  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.
