Minimize Control Inputs for Strong Structural Controllability Using Reinforcement Learning with Graph Neural Network

TL;DR

This work tackles the problem of minimizing control inputs to guarantee strong structural controllability (SSC) in directed networks under zero/nonzero/arbitrary structure. It reframes SSC as a graph coloring task guided by a color-change rule and solves the minimum-input problem via a reinforcement learning framework that employs a directed graph neural network in an actor-critic setup. The approach yields a general framework applicable to multiple structure types and reveals that minimal input sets tend to include nodes with low in-degree, with the required number of inputs scaling with the network's average degree. Empirical results on a social-influence network and Erdos–Renyi graphs show the RL method consistently outperforms degree-based heuristics, offering a scalable strategy for input placement in complex networks.

Abstract

Strong structural controllability (SSC) guarantees networked system with linear-invariant dynamics controllable for all numerical realizations of parameters. Current research has established algebraic and graph-theoretic conditions of SSC for zero/nonzero or zero/nonzero/arbitrary structure. One relevant practical problem is how to fully control the system with the minimal number of input signals and identify which nodes must be imposed signals. Previous work shows that this optimization problem is NP-hard and it is difficult to find the solution. To solve this problem, we formulate the graph coloring process as a Markov decision process (MDP) according to the graph-theoretical condition of SSC for both zero/nonzero and zero/nonzero/arbitrary structure. We use Actor-critic method with Directed graph neural network which represents the color information of graph to optimize MDP. Our method is validated in a social influence network with real data and different complex network models. We find that the number of input nodes is determined by the average degree of the network and the input nodes tend to select nodes with low in-degree and avoid high-degree nodes.

Figures (11)

• Figure 1: This figure shows how to convert the problem of controllability to a graph coloring problem. (a) The networked system is controlled by input signals $u_1$ and $u_2$, allowing the system to achieve a desired state from the initial state. Matrix $\textbf{A}$ describes the connections among nodes. Matrix $\textbf{B}$ shows nodes imposed signals. (b) Whether the system is fully controlled by the imposed signals $u_1, u_2$ can be converted to a graph coloring problem. Nodes $v_1, v_2, v_3$ in (b) represent variables $x_1, x_2, x_3$ in (a). If all nodes of the graph obtained by matrix $\textbf{A}$ can be colored black according to the color change rule in Definition (\ref{['def: color']}), then the system is controllable. Otherwise, the system is not controllable.
• Figure 2: This figure shows how to use reinforcement learning with directed graph neural networks to minimize the number of inputs to make the system controllable. The state represents the color of all nodes. The feature vector of each node is obtained by the color of nodes. Input the graph with feature vectors to the critic net to value the current state. Input the graph with feature vectors to the actor net to generate an action to color a node black. Coloring the graph according to the specific color change rule to get the next state. Repeat these steps until all nodes are colored black. Calculate the reward and state values to update parameters of Actor Nets and Critic Net.
• Figure 3: This figure shows the color-change rules. (a) Node 1 is the input node which is colored black at first. (b) node 1 colors 2. (c) node 2 colors node 3.
• Figure 4: (a) is the original graph $G(\textbf{A}, \textbf{B})$ and (b) is $G(\bar{\textbf{A}}, \textbf{B})$. The input node is $v_1$. According to the color change rule, the set of all black nodes in (a) and (b) is $(1, 2)$ and $(1, 2, 3, 5)$, respectively. The derived set ${\rm dset}(G, (1))$ is $(1, 2)$.
• Figure 5: This figure shows the algorithm based on Proposition \ref{['pro: 1']} and degree.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Theorem 2
• proof
• Theorem 3
• Proposition 1
• proof