  We study a simple exchange model in which price is fixed and the amount of a
good transferred between actors depends only on the actors' respective budgets
and the existence of a link between transacting actors. The model induces a
simply-connected but possibly multi-component bipartite graph. A trading
session on a fixed graph consists of a sequence of exchanges between connected
buyers and sellers until no more exchanges are possible. We deem a trading
session "feasible" if all of the buyers satisfy their respective demands. If
all trading sessions are feasible the graph is deemed "successful", otherwise
the feasibility of a trading session depends on the order of the sequence of
exchanges. We demonstrate that topology is important for the success of trading
sessions on graphs. In particular, for the case that supply equals demand for
each component of the graph, we prove that the graph is successful if and only
if the graph consists of components each of which are complete bipartite. For
the case that supply exceeds demand, we prove that the other topologies also
can be made successful but with finite reserve (i.e., excess supply)
requirements that may grow proportional to the number of buyers. Finally, with
computations for a small instance of the model, we provide an example of the
wide range of performance in which only the connectivity varies. These results
taken together place limits on the improvements in performance that can be
expected from proposals to increase the connectivity of sparse exchange
networks.
