  We study quantum entanglement distribution on networks with full-rank
bi-partite mixed states linking qubits on nodes. In particular, we use
entanglement swapping and purification to partially entangle widely separated
nodes. The simplest method consists of performing entanglement swappings along
the shortest chain of links connecting the two nodes. However, we show that
this method may be improved upon by choosing a protocol with a specific
ordering of swappings and purifications. A priori, the design that produces
optimal improvement is not clear. However, we parametrize the choices and find
that the optimal values depend strongly on the desired measure of improvement.
As an initial application, we apply the new improved protocols to the
Erd\"os--R\'enyi network and obtain results including low density limits and an
exact calculation of the average entanglement gained at the critical point.
