Is this a valid method for unipartite projection of a bipartite graph? I would like to know if a given method of projecting a bipartite graph exists, and if yes, if there is a formula for transforming the weight matrix.
Given a bipartite graph with edges' weights described by this matrix:
   x  y  z
A  0  1  3
B  2  4  6

The simple weighting method of projecting it as a unipartite graph would yield a matrix like this:
   A  B
A  0  2
B  2  0

Since A and B have common associations with 2 elements, y, and z.
I looking to keep some information about the weight of the original matrix, keeping the more little values between the two. For instance, A and B have a common association with z, with respective edges of weight 3 and 6, in that case 3 would be the value added to the unipartite matrix, instead of 1.
Finally, this method would yield this matrix for the projection:
   A  B
A  0  4
B  4  0

More formally:
Edge(A, B) = sum( min( Edge(A,y), Edge(B,y) ) + min( Edge(A,z), Edge(B,z) ) )
Edge(A, B) = sum( min( 1, 4 ) + min ( 3, 6 ) )
Edge(A, B) = sum( 1 + 3 )
Edge(A, B) = 4

My question is: is this method is documented or has a name ? If yes, does it exist a formula to transform the first matrix to the last ?
Thanks very much for you help !
 A: You may consider another method to project the bipartite graph to a unipartite one, called "affinity index". It is a simple modification to what you showed, but could be more sensible, as it is based on kinda rigorous probabilistic concepts. You just normalize your number of common neighbors by both A's and B's "popularities" in the following way. 
Let  $N_{total}$  be the total number of the nodes on the left part of the graph (x, y, z here, hence 3), and $N(A)$ be the A's "popularity" (degree in the bipartite graph, here equals 2). $N(A\cap B)$ is the number of common neighbors of A and B. Then:
$$
affinity(A,B) = \frac{N(A\cap B)*N_{total}}{N(A)*N(B)}
$$
It's not difficult to show that it's nothing else than an estimate of the following quantity:
$$
affinity(A,B) \equiv \frac{\hat{p}(A,B)}{\hat{p}(A)\hat{p}(B)}
$$
Here $p(A)$ is a probability of event "link with A" for every node on the left. If the events "link with A" and "link with B" were independent, there would be $p(A,B) = p(A)p(B)$. Then, affinity index measures how much the events "link with A" and "link with B" can tell about an occurrence of each other. Normalization makes the projection be aware of a fact that, say, 10 common left-neighbors for very "popular" nodes A and B say much less about their similarity than for "unpopular" nodes C and D. 
You may incorporate weights similarly, but in this case you would have to find some workaround computing $N(A\cap B)$
