  Connectivity patterns of relevance in neuroscience and systems biology can be
encoded in hierarchical modular networks (HMNs). Moreover, recent studies
highlight the role of hierarchical modular organization in shaping brain
activity patterns, providing an excellent substrate to promote both the
segregation and integration of neural information. Here we propose an extensive
numerical analysis of the critical spreading rate (or "epidemic" threshold)
--separating a phase with endemic persistent activity from one in which
activity ceases-- on diverse HMNs. By employing analytical and computational
techniques we determine the nature of such a threshold and scrutinize how it
depends on general structural features of the underlying HMN. We critically
discuss the extent to which current graph-spectral methods can be applied to
predict the onset of spreading in HMNs, and we propose the network topological
dimension as a relevant and unifying structural parameter, controlling the
epidemic threshold.
