复杂网络中的单纯度。拓扑数据分析在网络科学中的应用
Simplicial degree in complex networks. Applications of Topological Data Analysis to Network Science
摘要：
网络科学提供了一种通用的形式主义，用于基于代理之间的成对交互来建模和研究复杂系统。然而，社会、生物或计算机科学中的许多真实网络涉及两个以上代理之间的交互，因此具有单纯复形的固有结构。我们提出了新的概念，在单纯复形的单形的高阶邻接度，允许任何尺寸之间的比较，他们和他们的脸，据我们所知，缺乏文献。在有向单纯复形中引入了多参数边界算子和上边界算子，并定义了一种新的多参数组合Laplacian算子，推广了图和组合Laplacian算子.为了说明这些理论结果的潜在应用，我们进行了结构分析的高阶连通性的简单-复杂的网络，通过研究相关的分布与这些单纯度在17个现实世界的数据集来自不同的领域，如合著者网络，共同赞助国会法案，在学校的联系，药物滥用警告网络，电子邮件网络或出版物和用户在网上论坛。我们发现了丰富多样的高阶连通性结构，并观察到相同类型的数据集反映了类似的高阶协作模式。此外，我们表明，如果我们使用我们所谓的最大单纯度（它计算了包含我们的单纯形及其所有严格子社区的不同最大社区），那么它的度分布通常与经典节点度分布有惊人的不同。
英文摘要：
Network Science provides a universal formalism for modelling and studying complex systems based on pairwise interactions between agents. However, many real networks in the social, biological or computer sciences involve interactions among more than two agents, having thus an inherent structure of a simplicial complex. We propose new notions of higher-order degrees of adjacency for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces, which as far as we know were lacked in the literature. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined, which generalises the graph and combinatorial Laplacian. To illustrate the potential applications of these theoretical results, we perform a structural analysis of higher-order connectivity in simplicial-complex networks by studying the associated distributions with these simplicial degrees in 17 real-world datasets coming from different domains such as coauthor networks, cosponsoring Congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. We find rich and diverse higher-order connectivity structures and observe that datasets of the same type reflect similar higher-order collaboration patterns. Furthermore, we show that if we use what we have called the maximal simplicial degree (which counts the distinct maximal communities in which our simplex and all its strict sub-communities are contained), then its degree distribution is, in general, surprisingly different from the classical node degree distribution.
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。
