Controllability Of Complex Networks Research Articles

Quantitative Controllability Index of Complex Networks

In this paper, the controllability issue of complex network is discussed. A new quantitative index using knowledge of control centrality and condition number is constructed to measure the controllability of given networks. For complex networks with different controllable subspace dimensions, their controllability is mainly determined by the control centrality factor. For the complex networks that have the equal controllable subspace dimension, their different controllability is mostly determined by the condition number of subnetworks’ controllability matrix. Then the effect of this index is analyzed based on simulations on various types of network topologies, such as ER random network, WS small-world network, and BA scale-free network. The results show that the presented index could reflect the holistic controllability of complex networks. Such an endeavour could help us better understand the relationship between controllability and network topology.

Network structures and systems: II. Network and multiplex cores

The notion of the flow core of the network system is introduced and the functional advantages of its structure in comparison with the core of the complex network are analyzed. It has been shown that the flow cores allow us to construct a much more adequate with functional point of view the models of network systems than the cores of complex networks. This is demonstrated on the examples of practically important problems of controllability, observability and synchronization of complex networks. On the base of analysis of the structures of real systems, a generalized definition of the multiplex is proposed and the notion of its kernel is introduced. A few of modeling problems of multiplexes are considered and it is established that the effect of the "small world" in them increases. The flow adjacency matrix for the multiplex system is constructed. On its basis, a few of local and global characteristics of the multiplex system are determined and the properties of its flow core are investigated. Methods of reducing the dimension of models of multiplex systems with simultaneous monitoring of the quantitative measure of maintaining their adequacy are analyzed.

Stability and Hopf bifurcation of controlled complex networks model with two delays

This paper considers Hopf bifurcation of complex network with two independent delays. By analyzing the eigenvalue equations, the local stability of the system is studied. Taking delay as parameter, the change of system stability with time is studied and the emergence of inherent bifurcation is given. By changing the value of the delay, the bifurcation of a given system can be controlled. Numerical simulation results confirm the validity of the results found.

Minimum Cost Control of Directed Networks With Selectable Control Inputs.

The minimum cost control problem is one of the most important issues in controlling complex networks. Different from the previous works, in this paper, we consider the minimum cost control problem with selectable inputs by adopting the cost function summed over both quadratic terms of system input and system state with a weighting factor. To address such an issue, the orthonormal-constraint-based projected gradient method is proposed to determine the input matrix iteratively. Convergence of the proposed algorithm is established. Extensive simulation results are carried out to show the effectiveness of the proposed algorithm. We also investigate what kinds of nodes are most important for minimizing average control cost in directed stems/circles and small networks through simulation studies. The presented results in this paper bring meaningful physical insights in controlling the directed networks from an energy point of view.

Controllability of complex networks: Choosing the best driver set

Identifying the best driver set in a complex network is an unsolved problem in the application of pinning control methods. We choose the eigenratio of the augmented Laplacian matrix, and the best driver set is the subset of nodes providing the most effective pinning control strategy, i.e. the one for which synchronization of the whole network to the reference state is attained over the widest range of the coupling parameter. In this work, we propose a centrality measure based on sensitivity analysis of the Laplacian matrix of the connection graph to find an approximate solution to this problem. The proposed metric is computationally efficient as it requires only a single eigen-decomposition of the Laplacian matrix. Numerical results on a number of sample networks show that the proposed metric has significantly better accuracy than currently used heuristics, and in most cases can correctly identify the true optimal set, which is obtainable through combinatorial search.

Selecting pinning nodes to control complex networked systems

One of the fundamental problems in pinning control of complex networks is selecting appropriate pinning nodes, such that the whole system is controlled. This is particularly useful for complex networks with huge numbers of nodes. Recent research has yielded several pinning node selection strategies, which may be efficient.However, selecting a set of pinning nodes and identifying the nodes that should be selected first remain challenging problems. In this paper, we present a network control strategy based on left Perron vector. For directed networks where nodes have the same in- and out-degrees, there has so far been no effective pinning node selection strategy, but our method can find suitable nodes. Likewise, our method also performs well for undirected networks where the nodes have the same degree.In addition, we can derive the minimum set of pinning nodes and the order in which they should be selected for given coupling strengths. Our proofs of these results depend on the properties of non-negative matrices and $M$-matrices. Several examples show that this strategy can effectively select appropriate pinning nodes, and that it can achieve better results for both directed and undirected networks.

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.

Analysis of Critical and Redundant Vertices in Controlling Directed Complex Networks Using Feedback Vertex Sets.

Controlling complex networks through a small number of controller vertices is of great importance in wide-ranging research fields. Recently, a new approach based on the minimum feedback vertex set (MFVS) has been proposed to find such vertices in directed networks in which the target states are restricted to steady states. However, multiple MFVS configurations may exist and thus the selection of vertices may depend on algorithms and input data representations. Our attempts to address this ambiguity led us to adopt an existing approach that classifies vertices into three categories. This approach has been successfully applied to maximum matching-based and minimum dominating set-based controllability analysis frameworks. In this article, we present an algorithm as well as its implementation to compute and evaluate the critical, intermittent, and redundant vertices under the MFVS-based framework, where these three categories include vertices belonging to all MFVSs, some (but not all) MFVSs, and none of the MFVSs, respectively. The results of computational experiments using artificially generated networks and real-world biological networks suggest that the proposed algorithm is useful for identifying these three kinds of vertices for relatively large-scale networks, and that the fraction of critical and intermittent vertices is considerably small. Moreover, an analysis of the signal pathways indicates that critical and intermittent MFVSs tend to be enriched by essential genes.

Local [formula omitted] synchronization of uncertain complex networks via non-fragile state feedback control

This paper is concerned with local H∞ synchronization control of uncertain complex networks, in which there are nonlinear coupling functions between network nodes, and the coupling terms as well as network nodes are subject to time-varying delays. In order to make a complex network achieve H∞ synchronization, a non-fragile state feedback control scheme is used. By employing the Lyapunov functional method, a sufficient condition on the existence of H∞ non-fragile controllers is derived. This condition can not only guarantee the asymptotic stability of each network node, but also ensure a prescribed robust H∞ performance level for the synchronization error system under study. The effectiveness of the proposed method is finally verified through a numerical example.

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.

Enabling Controlling Complex Networks with Local Topological Information

Complex networks characterize the nature of internal/external interactions in real-world systems including social, economic, biological, ecological, and technological networks. Two issues keep as obstacles to fulfilling control of large-scale networks: structural controllability which describes the ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs; and optimal control, which is a typical control approach to minimize the cost for driving the network to a predefined state with a given number of control inputs. For large complex networks without global information of network topology, both problems remain essentially open. Here we combine graph theory and control theory for tackling the two problems in one go, using only local network topology information. For the structural controllability problem, a distributed local-game matching method is proposed, where every node plays a simple Bayesian game with local information and local interactions with adjacent nodes, ensuring a suboptimal solution at a linear complexity. Starring from any structural controllability solution, a minimizing longest control path method can efficiently reach a good solution for the optimal control in large networks. Our results provide solutions for distributed complex network control and demonstrate a way to link the structural controllability and optimal control together.

Effect of interaction strength on robustness of controlling edge dynamics in complex networks

Robustness plays a critical role in the controllability of complex networks to withstand failures and perturbations. Recent advances in the edge controllability show that the interaction strength among edges plays a more important role than network structure. Therefore, we focus on the effect of interaction strength on the robustness of edge controllability. Using three categories of all edges to quantify the robustness, we develop a universal framework to evaluate and analyze the robustness in complex networks with arbitrary structures and interaction strengths. Applying our framework to a large number of model and real-world networks, we find that the interaction strength is a dominant factor for the robustness in undirected networks. Meanwhile, the strongest robustness and the optimal edge controllability in undirected networks can be achieved simultaneously. Different from the case of undirected networks, the robustness in directed networks is determined jointly by the interaction strength and the network’s degree distribution. Moreover, a stronger robustness is usually associated with a larger number of driver nodes required to maintain full control in directed networks. This prompts us to provide an optimization method by adjusting the interaction strength to optimize the robustness of edge controllability.

Minimum energy control for complex networks

The aim of this paper is to shed light on the problem of controlling a complex network with minimal control energy. We show first that the control energy depends on the time constant of the modes of the network, and that the closer the eigenvalues are to the imaginary axis of the complex plane, the less energy is required for complete controllability. In the limit case of networks having all purely imaginary eigenvalues (e.g. networks of coupled harmonic oscillators), several constructive algorithms for minimum control energy driver node selection are developed. A general heuristic principle valid for any directed network is also proposed: the overall cost of controlling a network is reduced when the controls are concentrated on the nodes with highest ratio of weighted outdegree vs indegree.

Optimization of robustness of interdependent network controllability by redundant design.

Controllability of complex networks has been a hot topic in recent years. Real networks regarded as interdependent networks are always coupled together by multiple networks. The cascading process of interdependent networks including interdependent failure and overload failure will destroy the robustness of controllability for the whole network. Therefore, the optimization of the robustness of interdependent network controllability is of great importance in the research area of complex networks. In this paper, based on the model of interdependent networks constructed first, we determine the cascading process under different proportions of node attacks. Then, the structural controllability of interdependent networks is measured by the minimum driver nodes. Furthermore, we propose a parameter which can be obtained by the structure and minimum driver set of interdependent networks under different proportions of node attacks and analyze the robustness for interdependent network controllability. Finally, we optimize the robustness of interdependent network controllability by redundant design including node backup and redundancy edge backup and improve the redundant design by proposing different strategies according to their cost. Comparative strategies of redundant design are conducted to find the best strategy. Results shows that node backup and redundancy edge backup can indeed decrease those nodes suffering from failure and improve the robustness of controllability. Considering the cost of redundant design, we should choose BBS (betweenness-based strategy) or DBS (degree based strategy) for node backup and HDF(high degree first) for redundancy edge backup. Above all, our proposed strategies are feasible and effective at improving the robustness of interdependent network controllability.

Decentralized dynamic understanding of hidden relations in complex networks

Almost all the natural or human made systems can be understood and controlled using complex networks. This is a difficult problem due to the very large number of elements in such networks, on the order of billions and higher, which makes it impossible to use conventional network analysis methods. Herein, we employ artificial intelligence (specifically swarm computing), to compute centrality metrics in a completely decentralized fashion. More exactly, we show that by overlaying a homogeneous artificial system (inspired by swarm intelligence) over a complex network (which is a heterogeneous system), and playing a game in the fused system, the changes in the homogeneous system will reflect perfectly the complex network properties. Our method, dubbed Game of Thieves (GOT), computes the importance of all network elements (both nodes and edges) in polylogarithmic time with respect to the total number of nodes. Contrary, the state-of-the-art methods need at least a quadratic time. Moreover, the excellent capabilities of our proposed approach, it terms of speed, accuracy, and functionality, open the path for better ways of understanding and controlling complex networks.

Voltage and Reactive Power Control to Maximize the Energy Savings in Power Distribution System With Wind Energy

Commonly, voltage regulators and capacitor banks are used as control devices for voltage and reactive power (volt/VAr) control in distribution system. A specific mechanism can be developed using volt/VAr control for energy savings known as conservation voltage reduction (CVR). Active distribution system with high penetration of distribution generations (DG's) offers additional mechanism for volt/VAr control. The unique contribution of this paper is development of a computational algorithm for intelligent volt/VAr optimization and control of complex distribution networks with active participation from DG's to maximize overall network energy savings through CVR. Proposed volt/VAr optimization problem is solved to find the optimal voltage regulator settings, switched capacitor states, and voltage magnitude of DG controlled bus. This work presents a unique coordinated volt/VAr architecture that can find application in distribution management systems. Particle swarm optimization has been used given compatibility with combinatorial variables, robustness in solving nonlinear optimization problems, and ease of implementation. The results have been validated against an exhaustive search strategy for the IEEE 13 bus and IEEE 37 bus distribution system, and also using commercial distribution software SynerGEE. Results are also presented for a utility feeder with detailed analysis.

A novel algorithm for finding optimal driver nodes to target control complex networks and its applications for drug targets identification

BackgroundThe advances in target control of complex networks not only can offer new insights into the general control dynamics of complex systems, but also be useful for the practical application in systems biology, such as discovering new therapeutic targets for disease intervention. In many cases, e.g. drug target identification in biological networks, we usually require a target control on a subset of nodes (i.e., disease-associated genes) with minimum cost, and we further expect that more driver nodes consistent with a certain well-selected network nodes (i.e., prior-known drug-target genes).ResultsTherefore, motivated by this fact, we pose and address a new and practical problem called as target control problem with objectives-guided optimization (TCO): how could we control the interested variables (or targets) of a system with the optional driver nodes by minimizing the total quantity of drivers and meantime maximizing the quantity of constrained nodes among those drivers. Here, we design an efficient algorithm (TCOA) to find the optional driver nodes for controlling targets in complex networks. We apply our TCOA to several real-world networks, and the results support that our TCOA can identify more precise driver nodes than the existing control-fucus approaches. Furthermore, we have applied TCOA to two bimolecular expert-curate networks. Source code for our TCOA is freely available from http://sysbio.sibcb.ac.cn/cb/chenlab/software.htm or https://github.com/WilfongGuo/guoweifeng.ConclusionsIn the previous theoretical research for the full control, there exists an observation and conclusion that the driver nodes tend to be low-degree nodes. However, for target control the biological networks, we find interestingly that the driver nodes tend to be high-degree nodes, which is more consistent with the biological experimental observations. Furthermore, our results supply the novel insights into how we can efficiently target control a complex system, and especially many evidences on the practical strategic utility of TCOA to incorporate prior drug information into potential drug-target forecasts. Thus applicably, our method paves a novel and efficient way to identify the drug targets for leading the phenotype transitions of underlying biological networks.

Locally Optimal Control of Complex Networks.

It has recently been shown that the minimum energy solution of the control problem for a linear system produces a control trajectory that is nonlocal. An issue then arises when the dynamics represents a linearization of the underlying nonlinear dynamics of the system where the linearization is only valid in a local region of the state space. Here we provide a solution to the problem of optimally controlling a linearized system by deriving a time-varying set that represents all possible control trajectories parametrized by time and energy. As long as the control action terminus is defined within this set, the control trajectory is guaranteed to be local. If the desired terminus of the control action is far from the initial state, a series of local control actions can be performed in series, relinearizing the dynamics at each new position.

CytoCtrlAnalyser: a Cytoscape app for biomolecular network controllability analysis.

Studying the controllability of biomolecular networks can result in profound knowledge about molecular biological systems. However, there is no comprehensive and easy-to-use platform for analyzing controllability of biomolecular networks although various algorithms for analyzing complex network controllability have been proposed recently. In this application note, we develop the CytoCtrlAnalyser which is a Cytoscape app to provide a comprehensive platform for analyzing controllability of biomolecular networks. Nine algorithms have been integrated in CytoCtrlAnalyser. With network topologies and customized control settings imported into CytoCtrlAnalyser, users can identify the steering nodes which should be actuated by input control signals for achieving different control objectives as well as investigate the importance of nodes from different perspectives in the controllability of networks. CytoCtrlAnalyser offers a tool for many promising applications, such as identification of potential drug targets or biologically important nodes in biomolecular networks. Freely available for downloading at http://apps.cytoscape.org/apps/cytoctrlanalyser. faw341@mail.usask.ca. Supplementary data are available at Bioinformatics online.

Event-triggered synchronization control for complex networks with actuator saturation

This paper investigates the problem of event-triggered synchronization control for complex networks. Each node is subject to actuator saturation. Based on Lyapunov theory and algebraic graph theory, novel sufficient conditions are derived, under which the synchronization error system is asymptotically stable. Compared with the existing results, the proposed event-riggered synchronization control strategy can be applied to the network with the unknown coupling configuration matrix. A numerical example is presented to demonstrate the effectiveness of the proposed control strategy.