  Distributed average consensus is the main mechanism in algorithms for
decentralized computation. In distributed average consensus algorithm each node
has an initial state, and the goal is to compute the average of these initial
states in every node. To accomplish this task, each node updates its state by a
weighted average of its own and neighbors' states, by using local communication
between neighboring nodes. In the networks with fixed topology, convergence
rate of distributed average consensus algorithm depends on the choice of
weights. This paper studies the weight optimization problem in distributed
average consensus algorithm. The network topology considered here is a star
network where the branches have different lengths. Closed-form formulas of
optimal weights and convergence rate of algorithm are determined in terms of
the network's topological parameters. Furthermore generic K-cored star topology
has been introduced as an alternative to star topology. The introduced topology
benefits from faster convergence rate compared to star topology. By simulation
better performance of optimal weights compared to other common weighting
methods has been proved.
