关键词: 中心性, 超边, 影响力, 超网络

Abstract: In the real world, many systems can be abstracted as ordinary networks, which consist of nodes and edges. Nodes in a network represent the elements of a system, and edges connecting nodes represent the relationships between elements. The importance identification of nodes is a primary branch of network research aimed at recognizing nodes that play crucial roles in network structure and resources transfer processes. This is essential for a deeper understanding and optimization of networks, enabling effective management with significant value.
In ordinary networks, “edges” typically connect only two nodes. When faced with complex interactions among multiple elements, crucial higher-order information may be lost. This loss makes it difficult to effectively characterize relationships among internal nodes within the structure. Consequently, this difficulty leads to distortions in mapping to real-world scenarios. In such scenarios, hypernetworks have emerged as vital tools in the study of complex networks. Compared to ordinary networks, “hyperedges” in hypernetworks can include any number of nodes, reflecting high-order complex relationships among multiple nodes.
Generally, there are five metrics to evaluate the importance of nodes in a network: degree centrality, closeness centrality, betweenness centrality, K-shell index and eigenvector centrality. Specifically, in the context of identifying important nodes in hypernetworks, a common metric is node hyperdegree, which measures node importance based on the number of hyperedges to which the node is connected. Findings from research on identifying important nodes in hypernetworks and ordinary networks indicate that identifying important nodes in hypernetworks draws inspiration from the principles of node identification in ordinary networks, primarily starting from node attributes to identify important nodes. However, considering the changing nature of hyperedges in hypernetworks compared to ordinary networks, the identification of important nodes in hypernetworks must comprehensively consider the influence of hyperedges.
While some studies have considered the quantity/differences of hyperedges in identifying important nodes, they have overlooked the influence of hyperedges. For instance, variations in the topological structure of hyperedges in a network lead to differences in their influence, consequently affecting the importance of nodes within them. Failure to consider this scenario in identifying important nodes may result in distorted outcomes.
To address this issue, this study proposes a method for identifying important nodes in hypernetworks based on the influence of hyperedges. The method initially evaluates the influence of hyperedges in the network through external dominance and internal control. External dominance measures the ability of a hyperedge to establish connections with hyperedges that easily connect to others considered to have higher influence. This includes global dominance based on the distance between hyperedges and local dominance where nodes within the hyperedge act as bridges. Internal control measures the network control capability brought by the number of nodes within a hyperedge, with hyperedges containing more nodes considered to have greater influence. Subsequently, based on the set of hyperedge influence vectors obtained, the method calculates node importance using an adjacency matrix and identifies important nodes accordingly. Its advantage lies in considering not only the impact of the quantity of hyperedges on node importance but also the comprehensive influence of hyperedges on node importance. Finally, to validate the effectiveness of this research method, it is applied to sample hypernetworks and real hypernetworks and compared with other existing methods such as K-shell decomposition, hyperdegree value and core degree centrality value. Comparative results indicate that the proposed method can more accurately identify important nodes with higher distinctiveness, confirming the effectiveness of this research method. This study provides crucial references for a deeper understanding of hypernetwork structures, optimizing network layouts and resources allocation.

Key words: centrality, hyperedges, influence, hypernetworks