Figure 3

Controlling the states of the edges in a network. (a) A directed network with target edges(the red lines). (b) We locate the out-edges of the given nodes on its right, and the in-edges on the left. In this way, we construct a bipartite graph. We use SBD theory to obtain the driven edges and the driver nodes, which are marked with blue and green. The driven edges are {(1,2),(2,3),(4,5)} and the driver nodes are {1,2,4}. (c) By controlling the three nodes and the three driven edges, all the target edges can be controlled. Moreover, the whole system can be controlled. (d) We use the greedy algorithm to control the target edges. In the first iteration, the target edges are shown on the right (shown in red). For the tail node of each target edge, if in_degree(the number of edges on the left) is not less than out_degree(the number of edges on the right), we select the same number of out_edges on the left. Otherwise, the extra target edges are the driven edges according to SBD theory. In the first iteration, the target edge (1,2) is a driven edge(highlighted in blue). In the next iteration, the edges on the left of the bi_layer bipartite graph are the new target edges in the second iteration. After four rounds of iteration, we obtain the driver node 1,2 and the driven edge (1,2),(2,3). (e) By controlling the driver node calculated from the TEC, we can control the driven edge. Then, all the targets can be controlled.