The aim of the study was to investigate differences in cortical networks based on the state of consciousness. Five subjects performed a serial-awakening paradigm with electroencephalography (EEG) recordings. We considered four states of consciousness: (1) non-rapid eye movement (NREM) sleep with no conscious experience, (2) NREM sleep with conscious experience, (3) rapid eye movement (REM) sleep with conscious experience, and (4) wakefulness. We applied graph theoretical analysis to explore the cortical connectivity and network properties in five frequency bands. Connectivity between EEG channels was evaluated with the weighted phase lag index (wPLI). The characteristic path length, transitivity, and clustering coefficient were computed to evaluate functional integration and segregation of the associated brain network. There were no significant differences in wPLI among the four states of consciousness. In the beta band, functional integration in wakefulness was higher than in NREM sleep. Regarding functional segregation, in the theta band, transitivity and clustering coefficient in NREM sleep with no conscious experience were stronger than in wakefulness or REM sleep, but clustering in the beta band showed an opposite effect. The observed differences may be related to cortical bistability and add to previously observed neural correlates of consciousness.