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Transportation networks are inevitably selected with reference to their global cost which depends on the strengths and the distribution of the embedded currents. We prove that optimal current distributions for a uniformly injected d-dimensional network exhibit robust scale-invariance properties, independently of the particular cost function considered, as long as it is convex. We find that, in the limit of large currents, the distribution decays as a power law with an exponent equal to (2d-1)/(d-1). The current distribution can be exactly calculated in d=2 for all values of the current. Numerical simulations further suggest that the scaling properties remain unchanged for both random injections and by randomizing the convex cost functions.