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Open problems abound in the theory of complex networks, which has foundsuccessful application to diverse fields of science. With the aim of furtheradvancing the understanding of the brain's functional connectivity, we proposeto evaluate a network metric which we term the geodesic entropy. This entropy,in a way that can be made precise, quantifies the Shannon entropy of thedistance distribution to a specific node from all other nodes. Measurements ofgeodesic entropy allow for the characterization of the structural informationof a network that takes into account the distinct role of each node into thenetwork topology. The measurement and characterization of this structuralinformation has the potential to greatly improve our understanding of sustainedactivity and other emergent behaviors in networks, such as self-organizedcriticality sometimes seen in such contexts. We apply these concepts andmethods to study the effects of how the psychedelic Ayahuasca affects thefunctional connectivity of the human brain. We show that the geodesic entropyis able to differentiate the functional networks of the human brain in twodifferent states of consciousness in the resting state: (i) the ordinary wakingstate and (ii) a state altered by ingestion of the Ayahuasca. The entropy ofthe nodes of brain networks from subjects under the influence of Ayahuascadiverge significantly from those of the ordinary waking state. The functionalbrain networks from subjects in the altered state have, on average, a largergeodesic entropy compared to the ordinary state. We conclude that geodesicentropy is a useful tool for analyzing complex networks and discuss how and whyit may bring even further valuable insights into the study of the human brainand other empirical networks.

Characterizing complex networks using Entropy-degree diagrams: unveiling changes in functional brain connectivity induced by Ayahuasca