Towards the minimum-cost control of target nodes in directed networks with linear dynamics

Abstract

Determining an input matrix, i.e., locating predefined number of nodes (named “key nodes”) connected to external control sources that provide control signals, so as to minimize the cost of controlling a preselected subset of nodes (named “target nodes”) in directed networks is an outstanding issue. This problem arises especially in large natural and technological networks. To address this issue, we focus on directed networks with linear dynamics and propose an iterative method, termed as “L₀-norm constraint based projected gradient method” (LPGM) in which the input matrix B is involved as a matrix variable. By introducing a chain rule for matrix differentiation, the gradient of the cost function with respect to B can be derived. This allows us to search B by applying probabilistic projection operator between two spaces, i.e., a real valued matrix space R^N × M and a L₀ norm matrix space $R_{L_0}^{N\times M}$ by restricting the L₀ norm of B as a fixed value of M. Then, the nodes that correspond to the M nonzero elements of the obtained input matrix (denoted as $B^{L_0}$) are selected as M key nodes, and each external control source is connected to a single key node. Simulation examples in real-life networks are presented to verify the potential of the proposed method. An interesting phenomenon we uncovered is that generally the control cost of scale free (SF) networks is higher than Erdos-Renyi (ER) networks using the same number of external control sources to control the same size of target nodes of networks with the same network size and mean degree. This work will deepen the understanding of optimal target control problems and provide new insights to locate key nodes for achieving minimum-cost control of target nodes in directed networks.

Introduction

The control of complex networks has been widely applied in many fields, such as social systems [1], sensor networks [2], [3], biological networks [4], [5] and so on [6], [7], [8]. Previous studies mainly focus on the network controllability [9], [10], [11], [12], [13], [14], and recent efforts begin to consider how the cost can be as less as possible when controlling these networks, which has been known as the “minimum-cost control” problem. Regarding this matter, researchers explored the eigenvalue properties of controllability Gramian matrix to estimate the energy cost for directed networks [15], [16] with linear dynamics $\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$ where x(t), A, B and u(t) denote system states, adjacency matrix, input matrix and control signals, respectively. An effective approach termed “projected gradient method” (PGM) was proposed [17] to determine the input matrix B, which addresses how to select a node set (named “key nodes”) connected to external control sources that provide control signals, so as to minimize the cost of control. Especially, an importance index for each node was defined based on the final iteratively search B, and the nodes that correspond to the M largest indexes are selected as the key nodes which are connected to external control sources.

However, existing works [15], [16], [17], [18], [19] only consider the minimum-cost control problem for controlling the states of all nodes in the network. In large natural and technological networks, sometimes it is only necessary to control a target subset of nodes instead of the entire network [20]. This requires us to explore how to achieve minimum-cost control of a preselected subset of nodes, which is termed as the “minimum-cost control of target nodes” problem. To achieve this, two issues need to be addressed. The first one is to guarantee the target controllability of the subset of nodes, i.e., to guarantee the structural controllability of a subset of nodes instead of the entire network, which has recently been studied in [20], [21], [22]. The second one is to examine how to minimize the cost when the preselected subset of nodes are target controllable. This is a more physically important and critical issue. In [23], the authors started to consider exploring the relationship between the control cost and the number of target nodes as well as the number of driver nodes that are somehow the same as the key nodes considered in this work. However, to the best of our knowledge, extending the results to “minimum-cost control of target nodes” has been remaining as a largely open issue.

In this work, our objective is to determine the predefined number of key nodes that are connected to external control sources so as to minimize the cost of controlling the target nodes in directed networks. Firstly, the original key nodes selection problem for target control is formulated as non-convex optimization problem where a Boolean constraint is involved. Here the Boolean constraint means that we constrain the input matrix $B_{im}=1$ if an external control source m is connected to a key node i, otherwise, $B_{im}=0$. This non-convex optimization problem is difficult to be solved directly and efficiently. Through relaxing some constraints to their convex hull, an iterative algorithm, namely “projected gradient method” (PGM) [17] is revisited to iteratively search the global optimal solution. In addition, an importance index is defined for each node based on the obtained input matrix B, and the nodes that correspond to the M largest indexes are selected as the key nodes. This method is termed as “projected gradient method-extension” (PGME). One critical problem is that PGME may suffer from large performance penalty in real-life networks due to the diversity of different networks. To this end, we get back to the original formulated non-convex optimization problem and restrain that each external control source is only connected to a single node. This implies that we actually impose a L₀-norm constraint on the input matrix B based on the work in [18]. In reality, most mathematicians and engineers define the L₀-norm of a vector v as ${\left|v\right|}_0=♯\left(i\right|v_i\ne 0),$ which is the total number of non-zero elements of v. This is to say, we consider the number of non-zero elements of B as a constant. However, different from [18] considered controlling the entire nodes, we consider a totally different scenario which is how to control a preselected subset of nodes instead of the entire network nodes in directed networks. As a result, it is generally more difficult to optimal the cost model and obtain the gradient of the cost function with respect to the input matrix variable B. It has also been proven that the optimization model with L₀ norm constraint is composed of multiple sub convex optimization problems. By introducing a chain rule to obtain the derivative of optimization model. By doing this, we develop a L₀-norm constraint based projected gradient method (LPGM) to iteratively search B between the two spaces, i.e., a real valued matrix space R^N × M and a L₀ norm matrix space $R_{L_0}^{N\times M}$ by restricting the L₀ norm of B as a fixed value M. We have also implemented our method in scale free (SF) networks and Erdos-Renyi (ER) networks as well as real-life networks. Simulation results show the effectiveness of the proposed LPGM compared with PGME. An interesting phenomenon we uncovered is that generally the control cost of SF networks is higher than ER networks using the same number of external control sources to control the same size of target nodes of networks with the same network size and mean degree. This suggests target control of SF networks is generally more difficult than that of ER networks.

The main contributions of this work are summarized as follows:

• We formulate an optimization model (20) that constrains each external control source is only connected to a single key node to control preselected subset of nodes and minimize the energy cost in directed networks. Furthermore, it is proved that the proposed optimization model is composed of a set of sub convex problems.

• By introducing a chain rule, the derivative of the cost function $\mathbb{E}\left(B\right)$ with respect to a matrix variable B can be obtained.

• We develop an L₀-norm constraint based projected gradient method and design a probabilistic projection operator to iteratively search for the suboptimal solution. We test the algorithm in various applications and the results show that LPGM always gives a good enough solution through switching the solution among multiple convex optimization problems.

• We implement our method in scale free (SF) networks, Erdos–Renyi (ER) networks and real-life networks. An interesting phenomenon we uncovered is that generally the control cost of SF networks is higher than ER networks using the same number of external control sources to control the same size of target nodes of networks with the same network size and mean degree. This suggests that target control of SF networks is generally more difficult than that of ER networks.

The remaining part of the paper is organized as follows. In Section 2, we formulate a matrix function optimization model for the target control of directed networks. In Section 3, the L₀-norm constraint based projected gradient method (LPGM) is developed to solve the optimization problem and some techniques and theoretical analysis are provided. In Section 4, simulation results are presented to verify the potential of the proposed method. This work is concluded in Section 5.