Adaptive Weight Selection for Optimal Consensus Performance

We address the problem of allocating weights to edges in a given 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 semi-definite programming. We present a novel, adaptive method that can be implemented on-line to solve this problem. The presented strategy asymptotically converges to the optimal
solution for any feasible initial condition, and its continuous and smooth nature lends itself to Lyapunov stability analysis. We study the case where perfect global knowledge of the algebraic connectivity and its sensitivities is available to all nodes. Also we show, as a proof-of-concept, that the scheme can be extended
to so as to be implemented in a completely distributed manner. The theoretical derivations are illustrated via representative numerical examples.