# How can you construct a social network empirically from multiple sources of data

Let's say you have databases $A$ and $B$, and a population of users $\left\{u_1,\ldots,u_n\right\}$. You're building a social network graph of $\left\{u_1,\ldots,u_n\right\}$. From database A, you get an edge weight feature $a(i,j)$ which connects user $i$ to user $j$. From database B, you get a similar edge weight $b(i,j)$.

You want to build a single network incorporating all this data. I can think of a couple ways to do this.

1. Treat edges $a$ and $b$ separately, make a multigraph...

2. Combine the two edge weights together in some way. Suppose you define the edge weights $e(i,j) = a(i,j) + \alpha b(i,j)$, and $\alpha$ is a parameter.

I'm mostly interested in method $2$, but allow for the possibility that $1$ could work well. Is there a way to optimize the choice of $\alpha$? One idea would be: remove a small subset of the edges, and then try to predict where they would be (using a clustering/triangle approach, say), or fitting a likelihood value to where the removed edges are.

No, you can't do it in an unsupervised way. The reason is simple: somehow you need to incorporate a rule of when two nodes should be connected; or if the final graph is weighted, you need a rule about the weights you expect given the $a(i,j)$ and $b(i,j)$. In the first case you end up doing classification, in the later regression. The missing rule is the free parameter you try to find, that is how $a$ and $b$ are related.
If you can't make an arbitrary decision for the free parameter (e.g. just take their sum, that is free parameter equals to one), then you need to build a dataset with several different combinations of $a$ and $b$, the required result in each case and find the free parameter using a supervised approach. By the way, I don't know your data, but you already made some assumption about that rule: that it connects $a$ and $b$ in a linear way.