Are hub proteins essential because of the ratio?

We built the Saccharomyces cerevisiae interaction network from STRING v12, keeping links scored at least 700, and took essentiality from SGD null-mutant phenotypes: a protein is essential if any null allele is inviable. Redundancy is measured two ways — the mean degree of a protein's partners, and the local clustering coefficient of its neighbourhood. We compared five logistic models by five-fold cross-validated AUC: degree, redundancy, the posted ratio, degree plus redundancy, and all terms.

• Degree wins. Degree alone reaches AUC 0.729. Redundancy reaches 0.688. The posted ratio reaches 0.629, below both of its components.
• Redundancy adds little. Adding redundancy to degree moves AUC from 0.729 to 0.735, a gain of 0.006. Adding the ratio as well does not improve on that.
• The gradient is real. Essentiality rises from about 5 per cent in the lowest degree decile to 48 per cent in the highest. The association the claim wants to explain is there; the ratio is not what explains it.

The ratio is worse than degree because it divides by a noisy partner statistic that is itself correlated with degree. In this network, a hub's partners tend to be well connected, so the ratio compresses exactly the variation it was meant to isolate. The redundancy idea is biologically reasonable — a hub can be bypassed if its partners have alternatives — but neighbour degree and clustering are poor measures of that, and they do not earn their place beside degree.