  We analyse diffusion dynamics on weakly-coupled networks (interconnected
networks) by means of separation of time scales. Using an adiabatic
approximation we reduced the system dynamics to a Markov chain with aggregated
variables and derived a transport equation that is analogous to Fick's First
Law and includes a driving force. Entropy production is a sum of microscopic
entropy transport, which results from the particle's migration between networks
of different topologies and macroscopic entropy production of the Markov chain.
Equilibrium particles partition between different sub-networks depends only on
internal sub-network parameters. Our framework, confirmed by numerical
simulations, is also useful for considering diffusion in nested systems
corresponding to hierarchical networks with several different time scales thus
it can serve to uncover hidden hierarchy levels from observations of diffusion
processes.
