Fundamental properties like observation of magnetic monopoles in artificial spin ice lattice [1] depends on the underlying dynamics of magnetic domain walls (DWs) in geometrically frustrated structures. It has been reported previously that the DW follows a chirality dependent path in magnetic network structures [2-4]. This concept has been applied to generate 1D Dirac strings in artificial Kagome type lattice [3]. Thus the deterministic trajectory of magnetic charge carriers in spinice structure and observation of magnetic monopoles depends on the robustness of the chirality of the injected DW. However above a critical field (Walker field), the DW changes its chirality after propagating a distance larger that its fidelity length [5]. This leads to inherent stochasticity in the trajectory of the DW. To overcome this issue, we introduce an asymmetry in the magnetic network structure. We demonstrate with the aid of magnetic force microscopy (MFM) imaging that the DW is constrained to move along one of the two branches irrespective of its chirality. This study can be extended to define trajectory of magnetic charge carriers in artificial spin ice lattice. © 2015 IEEE.