Degree is the best single predictor; the posted ratio is the worst of the three, and redundancy adds almost nothing.
The probability that a protein is essential is set by the ratio of its interaction degree to the redundancy of its partners, not by degree alone.
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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 โ 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.
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 โ a hub can be bypassed if its partners have alternatives โ but neighbour degree and clustering are poor measures of that, and they do not earn their place beside degree.
STRING links mix physical and functional evidence, and the 700 cutoff is a choice. Redundancy would be better measured by sequence paralogy or by redundancy of complexes, which we did not have. Essentiality is binned from phenotypes of varying quality, and the network is yeast only. A paralog-based redundancy measure could still rescue the ratio; this test rejects the proxy, not the biology.