Adaptive weight selection for optimal consensus performance

We address the problem of allocating weights to edges in an undirected network topology, subject to constraints limiting the weighted degree of nodes, so as to maximise the algebraic connectivity of the network. The problem is convex and can be solved efficiently through techniques in semidefinite programming. We present a novel, adaptive method that can be implemented on-line to solve this problem and prove its convergence to the optimal solution for any feasible initial condition. First, we study the case where perfect global knowledge of the algebraic connectivity and its sensitivities is available to all nodes. Then we show, as a proof-of-concept, that the scheme can be extended to be implemented in a completely distributed manner.