  The problem of stochastic advection of passive particles by circulating
conserved flows on networks is formulated and investigated. The particles
undergo transitions between the nodes with the transition rates determined by
the flows passing through the links. Such stochastic advection processes lead
to mixing of particles in the network and, in the final equilibrium state,
concentration of particles in all nodes become equal. As we find, equilibration
begins in the subset of nodes, representing flow hubs, and extends to the
periphery nodes with weak flows. This behavior is related to the effect of
localization of the eigenvectors of the advection matrix for considered
networks. Applications of the results to problems involving spreading of
infections or pollutants by traffic networks are discussed.
