Hypergraphs have become an accurate and natural expression of high-order coupling relationships in complex systems. However, applying high-order information from networks to vital node identification tasks still poses significant challenges. This paper proposes a von Neumann entropy-based hypergraph vital node identification method (HVC) that integrates high-order information as well as its optimized version (semi-SAVC). HVC is based on the high-order line graph structure of hypergraphs and measures changes in network complexity using von Neumann entropy. It integrates s-line graph information to quantify node importance in the hypergraph by mapping hyperedges to nodes. In contrast, semi-SAVC uses a quadratic approximation of von Neumann entropy to measure network complexity and considers only half of the maximum order of the hypergraph's s-line graph to balance accuracy and efficiency. Compared to the baseline methods of hyperdegree centrality, closeness centrality, vector centrality, and sub-hypergraph centrality, the new methods demonstrated superior identification of vital nodes that promote the maximum influence and maintain network connectivity in empirical hypergraph data, considering the influence and robustness factors. The correlation and monotonicity of the identification results were quantitatively analyzed and comprehensive experimental results demonstrate the superiority of the new methods. At the same time, a key non-trivial phenomenon was discovered: influence does not increase linearly as the s-line graph orders increase. We call this the saturation effect of high-order line graph information in hypergraph node identification. When the order reaches its saturation value, the addition of high-order information often acts as noise and affects propagation.
Keywords: high-order line graph; hypergraph; saturation effect; vital nodes; von Neumann entropy.