Metabolic networks comprise thousands of enzymatic reactions functioning in a controlled manner and have been shaped by natural selection. Thanks to the genome data, the footprints of adaptive (positive) selection are detectable, and the strength of purifying selection can be measured. This has made possible to know where, in the metabolic network, adaptive selection has acted and where purifying selection is more or less strong and efficient. We have carried out a comprehensive molecular evolutionary study of all the genes involved in the human metabolism. We investigated the type and strength of the selective pressures that acted on the enzyme-coding genes belonging to metabolic pathways during the divergence of primates and rodents. Then, we related those selective pressures to the functional and topological characteristics of the pathways. We have used DNA sequences of all enzymes (956) of the metabolic pathways comprised in the HumanCyc database, using genome data for humans and five other mammalian species. We have found that the evolution of metabolic genes is primarily constrained by the layer of the metabolism in which the genes participate: while genes encoding enzymes of the inner core of metabolism are much conserved, those encoding enzymes participating in the outer layer, mediating the interaction with the environment, are evolutionarily less constrained and more plastic, having experienced faster functional evolution. Genes that have been targeted by adaptive selection are endowed by higher out-degree centralities than non-adaptive genes, while genes with high in-degree centralities are under stronger purifying selection. When the position along the pathway is considered, a funnel-like distribution of the strength of the purifying selection is found. Genes at bottom positions are highly preserved by purifying selection, whereas genes at top positions, catalyzing the first steps, are open to evolutionary changes. These results show how functional and topological characteristics of metabolic pathways contribute to shape the patterns of evolutionary pressures driven by natural selection and how pathway network structure matters in the evolutionary process that shapes the evolution of the system.