Self-organization of weighted networks for optimal synchronizability

We show that a network can self-organize its structure in a completely distributed manner in order to optimize its synchronizability whilst satisfying the local constraints: non-negativity of edge weights, and maximum weighted degree of nodes. A novel multilayer approach is presented which uses a distributed strategy to estimate two spectral functions of the graph Laplacian, the algebraic connectivity $\lambda_2$ and the eigenratio $r = \lambda_n / \lambda_2$ . These local estimates are then used to evolve the edge weights so as to maximize $\lambda_2$, or minimize $r$ and, hence, achieve an optimal structure.