How to biologically interpret path weight summation in weighted and directed gene regulation network?
There's quite a bit of use of graph theoretic approaches in gene regulation, so I'd suggest you start there and Stand on the Shoulders of Giants rather than reinventing gravity:

https://scholar.google.com/scholar?hl=en&amp;q=graph+theoretic+gene+regulation

Although this isn't my area of expertise, I think the thing most obvious to me as a biologist and statistician is:

Sums are misleading in non-linear systems

Biology is very non-linear. You can probably get closer to linear in the graph you are describing if the weights reflect log-transformed ratios: this transformation makes a multiplicative system linear. But even after this transformation, you are making substantial assumptions. The only way to verify those assumptions would be with either a more complex model of the underlying system (since the graph you describe is quite abstracted) or the old-fashioned way: by experiment.