Photonic and polaritonic lattices were recently theoretically proposed and experimentally realized as many-body simulators due to the rich behaviors exhibited by such systems at the macroscale. We show that the networks of polariton condensates encapsulate a large variety of behaviors of systems of coupled oscillators. By eliminating spatial degrees of freedom in a nonresonantly pumped polariton network, we establish that depending on the values of experimentally tunable parameters the networks of polariton condensates may represent Kuramoto, Sakaguchi-Kuramoto, Stuart-Landau, or Lang-Kobayashi oscillators and beyond. The networks of polariton condensates are therefore capable of implementing various regimes acting as analog spin Hamiltonian minimizers, producing complete and cluster synchronization, exotic spin glasses, and large-scale secondary synchronization of oscillations. We suggest that the recently implemented control of the system parameters for individual sites in polariton lattices allows addressing the interaction of sublattices that belong to different oscillatory classes.