Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron−Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.

网络科学家已经表明，研究系统中组件之间的成对相互作用具有重要价值。从线性代数的角度来看，这涉及定义和评估相关邻接矩阵的函数。最近的工作表明，直接考虑高阶交互还有更多好处，特别是通过超图表示，其中一条边可能涉及多个节点。基于这些想法，我们激发、定义和分析了一类谱中心性度量，用于识别超图中的重要节点和超边，概括现有的网络科学概念。通过利用非线性 Perron-Frobenius 理论的最新发展，我们展示了由此产生的约束非线性特征值问题如何具有独特的解决方案，可以通过非线性幂方法迭代有效地计算。我们说明了对现实数据集的测量。