and the second a correction
step. Several examples are provided to illustrate the effectiveness of the
approach.
