Minimum Number Of Driver Nodes Research Articles

On the minimum driver node set of k-uniform linear hypertree networks

The exact controllability research framework of complex networks points out that the controllability of the network has a great relationship with the minimum number of driver nodes. It is generally believed that the smaller the minimum number of driver nodes and the lower the cost of external control of the whole network to achieve the ideal state, the better the controllability of networks. In this paper, the minimum driver node problem of a hypernetwork is transformed into the maximum multiplicative problem of the eigenvalue of its 2-section graph. The minimum driver node numbers of two types of typical k-uniform hypertree networks are described, and their bounds are also given. By designing an algorithm, the method of characterizing the driver node set is obtained, and it is found that the selection of the minimum driver nodes of the network tends to the nodes with low hyperdegree. In addition, the paper verifies the minimum driver node set and the theoretical analysis results of controllability by simulation analysis.

Structural Controllability of Multiplex Networks with the Minimum Number of Driver Nodes

Structural Controllability of Multiplex Networks with the Minimum Number of Driver Nodes

Robustness of Network Controllability with Respect to Node Removals Based on In-Degree and Out-Degree

Network controllability and its robustness have been widely studied. However, analytical methods to calculate network controllability with respect to node in- and out-degree targeted removals are currently lacking. This paper develops methods, based on generating functions for the in- and out-degree distributions, to approximate the minimum number of driver nodes needed to control directed networks, during node in- and out-degree targeted removals. By validating the proposed methods on synthetic and real-world networks, we show that our methods work reasonably well. Moreover, when the fraction of the removed nodes is below 10% the analytical results of random removals can also be used to predict the results of targeted node removals.

Controllability of a class of swarm signalling networks

Abstract In this article, we propose closed-form analytical expressions to determine the minimum number of driver nodes that is needed to control a specific class of networks. We consider swarm signalling networks with regular out-degree distribution where a fraction $p$ of the links is unavailable. We further apply our method to networks with bi-modal out-degree distributions. Our approximations are validated through intensive simulations. Results show that our approximations have high accuracy when compared with simulation results for both types of out-degree distribution.

Structural controllability and time-to-control of directed scale-free networks with minimum number of driver nodes

In this paper we study structural controllability and time-to-control of directed scale-free networks, and propose algorithms to guarantee these properties with the minimum number of driver nodes. Structural controllability is a qualitative (i.e. topological) measure of the ability to steer the network to a desired state from an arbitrary initial state; while time-to-control measures how fast the above steering can be done. First, we develop an algorithm that generates directed scale-free networks which are provably structurally controllable with only one driver node. Moreover, considering the tradeoff between control cost (number of driver nodes) and control performance (time-to-control), we propose another algorithm that constructs directed scale-free networks which can be steered to a desired state within a prescribed time bound and with the minimum number of driver nodes.

The network asymmetry caused by the degree correlation and its effect on the bimodality in control

Our ability to control a whole network can be achieved via a small set of driver nodes. While the minimum number of driver nodes needed for control is fixed in a given network, there are multiple choices for the driver node set. A quantity used to investigate this multiplicity is the fraction of redundant nodes in the network, referring to nodes that do not need any external control. Previous work has discovered a bimodality feature characterized by a bifurcation diagram: networks with the same statistical property would stay with equal probability to have a large or small fraction of redundant nodes. Here we find that this feature is rooted in the symmetry of the directed network, where both the degree distribution and the degree correlation can play a role. The in-in and out-out degree correlation will suppress the bifurcation, as networks with such degree correlations are asymmetric under network transpose. The out-in and in-out degree correlation do not change the network symmetry, hence the bimodality feature is preserved. However, the out-in degree correlation will change the critical average degree needed for the bifurcation. Hence by fixing the average degree of networks and tuning out-in degree correlation alone, we can observe a similar bifurcation diagram. We conduct analytical analyses that adequately explain the emergence of bimodality caused by out-in degree correlation. We also propose a quantity, taking both degree distribution and degree correlation into consideration, to predict if a network would be at the upper or lower branch of the bifurcation. As is well known that most real networks are not neutral, our results extend our understandings of the controllability of complex networks.

Target controllability with minimal mediators in complex biological networks

Controllability of a complex network system is related to finding a set of minimum number of nodes, known as drivers, controlling which allows having a full control on the dynamics of the network. For some applications, only a portion of the network is required to be controlled, for which target control has been proposed. Often, along the controlling route from driver nodes to target nodes, some mediators (intermediate nodes) are also unwillingly controlled, which might cause various side effects. In controlling cancerous cells, unwillingly controlling healthy cells, might result in weakening them, thus affecting the immune system against cancer. This manuscript proposes a suitable candidate solution to the problem of finding minimum number of driver nodes under minimal mediators. Although many others have attempted to develop algorithms to find minimum number of drivers for target control, the newly proposed algorithm is the first one that is capable of achieving this goal and at the same time, keeping the number of the mediators to a minimum. The proposed controllability condition, based on path lengths between node pairs, meets Kalman's controllability rank condition and can be applied on directed networks. Our results show that the path length is a major determinant of in properties of the target control under minimal mediators. As the average path length becomes larger, the ratio of drivers to target nodes decreases and the ratio of mediators to targets increases. The proposed methodology has potential applications in biological networks. The source code of the algorithm and the networks that have been used are available from the following link: https://github.com/LBBSoft/Target-Control-with-Minimal-Mediators.git

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.

A novel network control model for identifying personalized driver genes in cancer.

Although existing computational models have identified many common driver genes, it remains challenging to identify the personalized driver genes by using samples of an individual patient. Recently, the methods of exploiting the structure-based control principles of complex networks provide new clues for identifying minimum number of driver nodes to drive the state transition of large-scale complex networks from an initial state to the desired state. However, the structure-based network control methods cannot be directly applied to identify the personalized driver genes due to the unknown network dynamics of the personalized system. Here we proposed the personalized network control model (PNC) to identify the personalized driver genes by employing the structure-based network control principle on genetic data of individual patients. In PNC model, we firstly presented a paired single sample network construction method to construct the personalized state transition network for capturing the phenotype transitions between healthy and disease states. Then, we designed a novel structure-based network control method from the Feedback Vertex Sets-based control perspective to identify the personalized driver genes. The wide experimental results on 13 cancer datasets from The Cancer Genome Atlas firstly showed that PNC model outperforms current state-of-the-art methods, in terms of F-measures for identifying cancer driver genes enriched in the gold-standard cancer driver gene lists. Furthermore, these results showed that personalized driver genes can be explored by their network characteristics even when they are hidden factors in transcription and mutation profiles. Our PNC gives novel insights and useful tools into understanding the tumor heterogeneity in cancer. The PNC package and data resources used in this work can be freely downloaded from https://github.com/NWPU-903PR/PNC.

Controlling edge dynamics in multilayer networks

Dynamical processes occurring on the edges of complex networks universally exists in many natural and engineered systems with multiple subsystems and layers. An edge dynamical system is controllable if, with appropriate external inputs, it can be driven from any initial state to any desired state in finite time. Here, we introduce the edge dynamics in multilayer networks, and analyze its controllability. The simulation results and analytic calculation show that the lower and upper bounds of the minimum number of driver nodes and driven edges for the edge dynamics are existed in multilayer networks, and the bound values significantly differ from that of single-layer networks with the same size. The edge dynamics in a multilayer network is more strongly structurally controllable than that in a single-layer network of the same size. Moreover, simulation results indicate that the degree correlation between layers plays an important role in the edge controllability of multilayer networks. These findings force us to go beyond single-layer network structure to offer a better understanding of the controllability of edge dynamics.

Identifying and using driver nodes in temporal networks

AbstractIn many approaches developed for defining complex networks, the main assumption is that the network is in a relatively stable state that can be approximated with a fixed topology. However, in several applications, this approximation is not adequate because (a) the system modelled is dynamic by nature, and (b) the changes are an essential characteristic that cannot be approximated. Temporal networks capture changes in the topology of networks by including the temporal information associated with their structural connections, that is, links or edges. Here, we focus on controllability of temporal networks, that is, the study of steering the state of a network to any desired state at deadline $t_f$ within $\Delta t=t_f - t_0$ steps through stimulating key nodes called driver nodes. Recent studies provided analytical approaches to find a maximum controllable subspace for an arbitrary set of driver nodes. However, finding the minimum number of driver nodes $N_c$ required to reach full control is computationally prohibitive. In this article, we propose a heuristic algorithm that quickly finds a suboptimal set of driver nodes with size $N_s \geq N_c$. We conduct experiments on synthetic and real-world temporal networks induced from ant colonies and e-mail communications of a manufacturing company. The empirical results in both cases show the heuristic algorithm efficiently identifies a small set of driver nodes that can fully control the networks. Also, as shown in the case of ants’ interactions networks, the driver nodes tend to have a large degree in temporal networks. Furthermore, we analyze the behavior of driver nodes within the context of their datasets, through which, we observe that queen ants tend to avoid becoming a driver node.

The evolution of network controllability in growing networks

The study of network structural controllability focuses on the minimum number of driver nodes needed to control a whole network. Despite intensive studies on this topic, most of them consider static networks only. It is well-known, however, that real networks are growing, with new nodes and links added to the system. Here, we analyze controllability of evolving networks and propose a general rule for the change of driver nodes. We further apply the rule to solve the problem of network augmentation subject to the controllability constraint. The findings strengthen our understandings of network controllability and shed light on controllability of real systems.

A multiple perspective method for urban subway network robustness analysis

Most network research studying the robustness of critical infrastructure networks focuses on a particular aspect and does not take the entire system into consideration. We develop a general methodological framework for studying network robustness from multiple perspectives, i.e., Robustness assessment based on percolation theory, vulnerability analysis, and controllability analysis. Meanwhile, We use this approach to examine the Shanghai subway network in China. Specifically, (1) the topological properties of the subway network are quantitatively analyzed using network theory; (2) The phase transition process of the subway network under both random and deliberate attacks are acquired (3) Critical dense areas that are most likely to be the target of terrorist attacks are identified, vulnerability values of these critical areas are obtained; (4) The minimum number of driver nodes for controlling the whole network is calculated. Results show that the subway network exhibits characteristics similar to a scale-free network with low robustness to deliberate attacks. Meanwhile, we identify the critical area within which disruptions produce large performance losses. Our proposed method can be applied to other infrastructure networks and can help decision makers develop optimal protection strategies.

Controllability and observability of cascading failure networks

Cascading failure analysis in previous existing works has mainly focused on random or intentional local attacks, and how to control these failures in complex networks with minimum input from the external signals is a new way of understanding cascades from a global view of network control. In this paper, we are motivated to develop a new framework, which first transforms the original networks (ONs) into cascading failure networks (CFNs) via the attack on each single node. Then, we study the controllability of the CFNs, and try to disclose the cascading failures of the ONs by network control. Furthermore, we analyze the impact of network structure dynamics on the framework. The results show that large-scale cascades are more likely to be triggered by hub node attacks in sparse and inhomogeneous networks, which need more driver nodes in order for them to be controlled. The driver nodes in the CFNs tend to avoid the high-degree nodes in the ONs, and this tendency enhances the homogeneous networks. In addition, networks with strong community strength and heterogeneous community sizes are more robust against attacks, and are easier to control. We also find that the degree distribution of the ONs mainly determines the size of the CFNs, as well as the minimum number of driver nodes, and the latter is also applicable to their observability. Furthermore, a similar impact on the CFNs can be observed by increasing the number of links as well as the capacity of the nodes in the ONs.

Critical Nodes Identification of Power Systems Based on Controllability of Complex Networks

This paper proposes a new method for assessing the vulnerability of power systems based on the controllability theories of complex networks. A novel controllability index is established, taking into consideration the full controllability of the power systems, for identifying critical nodes. The network controllability model is used to calculate the minimum number of driver nodes (ND), which can solve the computable problems of the controllability of power systems. The proposed approach firstly applies the network controllability theories to research the power systems' vulnerability, which can not only effectively reveal the important nodes but also maintain full control of the power systems. Meanwhile, the method can also overcome the limitation of the hypothesis that the weight of each link or transmission line must be known compared with the existing literature. In addition, the power system is considered as a directed network and the power system model is also redefined. The proposed methodology is then used to identify critical nodes of the IEEE 118 and 300 bus system. The results show that the failure of the critical nodes can clearly increase ND and lead a significant driver node shift. Thus, the rationality and validity are verified.

Controllability of deterministic networks with the identical degree sequence.

Controlling complex network is an essential problem in network science and engineering. Recent advances indicate that the controllability of complex network is dependent on the network's topology. Liu and Barabási, et.al speculated that the degree distribution was one of the most important factors affecting controllability for arbitrary complex directed network with random link weights. In this paper, we analysed the effect of degree distribution to the controllability for the deterministic networks with unweighted and undirected. We introduce a class of deterministic networks with identical degree sequence, called (x,y)-flower. We analysed controllability of the two deterministic networks ((1, 3)-flower and (2, 2)-flower) by exact controllability theory in detail and give accurate results of the minimum number of driver nodes for the two networks. In simulation, we compare the controllability of (x,y)-flower networks. Our results show that the family of (x,y)-flower networks have the same degree sequence, but their controllability is totally different. So the degree distribution itself is not sufficient to characterize the controllability of deterministic networks with unweighted and undirected.

Controllability of deterministic complex networks

Recently, the controllability of complex networks has attracted wide attention of researchers. The main contribution comes from Liu et al. who proposed the structural controllability as an analytical framework for making predictions regarding the control of directed networks in Nature. Since then, the controllability of many model and real networks has been deeply investigated except deterministic complex networks. In this paper, we focus on studying the controllability of deterministic complex networks. We examine six typical deterministic networks, the simulation results show that the minimum number of driver nodes grows linearly with network size. When the network size is large enough, the controllability approximates to a constant not more than 0.4, indicating that the deterministic networks are relatively easy to control. Furthermore, we investigate the characteristics of driver nodes in deterministic complex networks, finding that the driver nodes tend to avoid high degree nodes but to have high clustering coefficients.

Target control of complex networks.

Controlling large natural and technological networks is an outstanding challenge. It is typically neither feasible nor necessary to control the entire network, prompting us to explore target control: the efficient control of a preselected subset of nodes. We show that the structural controllability approach used for full control overestimates the minimum number of driver nodes needed for target control. Here we develop an alternate ‘k-walk’ theory for directed tree networks, and we rigorously prove that one node can control a set of target nodes if the path length to each target node is unique. For more general cases, we develop a greedy algorithm to approximate the minimum set of driver nodes sufficient for target control. We find that degree heterogeneous networks are target controllable with higher efficiency than homogeneous networks and that the structure of many real-world networks are suitable for efficient target control.

Theoretical progress and practical challenges in controlling complex networks

The issue of control has gained significant prominence in engineering, useful for aircraft control, manufacturing processes, communication systems and so on. Yet, the control of complex networks was barely studied, as we lack powerful theories to address it in a quantitative fashion. Fair recently, considerable efforts were made to address the controllability issue of complex networks [1]. As a key notion in control theory, controllability concerns our ability to drive a dynamic system from any initial state to any final state in finite time [2], which agrees well with our intuitive notion of control. To model complex networks as dynamic systems, we normally adopt the so-called nodal dynamics, i.e. we associate each node in a network with a state variable, whose time evolution crucially depends on the state variables of the node itself and/or its neighbors. Nodal dynamics is natural for modeling dynamic processes on many real systems, where the state variable of a node often has a clear physical meaning, e.g. the expression level of genes in transcriptional regulatory networks or the population of species in food webs. Thanks to a combination of tools from graph theory, control theory and statistical physics, a series of results about network controllability with nodal dynamics were obtained (see Fig. 1). For example, the minimum number of driver nodes, whose time-dependent control can guide the system’s dynamics, is mainly determined by the degree distribution of the network. And surprisingly, driver nodes tend to avoid hubs, i.e. the nodes with high connectivity. Sparse and heterogeneous networks are harder to be controllable than dense and homogeneous networks [1]. These

Controllability of fractal networks: An analytical approach

Fractal networks are ubiquitous in nature, ranging from river networks to vascular networks. The ultimate goal of exploring these fractal networked systems lies in controlling the dynamical processes that take place on them. We offer analytical results to exactly understand our ability to control the dynamics of regular fractal networks in terms of identifying the minimum number of driver nodes that are required to achieve full control. According to the exact controllability theory, the controllability of an undirected network is completely determined by the eigenvalue spectrum of the coupling matrix that captures the network structure. The self-similarity in the fractal networks allows us to solve exactly the eigenvalue spectrum from the growth unit and the steps of the iterations, enabling an analytical quantification of the controllability of the fractal networks via the eigenvalue spectrum. We validate our exact analytical results in three typical regular fractal networks. Our results have implications for the control of many real networked systems that have fractal characteristics.