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.
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.
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