We have generalized the network approach to include the effects of short-range imperfections in order to analyze recent experiments on mesoscopic superconducting double loops. The presence of weakly scattering imperfections causes gaps in the phase boundary B(T) or Phi(T) for certain intervals of T, which depend on the magnetic Bur penetrating each loop. This is accompanied by a critical temperature T-c(Phi), showing a smooth transition between symmetric and antisymmetric states. When the scattering strength of imperfections increases beyond a certain limit, gaps in the phase boundary T-c(B) or T-c(Phi) appear for values of magnetic flux lying in intervals around half-integer Phi(0) = hc/2e. The critical temperature corresponding to these values of magnetic flux is determined mainly by imperfections in the central branch. The calculated phase boundary is in good agreement with experiment.