How to biologically interpret path weight summation in weighted and directed gene regulation network?

Question

I've considered asking this question on Bioinformatics SE, Mathematics SE, and Stats SE, however I've judged that my question belongs on Biology SE because I am interested in a biological (domain-specific) interpretation rather than a further development of the mathematics or software.

My question pertains to applying graph theory to biology, so first I will describe the graph in its mathematical properties and how they reflect biological properties.

The graph

Let's consider a weighted and directed graph whose nodes represent genes and whose edges represent the binary relation gene A regulates gene B. The directionality of the edges is imposed in order to reflect that gene A regulates gene B does not imply that gene B regulates gene A. The magnitudes of the weights are to represent some measure of how "strongly" one gene is regulating another, and the signs of the weights are to represent whether the regulation is upregulation (positive) or downregulation (negative). For this graph, edges with a weight of zero are considered equivalent to 'no regulation' and should be excluded from the graph.

Path Analysis

Restricting ourselves to paths that do not visit the same node twice, what is the biological interpretation of the sum of the weights (of the edges) of a path between two nodes in the aforementioned graph?

I have conflicting intuitions. For example, why would these weights be additive? I'm not imposing that such weights have to be correlation coefficients, for example, but they are commonly used weights in networks that are not additive so it warrants some consideration in the more general case. If they're not additive, then their summation is not necessarily meaningful. Conditioned on being meaningful in the first place, my other inuition is that the meaning of such a summation would reflect the strength of 'indirect' regulation

Answer 1

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&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.