Weak polarization dependence exists in most fiber components. We numerically demonstrated passive mode locking in fiber lasers based on weak polarization dependence down to polarization-dependent loss of 1.76 dB. Different polarization dependences are investigated to unveil its role for achieving passive mode locking. The anomalous dispersion region and the normal dispersion regime are both tested. It is found that, independent of operating dispersion regime, stronger gain is required for achieving mode locking with lower polarization dependence. Our numerical demonstration confirms previous experimental results and broadens the knowledge on additive-pulse mode locking.