Visibility Graph and Weak Ergodicity Breaking

We apply the Visibility Graph (VG) technique to analyze time sequences generating non-

stationary correlation functions but recovering the stationary condition in the long-time limit. We

prove that with the mere use of the VG technique it is virtually impossible to distinguish these

sequences from the idealized condition of permanently stationary correlation, which may be a sign

of pathology for physiological processes. The departure from the stationary condition is due to the

occurrence of events that we call crucial events.The VG technique is based on the assumption that

crucial events do not exist, and we prove that the adoption of a technique, called Diffusion Entropy

Analysis (DEA), allows us to distinguish processes hosting crucial events from complex systems

lacking crucial events. DEA allows us also to establish when complexity is generated by the super-

position of stationary and non-stationary processes. These results offer a new way to predict the risk

of heart failure and can also be used to shed light into the meditation-induced physiological effects,

leading us to extend the concept itself of complexity from the network structure to the challenging

field of cognition, leading to the concept of temporal oscillations and coherence.