Human consciousness represents one the most fascinating topics in the field of Neuroscience. In addition, beyond all the theoretical advances that can result from investigations in this area, further relevant implications at clinical level can be identified. For instance, a quantitative method for detecting the transition from conscious to unconscious states might constitute a strong support to clinicians during surgeries (e.g. providing patients the optimal amount of anesthetic). At the same time, quantitative descriptions of the transition between different awareness states might lead to obtain further insights on the functionalities of the brain. These studies require the utilization of data that can be achieved by clinical technologies as fMRI and EEG. Here, our approach focuses on EEG signals, since the related data can be obtained by a cheap and non-invasive device. Investigations are then developed according to the following strategy. We generate networks by using correlation matrices obtained by filtering and spectrally correlating high-density EEG signals recorded on patients undergoing anesthesia. So, we analyze their topological features, and the relations with the different states of our individuals (i.e. the transition from awareness to sedation, and recovery back to consciousness). Moreover, we consider a number of stochastic processes on the resulting networks, as activation processes, and we apply methods derived from physics of spin models (e.g the Curie-Weiss) for a deeper understanding of the related outcome. Eventually, networks are generated considering different brain oscillations in the canonical delta, theta, gamma, alpha and beta bands. Our preliminary findings show that some network properties are conserved, while others are significantly modulated during the transition to unconsciousness. The latter, from a topological point of view, result from the variation of connectivity recorded in EEG signals during sedation.