Minimum Driver Node Research Articles

Minimum steering node set of complex networks and its applications to biomolecular networks.

Many systems of interests in practices can be represented as complex networks. For biological systems, biomolecules do not perform their functions alone but interact with each other to form so-called biomolecular networks. A system is said to be controllable if it can be steered from any initial state to any other final state in finite time. The network controllability has become essential to study the dynamics of the networks and understand the importance of individual nodes in the networks. Some interesting biological phenomena have been discovered in terms of the structural controllability of biomolecular networks. Most of current studies investigate the structural controllability of networks in notion of the minimum driver node sets (MDSs). In this study, the authors analyse the network structural controllability in notion of the minimum steering node sets (MSSs). They first develop a graph-theoretic algorithm to identify the MSS for a given network and then apply it to several biomolecular networks. Application results show that biomolecules identified in the MSSs play essential roles in corresponding biological processes. Furthermore, the application results indicate that the MSSs can reflect the network dynamics and node importance in controlling the networks better than the MDSs.

Controllability of the better chosen partial networks

How to control large complex networks is a great challenge. Recent studies have proved that the whole network can be sufficiently steered by injecting control signals into a minimum set of driver nodes, and the minimum numbers of driver nodes for many real networks are high, indicating that it is difficult to control them. For some large natural and technological networks, it is impossible and not feasible to control the full network. For example, in biological networks like large-scale gene regulatory networks it is impossible to control all the genes. This prompts us to explore the question how to choose partial networks that are easy for controlling and important in networked systems. In this work, we propose a method to achieve this goal. By computing the minimum driver nodes densities of the partial networks of Erdös–Rényi (ER) networks, scale-free (SF) networks and 23 real networks, we find that our method performs better than random method that chooses nodes randomly. Moreover, we find that the nodes chosen by our method tend to be the essential elements of the whole systems, via studying the nodes chosen by our method of a real human signaling network and a human protein interaction network and discovering that the chosen nodes from these networks tend to be cancer-associated genes. The implementation of our method shows some interesting connections between the structure and the controllability of networks, improving our understanding of the control principles of complex systems.

Co-controllability of drug-disease-gene network

Controllability of a single network often focuses on the determination of the network’s minimum dominating set, which aims to elaborate how to control the whole network with minimum driver nodes. This paper proposes a new framework, co-controllability of multiple networks, which stresses the control of one network by another network as well as the mutual control characteristics of multiple networks based on minimum dominating sets. We take a drug–disease–gene network that consists of a drug–drug network, a disease–disease network and a gene–gene network as an example to study co-controllability of multiple networks. The results show that driver nodes tend to be conserved, e.g. diseases highly associated with driver nodes of the drug–drug network tend to be driver nodes in the disease–disease network compared with random networks. In addition, co-controllability of multiple networks is probably associated with the networks’ node degree, which is more stringent than controllability of a single network that is mainly determined by the network’s degree distribution. We also find that diseases and drugs tend to be mapped as two different subnetworks of human protein–protein interaction (PPI) network, drugs are inclined to dominate diseases by controlling the PPI network, and the coded proteins of disease-related genes exhibit a low tendency to be drug targets for the control of diseases. The results in this paper not only play an important role in understanding co-controllability of multiple networks, but also are helpful for understanding the mechanisms of drug–disease–gene, disease treatments and drug design in a network-based framework.

Algorithm for Identifying Minimum Driver Nodes Based on Structural Controllability

Existing methods on structural controllability of networked systems are based on critical assumptions such as nodal dynamics with infinite time constants and availability of input signals to all nodes. In this paper, we relax these assumptions and examine the structural controllability for practical model of networked systems. We explore the relationship between structural controllability and graph reachability. Consequently, a simple graph-based algorithm is presented to obtain the minimum driver nodes. Finally, simulation results are presented to illustrate the performance of the proposed algorithm in dealing with large-scale networked systems.

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.

Studies on controllability of directed networks with extremal optimization

Almost all natural, social and man-made-engineered systems can be represented by a complex network to describe their dynamic behaviors. To make a real-world complex network controllable with its desired topology, the study on network controllability has been one of the most critical and attractive subjects for both network and control communities. In this paper, based on a given directed–weighted network with both state and control nodes, a novel optimization tool with extremal dynamics to generate an optimal network topology with minimum control nodes and complete controllability under Kalman’s rank condition has been developed. The experimental results on a number of popular benchmark networks show the proposed tool is effective to identify the minimum control nodes which are sufficient to guide the whole network’s dynamics and provide the evolution of network topology during the optimization process. We also find the conclusion: “the sparse networks need more control nodes than the dense, and the homogeneous networks need fewer control nodes compared to the heterogeneous” (Liu et al., 2011 [18]), is also applicable to network complete controllability. These findings help us to understand the network dynamics and make a real-world network under the desired control. Moreover, compared with the relevant research results on structural controllability with minimum driver nodes, the proposed solution methodology may also be applied to other constrained network optimization problems beyond complete controllability with minimum control nodes.

Control Capacity and A Random Sampling Method in Exploring Controllability of Complex Networks

Controlling complex systems is a fundamental challenge of network science. Recent advances indicate that control over the system can be achieved through a minimum driver node set (MDS). The existence of multiple MDS's suggests that nodes do not participate in control equally, prompting us to quantify their participations. Here we introduce control capacity quantifying the likelihood that a node is a driver node. To efficiently measure this quantity, we develop a random sampling algorithm. This algorithm not only provides a statistical estimate of the control capacity, but also bridges the gap between multiple microscopic control configurations and macroscopic properties of the network under control. We demonstrate that the possibility of being a driver node decreases with a node's in-degree and is independent of its out-degree. Given the inherent multiplicity of MDS's, our findings offer tools to explore control in various complex systems.