  Topological states are useful because they are robust against disorder and
imperfection. In this study, we consider the effect of disorder and the
breaking of parity symmetry on a topological network system in which the edge
states are protected by Chern numbers. In the absence of periodicity, the local
Chern number is adopted to characterize the topological features of the
network. Our numerical results show that the local Chern number and the edge
states are very robust against onsite disorder as long as the gap of the bulk
state continuum remains open and survives even when the bulk band gap is
closed. Breaking the parity symmetry can destroy the quantization of local
Chern numbers, compromising the existence of edge modes. We observed
non-integer local Chern number peaks that are non-zero inside the bulk bands
but these non-zero non-integral local Chern numbers are not associated with the
existence of robust edge states.
