单纯形复形上的随机游动与正规化Hodge 1-Laplacian
Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian
摘要：
专注于边之间的耦合，我们推广规范化的图形拉普拉斯和随机游走之间的关系，通过设计一个适当的规范化的霍奇拉普拉斯-单纯复形的图形拉普拉斯的推广-并将其与随机游走的边缘。重要的是，这些随机游动与单纯复形的拓扑紧密相连，就像图上的随机游动与图的拓扑有关一样。这是将基于拉普拉斯的分析纳入高阶交互的基础性步骤。我们演示了如何使用这些动态数据分析，提取信息的边空间的单纯复杂的补充和扩展基于图形的分析。具体来说，我们使用我们的归一化霍奇拉普拉斯算子来推导谱嵌入，用于检查马达加斯加附近的海洋漂流者的轨迹数据，并为单纯形复合体的边缘空间开发个性化PageRank的泛化，以分析图书共同购买数据集。
英文摘要：
Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph Laplacian for simplicial complexes -- and relate this to a random walk on edges. Importantly, these random walks are intimately connected to the topology of the simplicial complex, just as random walks on graphs are related to the topology of the graph. This serves as a foundational step towards incorporating Laplacian-based analytics for higher-order interactions. We demonstrate how to use these dynamics for data analytics that extract information about the edge-space of a simplicial complex that complements and extends graph-based analysis. Specifically, we use our normalized Hodge Laplacian to derive spectral embeddings for examining trajectory data of ocean drifters near Madagascar and also develop a generalization of personalized PageRank for the edge-space of simplicial complexes to analyze a book co-purchasing dataset.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
