  We discuss polariton graphs as a new platform for simulating the classical XY
and Kuramoto models. Polariton condensates can be imprinted into any
two-dimensional graph by spatial modulation of the pumping laser. Polariton
simulators have the potential to reach the global minimum of the XY Hamiltonian
in a bottom-up approach by gradually increasing excitation density to the
threshold or to study large-scale synchronisation phenomena and dynamical phase
transitions when operating above the threshold. We consider the modelling of
polariton graphs using the complex Ginzburg-Landau model and derive analytical
solutions for a single condensate, the XY model, two-mode model and the
Kuramoto model establishing the relationships between them.
