Reduce size of a network graph for bipartite projection
I think the problem is not with the data size, but with the presence of large degree nodes (the degree of a node is its number of neighbours).
Indeed all neighbours of a node in the bipartite graph are linked together in the projection; they form a clique (complete sub-graph) in the projection. This is terrible, as the number of links in a clique of $d$ nodes is $\frac{d\cdot(d-1)}{2} \sim d^2$.
In concrete words, a node of degree 2,000 in the bipartite graph will induce almost two million links in the projection... A node of degree 20,000 will induce almost two hundred million links...
For more discussion of the problems with projections of bipartite graphs, you may have a look for instance at this paper: Basic Notions for the Analysis of Large Affiliation Networks / Bipartite Graphs