  We analyze the existence and the size of the giant component in the
stationary state of a Markovian model for bipartite multigraphs, in which the
movement of the edge ends on one set of vertices of the bipartite graph is a
zero-range process, the degrees being static on the other set. The analysis is
based on approximations by independent variables and on the results of Molloy
and Reed for graphs with prescribed degree sequences. The possible types of
phase diagrams are identified by studying the behavior below the zero-range
condensation point. As a specific example, we consider the so-called Evans
interaction. In particular, we examine the values of a critical exponent,
describing the growth of the giant component as the value of the dilution
parameter controlling the connectivity is increased above the critical
threshold. Rigorous analysis spans a large portion of the parameter space of
the model exactly at the point of zero-range condensation. These results,
supplemented with conjectures supported by Monte Carlo simulations, suggest
that the phenomenological Landau theory for percolation on graphs is not broken
by the fluctuations.
