  We discuss two different regimes of condensate formation in zero-range
processes on networks: on a q-regular network, where the condensate is formed
as a result of a spontaneous symmetry breaking, and on an irregular network,
where the symmetry of the partition function is explicitly broken. In the
latter case we consider a minimal irregularity of the q-regular network
introduced by a single Q-node with degree Q>q. The statics and dynamics of the
condensation depends on the parameter log(Q/q), which controls the exponential
fall-off of the distribution of particles on regular nodes and the typical time
scale for melting of the condensate on the Q-node which increases exponentially
with the system size $N$. This behavior is different than that on a q-regular
network where log(Q/q)=0 and where the condensation results from the
spontaneous symmetry breaking of the partition function, which is invariant
under a permutation of particle occupation numbers on the q-nodes of the
network. In this case the typical time scale for condensate melting is known to
increase typically as a power of the system size.
