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We show that a network can self-organize its structure in a completelydistributed manner in order to optimize its synchronizability whilst satisfyingthe local constraints: non-negativity of edge weights, and maximum weighteddegree of nodes. A novel multilayer approach is presented which uses adistributed 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 soas to maximize $\lambda_2$, or minimize $r$ and, hence, achieve an optimalstructure.

Self-organization of weighted networks for optimal synchronizability