We consider robust stability analysis of a class of spatially interconnected systems. The individual subsystems may be different but they are assumed to share some properties that can be characterized by an integral quadratic constraint. The main contribution of the paper is to show that, for the case where the network interconnection matrix is normal, (robust) stability verification can be simplified to a simple problem of checking the location of the eigenvalues of the interconnection matrix. Most interestingly, we also identify a class of networks for which this characterization on eigenvalues is necessary and sufficient for robust stability.