Assessing the synthesizability of inorganic materials is a grand challenge for accelerating their discovery using computations. Synthesis of a material is a complex process that depends not only on its thermodynamic stability with respect to others, but also on factors from kinetics, to advances in synthesis techniques, to the availability of precursors. This complexity makes the development of a general theory or first-principles approach to synthesizability currently impractical. Here we show how an alternative pathway to predicting synthesizability emerges from the dynamics of the materials stability network: a scale-free network constructed by combining the convex free-energy surface of inorganic materials computed by high-throughput density functional theory and their experimental discovery timelines extracted from citations. The time-evolution of the underlying network properties allows us to use machine-learning to predict the likelihood that hypothetical, computer-generated materials will be amenable to successful experimental synthesis.