Semi-Lipschitz functions and machine learning for discrete dynamical systems on graphs

Abstract

Consider a directed tree {mathcal {U}} and the space of all finite walks on it endowed with a quasi-pseudo-metric—the space of the strategies {mathcal {S}} on the graph,—which represent the possible changes in the evolution of a dynamical system over time. Consider a reward function acting in a subset {mathcal {S}}_0 subset {mathcal {S}} which measures the success. Using well-known facts of the theory of semi-Lipschitz functions in quasi-pseudo-metric spaces, we extend the reward function to the whole space {mathcal {S}}. We obtain in this way an oracle function, which gives a forecast of the reward function for the elements of {mathcal {S}}, that is, an estimate of the degree of success for any given strategy. After explaining the fundamental properties of a specific quasi-pseudo-metric that we define for the (graph) trees (the bifurcation quasi-pseudo-metric), we focus our attention on analyzing how this structure can be used to represent dynamical systems on graphs. We begin the explanation of the method with a simple example, which is proposed as a reference point for which some variants and successive generalizations are consecutively shown. The main objective is to explain the role of the lack of symmetry of quasi-metrics in our proposal: the irreversibility of dynamical processes is reflected in the asymmetry of their definition.