It has been argued that much of evolution takes place in the absence of fitness gradients. Such periods of evolution can be analysed by examining the mutational network formed by sequences of equal fitness, that is, the neutral network. It has been demonstrated that, in large populations under a high mutation rate, the population distribution over the neutral network and average mutational robustness are given by the principal eigenvector and eigenvalue, respectively, of the network’s adjacency matrix. However, little progress has been made towards understanding the manner in which the topology of the neutral network influences the resulting population distribution and robustness. In this work, we use numerical methods and network models to enhance our understanding of how populations distribute themselves over neutral networks. We demonstrate that, in the presence of certain topological features, the population will undergo an exploration catastrophe and become confined to a small portion of the network. These results provide insight into the behaviour of populations on neutral networks, demonstrating that neutrality does not necessarily lead to an exploration of genotype/phenotype space or an associated increase in population diversity.