Training a Graph model like an Artificial Neural Network
First, a note: don't let the graph structure make you think that this can be addressed with neural networks; this structure only suggests that this is a hierarchical model.

Second, let's take into account the following:


In your first figure, you don't connect A, B or C to the output. I assume that there is a relationship from A,B,C to output.
You mention that you have expert a priori knowledge on the relationships from A to B and from A to C.
I also assume that A, B, C and the output are numeric variables.
We don't know whether A, B, C and the output can be observed or not.
We don't know why there are 3 lines from A to C and 2 lines from A to B, so I will not take this into account.


You may want to provide more information if any of the assumptions above is not correct.

Now:

If only A and the output are observable, we are talking about a latent variable model. Latent variables are best suited for Hierarchical Bayesian methods. If the relations between variables are linear, you can model this as a bayesian linear model, e.g. hierarchical bayesian regression to model your variables and inject your prior knowledge as prior distributions over the weights. Here you can see the application of hierarchical bayesian regression to a latent variable model. This kind of methods are simple and interpretable, they handle prior knowledge graciously and they handle uncertainty.

If all variables are observable, you can define a combination of different extremely simple "neural network" models to approach your problem: Network 1 would receive A and output C; Network 2 would receive A and output B; Network 3 would receive A, B and C and generate the output. If the relationships among variables are linear, Network 1 and Network 2 are simple multiplications. You can then initialize their weights with your prior knowledge. The training would adjust the weights for everything to fit, but will not give you insight about the relationships except the fitted weights.

I suggest you think about whether bayesian hierarchical models can fit your problem.