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

Abstract Consider a directed tree $${\mathcal {U}}$$ U and the space of all finite walks on it endowed with quasi-pseudo-metric—the strategies {S}}$$ S graph,—which represent possible changes in evolution dynamical system over time. reward function acting subset {S}}_0 \subset {\mathcal 0 ? which measures success. Using well-known facts theory semi-Lipschitz functions quasi-pseudo-metric spaces, we extend to whole {S}}.$$ . We obtain this way an oracle function, gives forecast for elements , that is, estimate degree success any given strategy. After explaining fundamental properties specific define (graph) trees (the bifurcation quasi-pseudo-metric), focus our attention analyzing how structure can be used systems graphs. begin explanation method simple example, is proposed as reference point some variants successive generalizations are consecutively shown. The main objective explain role lack symmetry quasi-metrics proposal: irreversibility processes reflected asymmetry their definition.