  We consider nonequilibrium phase transitions in weighted scale-free networks,
in which highly connected nodes, which are created earlier in time are
partially immunized. For epidemic spreading we solve the dynamical mean-field
equations and discuss finite-size scaling theory. The theoretical predictions
are confronted with the results of large scale Monte Carlo simulations on the
weighted Barab\'asi-Albert network. Local scaling exponents are found different
at a typical site and at a node with very large connectivity.
