SIAM Review, Volume 62, Issue 2, Page 353-391, January 2020.
Using graphs to model pairwise relationships between entities is a ubiquitous framework for studying complex systems and data. Simplicial complexes extend this dyadic model of graphs to polyadic relationships and have emerged as a model for multinode relationships occurring in many complex systems. For instance, biological interactions occur between sets of molecules and communication systems include group messages that are not pairwise interactions. While Laplacian dynamics have been intensely studied for graphs, corresponding notions of Laplacian dynamics beyond the node-space have so far remained largely unexplored for simplicial complexes. In particular, diffusion processes such as random walks and their relationship to the graph Laplacian---which underpin many methods of network analysis, including centrality measures, community detection, and contagion models---lack a proper correspondence for general simplicial complexes. 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 toward 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 copurchasing dataset.

中文翻译：

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随机游走于简单复合体和规范化Hodge 1-Laplacian

SIAM评论，第62卷，第2期，第353-391页，2020年1月。
使用图来建模实体之间的成对关系是研究复杂系统和数据的普遍存在的框架。简单复形将图的二元模型扩展到多元关系，并已成为许多复杂系统中发生的多节点关系的模型。例如，生物相互作用发生在分子组之间，并且通信系统包括不是成对相互作用的组消息。尽管对图进行了拉普拉斯动力学的深入研究，但到目前为止，对于单纯形复形，在节点空间以外的拉普拉斯动力学的相应概念仍未得到充分探索。尤其是扩散过程，例如随机游走及其与图形Laplacian的关系，这是许多网络分析方法的基础，包括集中度测量，社区检测，和传染模型-缺乏对一般简单复合体的适当对应。着重于边缘之间的耦合，我们通过为Hodge Laplacian设计适当的归一化来概括归一化图拉普拉斯算子与图上随机游走之间的关系-简单化图的拉普拉斯算子的归纳化-并将其与随机走在边缘。重要的是，这些随机游走与简单复合体的拓扑紧密相连，就像图上的随机游走与图的拓扑有关。这是将基于Laplacian的分析纳入更高阶交互的基础步骤。我们演示了如何使用这些动态信息进行数据分析，以提取有关简单复杂物边缘空间的信息，以补充和扩展基于图的分析。具体来说，我们使用归一化的Hodge Laplacian派生光谱嵌入来检查马达加斯加附近海洋漂流者的轨迹数据，还开发了针对简单复杂区域边缘空间的个性化PageRank的泛化，以分析书籍共同购买数据集。