标题： 多尺度节点嵌入用于图建模和生成 显示英文标题

标题： Multi-Scale Node Embeddings for Graph Modeling and Generation

摘要： 位于网络科学和机器学习的交叉点上，节点嵌入算法以图作为输入，并将其结构编码到输出向量中，这些向量在抽象几何空间中表示节点，从而使得各种基于向量的下游任务成为可能，例如网络建模、数据压缩、链接预测和社区检测。 有两个看似无关的限制影响着这些算法。 一方面，不清楚定义向量空间的基本操作，即向量求和，在网络中的原始节点中对应什么。 另一方面，虽然同一输入网络可以通过将组成节点粗粒化为任意的块节点，在多个分辨率级别上进行表示，但不同层次上的节点嵌入之间的关系尚不明确。 在这里，我们基于网络重整化理论的最新成果，同时解决这两个限制，并定义一种多尺度节点嵌入方法，该方法在任意粗粒化情况下，确保块节点的嵌入向量与组成节点的嵌入向量之和统计一致。 我们在两个可以自然地在多个分辨率级别上表示的经济网络上展示了这种方法的力量：即国家（组）之间的国际贸易以及荷兰（组）行业之间的投入产出流动。 我们确认了从粗粒化节点向量检索的网络与从细粒化节点向量之和检索的网络之间的统计一致性，这一结果是其他方法无法实现的。 几个关键的网络属性，包括大量的三角形，已经可以从非常低维度的嵌入中成功复制，从而允许在任意分辨率级别上生成忠实的原始网络副本。 显示英文摘要

摘要： Lying at the interface between Network Science and Machine Learning, node embedding algorithms take a graph as input and encode its structure onto output vectors that represent nodes in an abstract geometric space, enabling various vector-based downstream tasks such as network modelling, data compression, link prediction, and community detection. Two apparently unrelated limitations affect these algorithms. On one hand, it is not clear what the basic operation defining vector spaces, i.e. the vector sum, corresponds to in terms of the original nodes in the network. On the other hand, while the same input network can be represented at multiple levels of resolution by coarse-graining the constituent nodes into arbitrary block-nodes, the relationship between node embeddings obtained at different hierarchical levels is not understood. Here, building on recent results in network renormalization theory, we address these two limitations at once and define a multiscale node embedding method that, upon arbitrary coarse-grainings, ensures statistical consistency of the embedding vector of a block-node with the sum of the embedding vectors of its constituent nodes. We illustrate the power of this approach on two economic networks that can be naturally represented at multiple resolution levels: namely, the international trade between (sets of) countries and the input-output flows among (sets of) industries in the Netherlands. We confirm the statistical consistency between networks retrieved from coarse-grained node vectors and networks retrieved from sums of fine-grained node vectors, a result that cannot be achieved by alternative methods. Several key network properties, including a large number of triangles, are successfully replicated already from embeddings of very low dimensionality, allowing for the generation of faithful replicas of the original networks at arbitrary resolution levels.