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.
Categories: cond-mat.stat-mech cond-mat.dis-nn physics.soc-ph