  Average consensus algorithms can be implemented over wireless sensor networks
(WSN), where global statistics can be computed using communications among
sensor nodes locally. Simple execution, robustness to global topology changes
due to frequent node failures and underlying distributed philosophy has made
consensus algorithms more suitable to WSNs. Since these algorithms are
iterative in nature, their performance is characterized by convergence speed.
We study the convergence of the average consensus algorithms for WSNs using
regular graphs. We obtained the analytical expressions for optimal consensus
and convergence parameters which decides the convergence time for r-nearest
neighbor cycle and torus networks. We have also derived the generalized
expression for optimal consensus and convergence parameters for m-dimensional
r-nearest neighbor torus networks. The obtained analytical results agree with
the simulation results and shown the effect of network dimension, number of
nodes and transmission radius on convergence time. This work provides the basic
analytical tools for managing and controlling the performance of average
consensus algorithm in the finite sized practical networks.
