Flow on regular and disordered networks

The paper concerns processes on networks. The processes considered are linear and collective flow since they are the prime function of many real networks. The study has bearing on and how a network is probed by a process, and in particular how it may be categorised through its dynamic and steady-state flow characteristics. The linear flow processes considered include both symmetric and biassed flow, e.g. of single `particles', or `currents' related linearly to `density', and for these we exploit existing techniques to deal with background geometry and disorder. The nonlinear/collective flow considered is `driven' flow of vehicles and other excluding `particles' as it is captured by a biased hard core hopping model. This stochastic nonequilibrium process has been much studied in the one-dimensional case, where it provides the simplest example of a nonequilibrium phase transition. Recent work has given some insights into the typically severe effects of disorder in such collective models, mainly in one dimension. Correspondingly, striking effects are expected for such processes on structured networks, but so far little has been published, even for the pure case. The paper uses linear flow to provide a basic introduction to the subject, then briefly discusses disorder effects before moving on to nonlinear flow on pure structured networks, including the Cayley tree, and finally comments on the combined effects of structure and disorder.