For analysis, we use tools and concepts from the field of complex networks, a brief history of which follows. The application of graph theory to phase transitions and complex systems led to significant progress in understanding a variety of cooperative phenomena over a period of several decades. In the 1960s, the books by Harary, especially Graph Theory and Theoretical Physics 34, introduced readers to powerful mathematical techniques. The chapter by Kastelyn, still considered to be a classic, showed that difficult combinatorial problems of exact enumeration could be attacked via graph theory, including the exact solution of the two-dimensional Ising model (e.g., see Feynman35). In the 1980s, certain families of neural network models were shown to be equivalent to Ising systems, e.g., the Hopfield network36 is a content-addressable memory which is isomorphic to a generalized Ising model37. Beginning in the 1990s, new approaches to networks, giving emphasis to concepts such as the node degree distribution, clustering, assortativity, small-worldliness; and network efficiencies, led eventually to what has become the new field of complex networks38, 39. These new tools and concepts40,41,42 have found successful application in the study of diverse phenomena, such as air transportation networks43, terrorist networks44, gene regulatory networks45, and functional brain networks46,47,48,49,50. We approach the human brain from this perspective of complex networks51, 52.
The key technical innovation is the measurement of the Shannon entropy of the degree distribution of the complex networks that represent the functional connectivity of the human brain. Moreover, the Shannon entropy is also very closely related to the Boltzmann-Gibbs entropy used in statistical mechanics. Hence, our approach to studying the brain experimentally is grounded in two strong theoretical traditions: graph theory and complex networks on the one hand, and information theory and statistical physics on the other. Our study also represents a significant advance for the following additional reasons: (i) our results unveil how Ayahuasca (and likely most other tryptamine psychedelics) alter brain function, both locally and globally; (ii) it is the first time this specific approach has been applied to characterize functional brain networks in altered states of consciousness; (iii) our study of Ayahuasca covers all brain regions; and (vi) the method we have developed can be immediately applied to study a variety of other phenomena (e.g., the effects of medication for mental health disorders).
An effort has been made to present the various topics in the theory of graphs in a logical order, to indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results. In addition, there are three appendices which provide diagrams of graphs, directed graphs, and trees. The emphasis throughout is on theorems rather than algorithms or applications, which however are occaisionally mentioned. 2b1af7f3a8