  We study statistical properties of a zero-range process (ZRP) on random
networks. We derive an analytic expression for the distribution of particles
(also called node occupation distribution) in the steady state of the ZRP in
the ensemble of uncorrelated random graphs. We analyze the dependence of this
distribution on the node-degree distribution. In particular, we show that when
the degree distribution is tuned properly, one can obtain scale-free
fluctuations in the distribution of particles. Such fluctuations lead to a
power law in the distribution of particles, just like in the ZRP with the
hopping rate u(m)=1+b/m on homogeneous graphs.
