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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

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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.

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  • $\begingroup$ We have the same concern about the weights, as additivity is one of the conditions of linearity. Since posting the question I've been thinking about how to refine my question to be particular weight function, otherwise my question has multiple answers depending on what properties we can assume about these weights. $\endgroup$
    – Galen
    Apr 14 '20 at 21:19
  • $\begingroup$ Yes, there are transformations that can linearize. A good point in general, although it doesn't answer my question. Partly my fault as I have not specified what operations are valid on these weights. $\endgroup$
    – Galen
    Apr 14 '20 at 21:22
  • $\begingroup$ @Galen Overall, I think the question is a bit ill-posed. Rather than starting from graph theory and thinking about how graph theoretic measures should be interpreted biologically, I think you want to start from biology and figure out what biological question you want to answer. Then, the hard step that represents an intellectual contribution to science is deciding how to use the graph theoretic tools you have. $\endgroup$
    – Bryan Krause
    Apr 14 '20 at 21:25
  • $\begingroup$ I think my question is ill-posed because it isn't specific enough. I think more would have to be said about the weights before an interpretation is possible. $\endgroup$
    – Galen
    Apr 14 '20 at 21:39

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