  Manipulating quantum thermal transport relies on uncovering the principle
working cycles of quantum devices. Here, we apply the cycle flux ranking of
network analysis to nonequilibrium thermal devices described by graphs of
quantum state transitions. To excavate the principal mechanism out of complex
transport behaviors, we decompose the quantum-transition network into cycles,
calculate the cycle flux by algebraic graph theory, and pick out the dominant
cycles with top-ranked fluxes, i.e., the cycle trajectories with highest
probabilities. We demonstrate the cycle flux ranking in typical quantum device
models, such as a thermal-drag spin-Seebeck pump, and a quantum thermal
transistor as thermal switch or heat amplifier. The dominant cycle trajectories
indeed elucidate the principal working mechanisms of those quantum devices. The
cycle flux analysis provides an alternative perspective that naturally
describes the working cycle corresponding to the main functionality of quantum
thermal devices, which would further guide the device optimization with desired
performance
