  Motivated by synchronization problems in noisy environments, we study the
Edwards-Wilkinson process on weighted uncorrelated scale-free networks. We
consider a specific form of the weights, where the strength (and the associated
cost) of a link is proportional to $(k_{i}k_{j})^{\beta}$ with $k_{i}$ and
$k_{j}$ being the degrees of the nodes connected by the link. Subject to the
constraint that the total network cost is fixed, we find that in the mean-field
approximation on uncorrelated scale-free graphs, synchronization is optimal at
$\beta^{*}$$=$-1. Numerical results, based on exact numerical diagonalization
of the corresponding network Laplacian, confirm the mean-field results, with
small corrections to the optimal value of $\beta^{*}$. Employing our recent
connections between the Edwards-Wilkinson process and resistor networks, and
some well-known connections between random walks and resistor networks, we also
pursue a naturally related problem of optimizing performance in queue-limited
communication networks utilizing local weighted routing schemes.
