  We analyze the solutions, on single network instances, of a recently
introduced class of constraint-satisfaction problems (CSPs), describing
feasible steady states of chemical reaction networks. First, we show that the
CSPs generalize the scheme known as Network Expansion, which is recovered in a
specific limit. Next, a full statistical mechanics characterization (including
the phase diagram and a discussion of physical origin of the phase transitions)
for Network Expansion is obtained. Finally, we provide a message-passing
algorithm to solve the original CSPs in the most general form.
