  The spectral dimension is a generalization of the Euclidean dimension and
quantifies the propensity of a network to transmit and diffuse information. We
show that, in hierarchical-modular network models of the brain, dynamics are
anomalously slow and the spectral dimension is not defined. Inspired by
Anderson localization in quantum systems, we relate the localization of neural
activity - essential to embed brain functionality - to the network spectrum and
to the existence of an anomalous "Lifshitz dimension". In a broader context,
our results help shedding light on the relationship between structure and
function in biological information-processing complex networks.
