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Fig. 2 | BMC Genomics

Fig. 2

From: A novel algorithm for finding optimal driver nodes to target control complex networks and its applications for drug targets identification

Fig. 2

Demonstration of identifying the target controllable subspace a A directed network. The target control nodes set is {v3,v4,v6} (highlighted in green) and the constrained nodes set is {v1} (shape in hexagon). b Construct the “linking and dynamic graph”. Initialize a bipartite graph B0, where the right side R0 consists of the target nodes {v3,v4,v6}, and the left side L0 consists of the nodes that can reach the target nodes. Identify the maximum matching m 0  = {(v 2 ,v 3 ),(v 3 ,v 4 ),(v 5 ,v 6 )} in the initialized bipartite graph B0. Let the matched nodes {v2,v3,v5} in L0 to be R1 set and get a new bipartite graph B1. In the new bipartite graph B1, we can obtain the corresponding maximum matching m 1  = {(v 1 ,v 3 ),(v 3 ,v 5 )}. Repeat this process and we obtain the maximum matching m 2  = {(v 1 ,v 3 )} in the new bipartite graph B2, which result in the “linking and dynamic graph” {m 0 , m 1 , m 2 } c Construct the target control tree from the “linking and dynamic graph”. In the sub-graph CF(R0 + L0,m0) and subgraph CF(L0 + L1,m1) and CF(L1 + L2,m2), add edges set E 0  = {(v 2 ,v 3 ), (v 2 ,v 4 ), (v 3 ,v 4 ),(v 5 ,v 6 )} and E 1  = {(v 1 ,v 3 ), (v 1 ,v 5 ), (v 3 ,v 5 )} and E 2  = {(v 1 ,v 3 )} to TCT, which result in the target control tree. d We first form a new bipartite graph, in which the up layer consist of all the nodes in the network, and the bottom layer consist of the target nodes. Based on the target control subspace theorem, we can show that the node v i in the L 0 , L 1 , L 2 can control node v j in R 0 , if there exist a path from v i to the target node v j in the target control tree TCT. And then we add edges from the node v i in the up layer to node v j in the bottom layer for the new formed bipartite graph. Finally we can identify the target controllable subspace of each node from the formed bipartite graph

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