Fig. 4

Benefit-damage correlation as an instance difficulty measure. a Value-weight correlation in classical knapsack instances; the stronger the correlation the computationally harder the instances (Pisinger 2005). b Benefit-damage correlation in NEP instances from PPI networks and corresponding synthetic analogs. Dots represent the % of genes having a given (benefit,damage) score, (b,d), in an NEP instance (averaged over 1-5K instances). Size and colour of each dot reflects frequency of that (b,d) pair (see bottom-right legend). ∼50% of all (b,d) pairs in Fly PPI network for example are unambiguous (b=0,d≠0 or b≠0,d=0) largely due to leaf genes of degree 1 (alone contributing on average ∼43%, large yellow dots in Fly subplot). In contrast, the fraction of unambiguous (b,d) pairs in NH, NL, and RN are ∼15.6,∼3.1, and ∼28%, respectively, as their most frequent (b,d) pairs cluster around dominant degrees (a (b,d) pair is contributed by nodes of degree d=b+d). Leaf-deprived NL network manifests the strongest (b,d) correlation given the range of ambiguity that most of its nodes (clustered they are around NL’s relatively higher mean of ∼12) can assume. The equal likelihood of an interaction being beneficial or damaging results in symmetry along the diagonal (see text for details)