The Controllability Analysis Of Complex Networks And The Topological Properties Research Of Driver Nodes Sets

In the real world, many systems exist in the form of directed complex networks. To ensure the normal operation of these systems, it is necessary to control the whole system. The system is said to be controllable if it can be driven from any initial state to any desired final state in finite time. If each node in the network can be controlled individually, the whole network is controllable, but it is difficult to achieve for a large complex systems. We hope to find the minimum number of nodes to control the whole network. The complex network is mapped to a linear system, using the algorithm of bipartite graph maximum matching, and the non-matching node act as the driver nodes. We can control the complex networks by inputting the external signal to the driver nodes.This paper studies the controllability of complex networks based on the algorithm of bipartite graph maximum matching, and mainly analyzes the characteristics of the sets of the driver nodes and the core driver nodes. Firstly, we study the change of the average degree of the driver nodes sets in the real networks and the model networks, and show the topological characteristics of the different sets of driver nodes, using the algorithm of the high-degree nodes as the initial matching and the algorithm of the low-degree nodes as the initial matching. Secondly, we find the sets of driver nodes repeatedly using random sampling method through ordering the network nodes randomly, then study the number of the different sets and the characteristics of them, and compare the characteristics with the degree of the network. Then research the characteristics and effect on the network controllability of the intersection of the driver nodes sets, and summarize the principle of network controllability. Thirdly, we classify each link and each node into three different types according to its role in maintaining controllability, and analyze them on the network control effects. Finally, we count the numbers of occurrences of each node in the sets of driver nodes, then give the definition and solution method of the core driver nodes, and analyze their characteristics and role in the network controllability. Through analyzing the characteristics of the driver nodes sets, we find that the average degree of the driver nodes sets have a wide range, and the average degree of driver nodes sets is big than that of network in some networks. So we think the conclusion that the driver nodes tend to avoid the high-degree nodes is unreasonable. We find the principle of network controllability is that to affect the high-degree nodes to control the whole network by controlling the low-degree nodes.