Free zero-range processes on networks

A free zero-range process (FRZP) is a simple stochastic process describing the dynamics of a gas of particles hopping between neighboring nodes of a network. We discuss three different cases of increasing complexity: (a) FZRP on a rigid geometry where the network is fixed during the process, (b) FZRP on a random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical network whose topology continuously changes during the process in a way which depends on the current distribution of particles. The case (a) provides a very simple realization of the phenomenon of condensation which manifests as the appearance of a condensate of particles on the node with maximal degree. The case (b) is very interesting since the averaging over typical ensembles of graphs acts as a kind of homogenization of the system which makes all nodes identical from the point of view of the FZRP. In the case (c), the distribution of particles and the dynamics of network are coupled to each other. The strength of this coupling depends on the ratio of two time scales: for changes of the topology and of the FZRP. We will discuss a specific example of that type of interaction and show that it leads to an interesting phase diagram.