Research On Controllability Of Complex Networks

Controllability of complex networks has been a hot topic in recent years. With the deep research of complex networks, more and more attention have been attracted to the point which is how to control the system from the arbitrary state to any desired state. To fully control the system, we should firstly determine whether the system can be controlled within the existent inputs. Controllability is the final goal of the research of complex networks, and it is also practically significant. Though the traditional control theory is well developed, it does not apply to the controllability of the large scale of complex networks.An article published in2011of the journal Nature has done the groundbreaking work of the controllability of complex networks. It combined the directed networks with traditional control theory and analytically calculated the minimial number of driver n-odes based on the structural controllability. Then, the exact controllability provided the theoretical framework for the study of arbitrary networks with weights. The structural controllability and exact controllability opened up a new way to explore the controlla-bility of complex networks and attracted growing attention. Based on these, a series of following works exploring these problem deeper and more diversified.From the point of topogical structure, we firstly discuss the change of controllability in cascading failure. Due to the external attack, some nodes or edges will fail in systems, then the load in failed nodes or edges will be transferred to the left parts in system, which causes the cascading failure of more nodes or edges. We discuss the change of controllability in cascading failure based on the edge-attack. Under the removal of the highest load edge, the minimal number of driver nodes increases in cascading failure for ER networks with moderate average degree, while the controllability of ER networks with small or large average degree hold unchanged in cascades. The number of driver nodes corresponds to the amount of failed edges in strongly connected components of networks. For scale-free networks with small average degree, the amount of failed edges are different in networks with varied degree exponents, however, the increment of driver nodes are nearly the same. Under the removal of a fraction of edges, the number of driver nodes under random and intentional attacks rise alternately with the increasing of removal fraction. In addition, the change of driver nodes depends on the control category of removed edges as large removal fraction.Considering the real networks are usually the multiplex networks rather than the single ones, we analyze the effect of interactions among the layers on the controllability of multiplex networks. Different from the first part of this paper, we consider the weights on edges and calculate the minimal number of driver nodes in the method of exact con-trollability. We find that the minimal number of driver nodes decreases with correlation for lower density of interconnections. However, the controllability of networks with higher density of interconnections shows the contrary tendency. For different intercon-nections’correlations, controllability of multiplex networks depicts transition with the density of interconnections. For lower interconnections density, the networks with dis-assortative coupling patterns are harder to control. Whereas, for higher interconnections density, the networks with assortative coupling patterns are harder to control. The result shows the dense interactions is not always in favor of the control, and there exists the optimal mode of interactions for controllability of multiplex networks.Structural controllability and exact controllability could identify the minimal num-ber of driver nodes, but they ignore the length of control trajectory, energy consumption and the calculation precision. The Kalman’s rank condition is equivalent to the Grami-an’s matrix condition in traditional control theory, however, it has been demonstrated that the system with a well-conditioned Kalman’s controllability matrix may have an ill-conditioned controllability Gramian. The number of driver nodes needed to achieve full control of system in real control is larger than the value calculated by the method of structural controllability. With the increasing of the number of driver nodes, the numer-ical success rate increases sharply from zero to one which leads to a transition point of driver nodes. Based on these, we explore the effect of degree correlation on transition point and find out the result is quite different from that in structural controllability. The transition point depends linearly on the network size, and its value can not be reduced by the increasing of average degree in dense networks. The result is significant to the control of real systems.This paper focus on the effect of topogical structure on the controllability of com-plex networks. Under the method of structural controllability, exact controllability and the Gramian’s matrix controllability, we discuss the relationship between robustness and controllability, considering the impact of interconnections on controllability of multi-plex networks and explore the real control of large systems.