Systems of weakly coupled networks offer an interesting feature of interaction strength separation. For Ising models it makes possible analytical description of phase transition using proper order parameters of individual subnetworks [1]. Similarly a diffusion process at a special class of multi-networks consisting of weakly coupled networks can analytically solved by an appropriate separation of time scales and by reducing the system dynamics to a Markov chain for aggregated variables. A presence of fitness factors describing attractiveness of individual nodes can be taken into account. In the case of system of two coupled networks an equation analogous to the First Fick Law with an additional driving force and a corresponding diffusion constant are found. The entropy production is a sum of entropy changes resulting from a network heterogeneity and the entropy of the Markov chain [2].
[1] K. Suchecki and J.A. Holyst, Phys. Rev. E. 74, 011122, (2006) and ibid 80, 031110, (2009).
[2] G. Siudem and J.A. Holyst,
arXiv:1303.2650