In this paper we consider a distributed consensus problem over a d-dimensional network whose each dimension is connected in the same circulant pattern. We call this network as a d-dimensional uniformly circulant network (UCN). All the communication links in a UCN are modeled as erasure channels and our objective is to analyze the mean square (MS) stability of this system. Generally speaking, the complexity of checking the MS stability does not scale nicely with the system's size and dimension. Previous work has shown that for systems with spatially invariant structures, their MS stability indices can be calculated in a closed form. In this paper, we further exploit the structures of UCN-based distributed consensus systems and obtain a more efficient result for their MS stability analysis. Moreover, further discussions gives us an insight into how system parameters will affect the MS stability. Simulations are provided to illustrate our results.