  We develop a general framework for studying the equilibrium and
non-equilibrium properties of arbitrary networks of Sachdev-Ye-Kitaev clusters
coupled to thermal baths. We proceed to apply this technique to the problem of
energy transport, which is known to be diffusive due to the strange metal
behavior of these models. We use the external baths to impose a temperature
gradient in the system and study the emerging non-equilibrium steady state
using the Schwinger-Keldysh formalism. We consider two different configurations
for the baths, implementing either a boundary or bulk driving, and show that
the latter leads to a significantly faster convergence to the steady state.
This setup allows us to compute both the temperature and frequency dependence
of the diffusion constant. At low temperatures, our results agree perfectly
with the previously known values for diffusivity in the conformal limit. We
also establish a relationship between energy transport and quantum chaos by
showing that the diffusion constant is upper bounded by the chaos propagation
rate at all temperatures. Moreover, we find a simple analytical form for the
non-equilibrium Green's functions in the linear response regime and use it to
derive exact closed-form expressions for the diffusion constant in various
limits. We mostly focus on uniform one-dimensional chains, but we also discuss
higher-dimensional generalizations.
