Network representation of dynamical systems: Connectivity patterns, information and predictability

The present work elaborates on predictability and information aspects of dynamical systems, in connection with the connectivity features of their network representation. The basic idea underlying this work is to map the set of coarse-grained states of a dynamical system onto a set of network nodes and transitions between them onto a set of network links. Based on the vertex centrality of these nodes, we define (a) a local indicator of predictability, (b) a measure of the information that is available about the state of the system after one transition occurring within an arbitrary long time window and (c) an upper bound for the time horizon of predictability. We address the cases of the tent and the cusp maps, as representative examples of Markov and non-Markov processes. An analytical exact result for the horizon of predictability is obtained for the tent map, as well as for its higher iterates, and its connection with the corresponding network diameters is discussed. Similarly, analytical expressions are derived for the bounds of the predictability horizon in the case of the cusp map.