We explore the relation between the topological relevance of a node in a complex network and the individual dynamics it exhibits. When the system is weakly coupled, the effect of the coupling strength against the dynamical complexity of the nodes is found to be a function of their topological role, with nodes of higher degree displaying lower levels of complexity. We provide several examples of theoretical models of chaotic oscillators, pulse-coupled neurons and experimental networks of nonlinear electronic circuits evidencing such a hierarchical behavior. Importantly, our results imply that it is possible to infer the degree distribution of a network only from individual dynamical measurements.

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We explore the consequences of introducing higher-order interactions in a geometric complex network of Morris-Lecar neurons. We focus on the regime where travelling synchronization waves are observed out of a first-neighbours based coupling, to evaluate the changes induced when higherorder dynamical interactions are included. We observe that the travelling wave phenomenon gets enhanced by these interactions, allowing the information to travel further in the system without generating pathological full synchronization states. This scheme could be a step towards a simple modelization of neuroglial networks.