Many complex systems involve interactions between more than two agents. Hypergraphs capture these higher-order interactions through hyperedges that may link more than two nodes. We consider the problem of embedding a hypergraph into low-dimensional Euclidean space so that most interactions are short-range. This embedding is relevant to many follow-on tasks, such as node reordering, clustering, and visualization. We focus on two spectral embedding algorithms customized to hypergraphs which recover linear and periodic structures respectively. In the periodic case, nodes are positioned on the unit circle. We show that the two spectral hypergraph embedding algorithms are associated with a new class of generative hypergraph models. These models generate hyperedges according to node positions in the embedded space and encourage short-range connections. They allow us to quantify the relative presence of periodic and linear structures in the data through maximum likelihood. They also improve the interpretability of node embedding and provide a metric for hyperedge prediction. We demonstrate the hypergraph embedding and follow-on tasks -- including structure quantification, clustering and hyperedge prediction -- on synthetic and real-world hypergraphs. We find that the hypergraph approach can outperform clustering algorithms that use only dyadic edges. We also compare several triadic edge prediction methods on high school contact data where our algorithm improves upon benchmark methods when the amount of training data is limited.
翻译:许多复杂的系统涉及两个以上代理人之间的相互作用。 超格通过可能连接两个以上节点的顶端, 捕捉到这些较高级的交互作用。 我们考虑将高光谱嵌入到低维的欧几里德空间的问题, 以便大多数互动都是短距离的。 这种嵌入与许多后续任务有关, 例如节点重新排序、 集群和可视化。 我们侧重于两种适合高光谱的光谱嵌入算法, 分别恢复线性和定期结构。 在定期案例中, 节点位于单位圆上。 我们显示, 两种光谱高光谱嵌入算法与一种新的基因化高光谱模型相联。 这些模型产生超高亮度, 根据嵌入空间中的节点位置, 并鼓励短距离连接。 它们使我们能够通过最大的可能性量化数据中定期和线性结构的相对存在。 我们还注重高光谱嵌入和高端预测的测量值。 我们展示高光谱嵌入和后续任务, 包括结构量化、 组合和高端预测 -- 这些模型根据嵌入法产生高空和高端的高级分析方法, 我们只能在高端的高级和高端预测方法上进行。