Complex networks encode a rich repertoire of functional states that unfold once a specific dynamics is established. This is the case of the human brain, a glaring example of a complex system. Here, we investigate the relationship between structure and function in complex networks, with a focus on linear dynamical models and their application to the human brain. We first propose a model of elastic interactions between brain areas at equilibrium, corresponding to a Gaussian distribution for the brain activity time series. Thus, we tackle the problem of inferring such a model from typical human brain activity time series retrieved by functional Magnetic Resonance Imaging (fMRI).We compare several regularization (or noise-cleaning) algorithms addressing this task, finding that the quality of the inferred matrices crucially depends on regularization. Hence, we analyze the relationship between the inferred Gaussian couplings and the actual structure of the anatomical connection between brain areas, observing that the two networks share a similar hierarchical modular organization. Finally, we step out of equilibrium to study how the information encoded in the nodes of a network diffuses through it, as the system relaxes towards the stationary state. In particular, we present a framework inspired by the Renormalization Group in physics and apply it to unveil the characteristic scales and interface structures at different resolutions of both synthetic and real networks, including the human brain structural network.
Santucci, Francesca (Mon,) studied this question.