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Network models of neural connectivity and function often invoke the ability of the brain to localize activity in distinct modules simultaneously. The propensity of a network to do the opposite instead, that is to transmit and diffuse information homogeneously, is measured by its spectral dimension, a quantity that is easily accessible through analyses of random walks, or equivalently diffusion processes. Here we show that diffusive dynamics in hierarchical modular network models, representing brain connectivity patterns, exhibit a strongly anomalous features, pointing to a global asymptotic slowdown at large times and to the emergence of localization phenomena. Using theoretical modeling and very-large-scale computer simulations, we demonstrate that the spectral dimension is not defined in such systems and that the observed anomalous dynamical features stem from the existence of Lifshitz tails in the lower spectral edge of the Laplacian matrix. We are able to derive the correct scaling laws relating the spectral density of states and anomalous dynamics, emphasizing the fundamental role played by the Lifshitz dimension. Our work contributes to establishing a theoretical framework for anomalous dynamical features, such as activity localization and frustrated synchronization in hierarchical and hierarchical-modular networks and helps contextualize previous observations of sub-diffusive behavior and rare-region effects in brain networks. More in general, our results, help shedding light on the relation between structure and function in biological information-processing complex networks.

Anomalous Lifshitz dimension in hierarchical networks of brain connectivity