Statistical network modelling has focused on representing the graph as a discrete structure, namely the adjacency matrix. When assuming exchangeability of this array-which can aid in modelling, computations and theoretical analysis-the Aldous-Hoover theorem informs us that the graph is necessarily either dense or empty. We instead consider representing the graph as an exchangeable random measure and appeal to the Kallenberg representation theorem for this object. We explore using completely random measures (CRMs) to define the exchangeable random measure, and we show how our CRM construction enables us to achieve sparse graphs while maintaining the attractive properties of exchangeability. We relate the sparsity of the graph to the Lévy measure defining the CRM. For a specific choice of CRM, our graphs can be tuned from dense to sparse on the basis of a single parameter. We present a scalable Hamiltonian Monte Carlo algorithm for posterior inference, which we use to analyse network properties in a range of real data sets, including networks with hundreds of thousands of nodes and millions of edges.

中文翻译：

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使用可交换随机度量的稀疏图。

统计网络建模专注于将图表示为离散结构，即邻接矩阵。当假设此数组的可交换性（有助于建模，计算和理论分析）时，Aldous-Hoover定理告诉我们该图必然是稠密的或空的。相反，我们认为将图表示为可交换的随机度量，并且对该对象的Kallenberg表示定理具有吸引力。我们探索使用完全随机度量（CRM）定义可交换的随机度量，并展示我们的CRM构造如何使我们能够获得稀疏图，同时保持可交换性的吸引力。我们将图的稀疏性与定义CRM的Lévy度量相关联。对于特定的CRM选择，我们的图可以根据单个参数从稠密调整为稀疏。我们提出了一种可扩展的哈密顿蒙特卡洛算法用于后验推理，我们使用它来分析一系列实际数据集中的网络属性，包括一组具有数十万个节点和数百万个边的网络。