Weight Design for Consensus Algorithms in Static and Random Topologies: Finite Time Horizon

We address the problem of the weights design for consensus algorithms under random network topology. We differ two different cases: 1) high estimate precision is required, and there is no firm restriction on the number of available iterations; 2) there is only a small, limited budget of iterations available. For the first case, we show that minimizing the mean squared error contraction factor is a convex optimization problem and we globally solve it by the subgradient algorithm. For the case of limited budget of iterations, we consider both static and random topologies. For static topology, we define the objective as to minimize the sum of k largest eigenvalues of the consensus error contraction matrix. We show that the proposed optimization problem is convex which enables finding globally optimal solution. Further, it has been discovered that there is a trade off for the choice of the parameter k = 1, ..., N : larger k yields faster convergence in the transient regime and slower long run convergence. Thus, the parameter k should be tuned according to a specific application and requirements. All results are extended to the case of random topology, and the validity of the proposed weight choice is verified through simulations.