The semiconductor-microcavity polariton is the resulting quasi-particle from strong coupling between a quantum well exciton and cavity photon. With the cavity wavelength similar to that of the photon, the exciton-polariton (herein polariton) eigenstates come into play. Due to their light effective mass alongside the particle’s bosonic nature, polaritons undergo Bose-Einstein condensation (BEC) at high densities when below the critical condensation temperature. Unlike the traditional BEC phase transition of an atomic gas, the BEC of polaritons can be attained at much higher temperatures of 4 K up to room temperature. The work in this thesis specifically studies polariton condensates formed in inorganic semiconductor microcavities optically imprinted with ring or elliptical shaped traps. Such an excitation profile creates a potential minimum at the centre which is radially fed by polaritons. The polaritons build in population and undergo a quantum phase transition to form a polariton condensate spatially separated from the excitation profile. In this thesis, optically trapped polariton condensates are used to create a synchronised condensate network, which is initially characterised using two ground state condensates to map out regions of in-phase and anti-phase synchronisation, and through describing the system as a Stuart-Landau network, the system presents as a robust optical analogue simulator of the XY Hamiltonian. The system is then used to approximately solve the NP-hard max-3-cut graph problem through minimising the XY Hamiltonian, and the minor embedding technique is explored and characterised by mapping arbitrarily connected dense graphs to experimentally-achievable planar networks of Stuart-Landau oscillators. Next, the optically trapped first excited state polariton condensate is explored via a rotating quasi-elliptical trap where quantised vortices are observed with a deterministic rotation direction, as well as a continuous chain of excited-state condensates which exhibit geometric frustration described using Bloch’s theorem and likened to twisted states from the Kuramoto model, where the network of excited state condensates has potential as a 4D analogue simulator.