  We study the electronic and transport properties of a network of domain walls
between insulating domains with opposite valley Chern numbers. We find that the
network is semi-metallic with Dirac dispersion near the charge neutrality point
and the corresponding electronic states distribute along the domain walls. Near
the charge neutrality point, we find quantized conductance in nanoribbon with
sawtooth domain wall edges that propagates along the boundaries and is robust
against weak disorder. For a trident edged ribbon, we find a small energy gap
due to the finite size effect making the nanoribbon an insulator. When the
Fermi energy is away from charge neutrality point, all domain walls contribute
to the conduction of current. Our results provide a comprehensive analysis of
the electronic transport properties in a topological domain wall network that
not only agrees qualitatively with experiments on marginally twisted bilayer
graphene under a perpendicular electric field, but also can provide useful
insights for designing low-power topological quantum devices.
