Structural Controllability Of Complex Networks Research Articles

Input Structure Design for Structural Controllability of Complex Networks

This paper addresses the problem of the input design of large-scale complex networks. Two types of network components, redundant inaccessible strongly connected component (RISCC) and intermittent inaccessible strongly connected component (IISCC) are defined, and a subnetwork called a driver network is developed. Based on these, an efficient method is proposed to find the minimum number of controlled nodes to achieve structural complete controllability of a network, in the case that each input can act on multiple state nodes. The range of the number of input nodes to achieve minimal control, and the configuration method (the connection between the input nodes and the controlled nodes) are presented. All possible input solutions can be obtained by this method. Moreover, we give an example and some experiments on real-world networks to illustrate the effectiveness of the method.

Effect of the interdependence between subnets on the structural controllability of complex networks

AbstractControllability is a hot issue in complex networks. In this paper, without changing the number and the positions of the controllers, the effect of interdependence (viz., inter‐coupling relations) between two directed subnets that are controllable or uncontrollable on the structural controllability of complex networks is investigated. According to the Kalman rank criterion and PBH rank criterion, some sufficient conditions and necessary conditions for the structural controllability or uncontrollability of interdependent networks are presented. The results show that (1) if the uncontrollable and controllable nodes in one subnet are inter‐coupled with the controllable and uncontrollable nodes in the other subnet, respectively, and the directions are from controllable nodes to the uncontrollable nodes, then the interdependent network may be controllable; (2) the interdependent network can be obtained structurally controllable if there are simply directed paths covering all the uncontrollable nodes of the two subnets, separately; and (3) the interdependent networks are structurally controllable no matter what kind of inter‐couplings between two controllable subnets. All of these results are adaptive to undirected subnets. The final example verifies the effectiveness and correctness of the results provided in this paper.

Topology Prediction and Structural Controllability Analysis of Complex Networks Without Connection Information

In this article, we consider complex networks without connection information. The absence of global structure induces a great obstacle in the structural controllability analysis of these networks. Thus, a topology predicting method based on the connection probability matrix is proposed to provide a global structure for structural controllability analysis in this article. Furthermore, the modified principles of predicting global network topologies are established to acquire a more accurate global connection relationship. Eventually, the drive node set of these networks is determined by predicting global topologies. The accuracy of the proposed topology predicting method is verified by numerical simulations in the context of artificial networks and real networks. The results reveal that the global topology and structural controllability of complex networks with large scale and high edge density could be accurately predicted by utilizing the proposed method.

Some Combinatorial Problems in Power-Law Graphs

Abstract The power-law behavior is ubiquitous in a majority of real-world networks, and it was shown to have a strong effect on various combinatorial, structural and dynamical properties of graphs. For example, it has been shown that in real-life power-law networks, both the matching number and the domination number are relatively smaller, compared with homogeneous graphs. In this paper, we study analytically several combinatorial problems for two power-law graphs with the same number of vertices, edges and the same power exponent. For both graphs, we determine exactly or recursively their matching number, independence number, domination number, the number of maximum matchings, the number of maximum independent sets and the number of minimum dominating sets. We show that power-law behavior itself cannot characterize the combinatorial properties of a heterogenous graph. Since the combinatorial properties studied here have found wide applications in different fields, such as structural controllability of complex networks, our work offers insight in the applications of these combinatorial problems in power-law graphs.

Research Progress in Enhancing the Controllability of Complex Networks

At present, the controllability of complex networks is a hot issue in complex networks domain. From the perspective of graph theory, in the paper quite a few pioneering works on the controllability of complex networks were introduced, and the latest progress of the controllability of complex networks was emphatically summarized. In addition, the paper mainly discussed the basic ideas and effective methods to enhance the structural controllability of complex networks and analyzed the advantages and disadvantages of different methods. Finally, several problems and possible development directions of the controllability of complex networks were summarized.

Diversity of Structural Controllability of Complex Networks With Given Degree Sequence

We investigate to what extent the degree sequence of a directed network constrains the number of driver nodes. We develop a pair of algorithms that take a directed degree sequence as input and aim to output a network with the maximum or minimum number of driver nodes. We find an upper bound for the maximum and a lower bound for the minimum, and show that the algorithms achieve these bounds for all real and model networks, with few exceptions characterized by tiny system size and heterogeneous degree distributions. Applying these algorithms to a broad range of real networks, we show the gap between the upper and lower bounds can vary dramatically across different degree sequences. Thus, we introduce the notion of \bl{structural control} complexity to capture how much more difficult it is to control a specific network beyond what is required given its degree sequence, suggesting additional structure is present. Using model networks, we numerically and analytically investigate how typical features of the degree distribution affect the range of required driver nodes. We find that the minimum is determined by the number of sources or sinks, while the maximum is strongly affected by the presence of hubs.

Structural Controllability for a Class of Complex Networks With Root Strongly Connected Components

Structural controllability of complex networks has been an attractive research area, and Minimum Inputs Theorem was proposed to identify the minimum driver nodes for complex networks with one-dimensional node dynamics. Then, the Minimum Inputs Theorem was extended to the complex network with multidimensional node dynamics by looking the multidimensional node as a subnetwork. However, when the structures of these subnetworks possess the root Strongly Connected Components that have perfect matching, the minimum driver nodes of these subnetworks are not sufficient to guarantee the full control, and some extra nodes are needed to be controlled. Therefore, in this paper, we study the structural controllability of complex networks with multidimensional node dynamics whose corresponding subnetworks possess such root Strongly Connected Components. First, we apply the Maximum Matching principle to the network topology to obtain which subnetworks we need to control. Then, an algorithm is proposed to identify a feasible minimum controlled node set of the subnetwork. Finally, by analyzing the structural features of the whole network and synthetically applying the proposed algorithm, Maximum Matching principle and Graphical Approach, a flowchart is given for identifying the minimum controlled node set of the whole network. By duality, the above results can also apply to the structural observability problem of such complex networks.

Block-based minimum input design for the structural controllability of complex networks

Traditional control system design to complex networks is generally implemented by integrated structural analysis aiming at a global network. However, such a global method may be inefficient, in particular, when a massive network with a huge number of nodes and associations is considered. In this paper, motivated by the idea of “dividing and dealing”, we propose a block-based approach to the issue of minimum input design for structural controllability (MIDSC) of complex networks that potentially incurs in higher efficiency. Specifically, we consider a large-scale networked system that consists of several local blocks. The main challenge for control configuration design of this class of systems is how to find the minimum inputs of global network according to the local block information while maintaining system’s structural controllability. To this end, two block-based graphical algorithms are developed to meet the conditions required for achieving structural controllability, and meanwhile determine an optimal solution for addressing the MIDSC problem. The complexity of the proposed method is analyzed, which is also compared with existing algorithms designed mainly based on monolithic model. In particular, we show that, under some mild conditions on blocking structure, the complexity of the proposed algorithm is strictly lower than that of existing algorithms to the MIDSC problem.

Control principles for complex biological networks.

Networks have been widely used to model the structure of various biological systems. Currently, a series of approaches have been developed to construct reliable biological networks. However, the ultimate understanding of a biological system is to steer its states to the desired ones by imposing signals. The control process is dominated by the intrinsic structure and the dynamic propagation. To understand the underlying mechanisms behind the life process, the control theory can be applied to biological networks with specific target requirements. In this article, we first introduce the structural controllability of complex networks and discuss its advantages and disadvantages. Then, we review the effective control to meet the specific requirements for complex biological networks. Moreover, we summarize the existing methods for finding the unique minimum set of driver nodes via the optimal control for complex networks. Finally, we discuss the relationships between biological networks and structural controllability, effective control and optimal control. Moreover, potential applications of general control principles are pointed out.

Controllability Analysis of A Gene Network for Arabidopsis thaliana Reveals Characteristics of Functional Gene Families.

Based on structural controllability of complex networks and a constructed gene network with 9241 nodes for Arabidopsis thaliana, we classified nodes into five categories via their roles in control or node deletion, including indispensable, neutral, dispensable, driver and critical driver nodes. The indispensable nodes can increase the number of drivers after deletion, which are never drivers or critical drivers. About 10% nodes are indispensable. However, more than 60% nodes are neutral ones. More than 62% nodes are drivers, and indicates the gene network is very difficult to be fully controlled. Gene Ontology (GO) enrichment analysis reveals that different sets of nodes have preferred biological functions and processes.The indispensable nodes are significantly enriched as essential genes, drought responsive and abscisic acid (ABA) independent genes, transcriptional factors (TFs), core cell cycle genes, ABA and Gibberellin (GA) related genes. The critical drivers are enriched as receptor kinase-like genes, while shorted in WRKY TFs and functional genes that are enriched in the indispensable nodes. Robustness analysis based on node and edge additions, edge rewiring indicate the obtained conclusions are robust to network perturbations. Our investigations clarify control roles of some gene families and provide potential implications for identifying functional genes in other plant species, such as drought responsive genes and TFs.

Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

The Apollonian networks display the remarkable power-law and small-world properties as observed in most realistic networked systems. Their dual graphs are extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by adding a special vertex linked to all its three extreme vertices. In this paper, we study analytically maximum matchings and minimum dominating sets in Apollonian networks and their dual graphs, both of which have found vast applications in various fields, e.g. structural controllability of complex networks. For both networks, we determine their matching number, domination number, the number of maximum matchings, as well as the number of minimum dominating sets.

Spectral coarse grained controllability of complex networks

With the accumulation of interaction data from various systems, a fundamental question in network science is how to reduce the sizes while keeping certain properties of complex networks. Combined the spectral coarse graining theory and the structural controllability of complex networks, we explore the structural controllability of undirected complex networks during coarse graining processes. We evidence that the spectral coarse grained controllability (SCGC) properties for the Erdös–Rényi (ER) random networks, the scale-free (SF) random networks and the small-world (SW) random networks are distinct from each other. The SW networks are very robust, while the SF networks are sensitive during the coarse graining processes. As an emergent properties for the dense ER networks, during the coarse graining processes, there exists a threshold value of the coarse grained sizes, which separates the controllability of the reduced networks into robust and sensitive to coarse graining. Investigations on some real-world complex networks indicate that the SCGC properties are varied among different categories and different kinds of networks, some highly organized social or biological networks are more difficult to be controlled, while many man-made power networks and infrastructure networks can keep the controllability properties during the coarse graining processes. Furthermore, we speculate that the SCGC properties of complex networks may depend on their degree distributions. The associated investigations have potential implications in the control of large-scale complex networks, as well as in the understanding of the organization of complex networks.

Optimization of controllability and robustness of complex networks by edge directionality

Recently, controllability of complex networks has attracted enormous attention in various fields of science and engineering. How to optimize structural controllability has also become a significant issue. Previous studies have shown that an appropriate directional assignment can improve structural controllability; however, the evolution of the structural controllability of complex networks under attacks and cascading has always been ignored. To address this problem, this study proposes a new edge orientation method (NEOM) based on residual degree that changes the link direction while conserving topology and directionality. By comparing the results with those of previous methods in two random graph models and several realistic networks, our proposed approach is demonstrated to be an effective and competitive method for improving the structural controllability of complex networks. Moreover, numerical simulations show that our method is near-optimal in optimizing structural controllability. Strikingly, compared to the original network, our method maintains the structural controllability of the network under attacks and cascading, indicating that the NEOM can also enhance the robustness of controllability of networks. These results alter the view of the nature of controllability in complex networks, change the understanding of structural controllability and affect the design of network models to control such networks.

Effects of Edge Directions on the Structural Controllability of Complex Networks.

Recent advances indicate that assigning or reversing edge direction can significantly improve the structural controllability of complex networks. For directed networks, approaching the optimal structural controllability can be achieved by detecting and reversing certain “inappropriate” edge directions. However, the existence of multiple sets of “inappropriate” edge directions suggests that different edges have different effects on optimal controllability—that is, different combinations of edges can be reversed to achieve the same structural controllability. Therefore, we classify edges into three categories based on their direction: critical, redundant and intermittent. We then investigate the effects of changing these edge directions on network controllability, and demonstrate that the existence of more critical edge directions implies not only a lower cost of modifying inappropriate edges but also better controllability. Motivated by this finding, we present a simple edge orientation method aimed at producing more critical edge directions—utilizing only local information—which achieves near optimal controllability. Furthermore, we explore the effects of edge direction on the controllability of several real networks.

Structural controllability of complex networks based on preferential matching.

Minimum driver node sets (MDSs) play an important role in studying the structural controllability of complex networks. Recent research has shown that MDSs tend to avoid high-degree nodes. However, this observation is based on the analysis of a small number of MDSs, because enumerating all of the MDSs of a network is a #P problem. Therefore, past research has not been sufficient to arrive at a convincing conclusion. In this paper, first, we propose a preferential matching algorithm to find MDSs that have a specific degree property. Then, we show that the MDSs obtained by preferential matching can be composed of high- and medium-degree nodes. Moreover, the experimental results also show that the average degree of the MDSs of some networks tends to be greater than that of the overall network, even when the MDSs are obtained using previous research method. Further analysis shows that whether the driver nodes tend to be high-degree nodes or not is closely related to the edge direction of the network.

Nodal Dynamics, Not Degree Distributions, Determine the Structural Controllability of Complex Networks

Structural controllability has been proposed as an analytical framework for making predictions regarding the control of complex networks across myriad disciplines in the physical and life sciences (Liu et al., Nature:473(7346):167–173, 2011). Although the integration of control theory and network analysis is important, we argue that the application of the structural controllability framework to most if not all real-world networks leads to the conclusion that a single control input, applied to the power dominating set, is all that is needed for structural controllability. This result is consistent with the well-known fact that controllability and its dual observability are generic properties of systems. We argue that more important than issues of structural controllability are the questions of whether a system is almost uncontrollable, whether it is almost unobservable, and whether it possesses almost pole-zero cancellations.