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Consider a reward function acting in a subset<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {S}}_0 \\subset {\\mathcal {S}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msub><mml:mi>S<\/mml:mi><mml:mn>0<\/mml:mn><\/mml:msub><mml:mo>\u2282<\/mml:mo><mml:mi>S<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>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<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {S}}.$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>S<\/mml:mi><mml:mo>.<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>We obtain in this way an oracle function, which gives a forecast of the reward function for the elements of<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {S}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>S<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, 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.<\/jats:p>","DOI":"10.1007\/s10994-022-06130-x","type":"journal-article","created":{"date-parts":[[2022,3,23]],"date-time":"2022-03-23T19:03:06Z","timestamp":1648062186000},"page":"1765-1797","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Semi-Lipschitz functions and machine learning for discrete dynamical systems on graphs"],"prefix":"10.1007","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8733-1045","authenticated-orcid":false,"given":"H.","family":"Falciani","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8854-3154","authenticated-orcid":false,"given":"E. 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