Identifying Vital Nodes in Hypergraphs Based on Von Neumann Entropy

Abstract

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.

The Chromatic Entropy of Linear Supertrees and Its Application

“…Up until now, the research works on the chromatic entropy of a hypergraph are found in only one paper: we refer to the reader to [8]. In it, some tight upper and lower bounds of such graph entropy, as well as the corresponding extremal hypergraphs, are obtained.…”

“…Shannon entropy is of great importance in the field of graph structure information theory. Based on Shannon entropy and some graph variables, many graph entropies were proposed; we refer to the reader to [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. For graph entropy, there are lots of applications in chemistry, network, biology and so on; we refer to the reader to [16][17][18][19][20][21][22][23].…”