Introduction: Abstract: Advanced brain imaging techniques, including EEG, ECoG, fMRI, and MEG, indicate discontinuities in brain dynamics at theta rates (4-8 Hz). The observed neural processes can be interpreted as neural correlates of cognition. In particular, sudden synchronization-desynchronization transitions in brain dynamics have been identified as neural markers of higher cognition, decision making, and the so-called “aha” moment of sudden insight and understanding. We employ graph theoretical tools to interpret the experimental findings. Graph theoretical approaches have been extremely useful in the past decades to describe structural and functional properties on large-scale networks, including the world-wide-web, social networks, ecological networks, biological systems. Our focus here is on neural systems, in particular on brain networks. These studies lead to breakthrough of our understanding of cortical networks and dynamics. By developing random graph theory and percolation models, we provide a solid theoretical foundation of sudden changes in brains, and interpret them as cortical phase transitions. This leads to several crucial hypotheses, such as the presence of effects described as "Black Swan" and "Dragon King," and the predictability of the transient dynamics. Applications include not only cognitive engineering, but issues related to emergency response to catastrophic events, as well as transient dynamics in variable stars and supernovae formation.