Figure 5
From: Enabling Controlling Complex Networks with Local Topological Information
Interesting observations of ILQR. (a) Illustration that control nodes tend to divide elementary topologies averagely for consuming less energy cost as the cost is mainly dependent on the length of the longest control path. The located control nodes are marked in an elementary stem and circle in (a1–a3) and (a7–a9) (M = 1, 2, 3) and in an elementary dilation with (a4–a6) (M = 2, 3). When M = 2, the control node set converges to either {node 1, 5} in case a4 or {node 1, 2} in case a5 with different probabilities, around 34.19% and 65.81% respectively in 10000 rounds of experiments. When M = 3, the control node set approaches {node 1, 4, 5} and {node 1, 3, 5} with percentages 24.64% and 75.36%, respectively, in 10000 rounds of experiments. (b) Illustration that the control cost is proportional to the longest control path. Here M external control inputs are randomly allocated on a 100-node elementary circle, and the control cost vs the longest control path that is just the maximum values of all the number of edges between any two adjacent control nodes, are recorded. The experiments are simulated on Matlab with higher precision (130 significant digits) by using the Advanpix multi-precision computing toolbox. (c) Illustration that MLCP performs significantly better than Random Allocation Method (RAM), in low-degree networks while they become almost indistinguishable as the mean degree (mean in-/out-degree) of the networks becomes dense. The experiment is done on an ER network by adding edges randomly and persistently (SI, Section 4.4), with M being given by \(M={N}_{D}^{LM}+{m}_{0}\) (m0 = 100). The mean degree increases as more edges are continuously added, and the three fitting curves plotted for RAM, OPGM and MLCP respectively coincide with each other when mean degree is around 6.
