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Fig. 2 | Applied Network Science

Fig. 2

From: Computational intractability law molds the topology of biological networks

Fig. 2

The network evolution problem (NEP). a Left: an example network of seven genes G=g1,g2,..,g7 that is under some hypothetical evolutionary pressure described by an Oracle advice (OA) A=(0,+1,−1,+1,0,0,−1). The OA is indicating that genes g2,g4 and g3,g7 should be promoted and inhibited respectively (notice +1 and -1 values in OA; aiA is the Oracle’s opinion on gene gi). Centre: the network graph in an equivalent adjacency matrix representation; green or red entry indicates the interaction is in agreement or disagreement, respectively, with the OA (e.g. m14=−1 in disagreement with a4=+1). Right: benefit/damage scores equal the sum of beneficial/damaging interactions gi is projecting onto or attracting from other genes (counting along row i for projection and column i for attraction; g3’s scores highlighted by dotted frames as an example). b, an instance of the knapsack optimization problem (KOP); the challenge is to determine the objects to include in and exclude from the limited-capacity knapsack so as to maximize the overall total value while keeping the overall total weight of packed items under the knapsack’s capacity threshold (5 lb). This KOP instance is reducible to the NEP instance in (a); objects/values/weights correspond to the genes/benefits/damages in (a) (e.g. laptop in (b) corresponds to gene g2 in (a)). Capacity in KOP corresponds to a damage tolerance threshold in NEP (not depicted in (a), see text). Solving the question ‘which genes to conserve and which to delete’ in (a) translates into a solution to the corresponding KOP instance in (b). NEP instance optimal solution, with a threshold ≤5 damaging interactions, is to conserve genes g2,g4,g5, g6 and delete g1,g3g7, which respectively translate back to an optimal solution to the KOP instance: include in the knapsack the laptop, lunch box, pen and water bottle, and exclude from it the notebook, textbook, and candle. c and d, NEP can semantically be interpreted in the context of, respectively, regulation (which genes or interactions should be fine-tuned positively or negatively) or (evolution (which genes or interactions represent an asset or a liability to the system long-term). The formal definition of NEP and a generalized reduction of KOP to NEP are included in SI 3 and 4 respectively. See main text and SI 5 for interaction-targeting OA (as opposed to gene-targeting OA here in a)

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