We study mechanical fluctuations and transport in resonator networks that are fully induced and controlled through radiation pressure drives. The optical spring effect in a multimode nano-optomechanical cavity allows the programming of arbitrary quadratic bosonic Hamiltonians for mechanical resonator networks, as well as Duffing-like nonlinearities. We use this to create networks with varying symmetry, topology, and functionality. In particular, we study how broken time-reversal symmetry, i.e., synthetic magnet fields in mechanical circuits, impacts the refrigeration performance of a network. Enhanced cooling efficiency is associated with persistent heat currents arising from the induced chirality. Moreover, we construct nonlinear mechanical networks that function as minimal information processing systems, to enable fundamental investigations of computing performance near the thermal limit. We demonstrate programmable logic gates in a single nanomechanical resonator, where different logic operations can be dynamically selected by adjusting laser parameters. Additionally, by coupling multiple mechanical modes within one resonator structure, we achieve cascadable logic gates, paving the way to programmable optomechanical computing networks.