Abstract
Controlling regions in cortical networks, which serve as key nodes to control the dynamics of networks to a desired state, can be detected by minimizing the eigenratio R and the maximum imaginary part sigma of an extended connection matrix. Until now, optimal selection of the set of controlling regions is still an open problem and this paper represents the first attempt to include two measures of controllability into one unified framework. The detection problem of controlling regions in cortical networks is converted into a constrained optimization problem (COP), where the objective function R is minimized and sigma is regarded as a constraint. Then, the detection of controlling regions of a weighted and directed complex network (e. g., a cortical network of a cat), is thoroughly investigated. The controlling regions of cortical networks are successfully detected by means of an improved dynamic hybrid framework (IDyHF). Our experiments verify that the proposed IDyHF outperforms two recently developed evolutionary computation methods in constrained optimization field and some traditional methods in control theory as well as graph theory. Based on the IDyHF, the controlling regions are detected in a microscopic and macroscopic way. Our results unveil the dependence of controlling regions on the number of driver nodes l and the constraint r. The controlling regions are largely selected from the regions with a large in-degree and a small out-degree. When r = + infinity, there exists a concave shape of the mean degrees of the driver nodes, i.e., the regions with a large degree are of great importance to the control of the networks when l is small and the regions with a small degree are helpful to control the networks when l increases. When r 0, the mean degrees of the driver nodes increase as a function of l. We find that controlling sigma is becoming more important in controlling a cortical network with increasing l. The methods and results of detecting controlling regions in this paper would promote the coordination and information consensus of various kinds of real-world complex networks including transportation networks, genetic regulatory networks, and social networks, etc.