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.

翻译：节点嵌入算法位于网络科学与机器学习的交叉界面，它接收图作为输入，并将其结构编码到表示抽象几何空间中节点的输出向量上，从而支持多种基于向量的下游任务，如网络建模、数据压缩、链路预测和社区发现。这些算法受到两个看似无关的限制：一方面，定义向量空间的基本运算（即向量加法）对应于网络中原始节点的何种操作尚不明确；另一方面，尽管通过将组成节点粗粒化为任意块节点，同一输入网络可在多个分辨率层级上表示，但不同层次结构层级获得的节点嵌入之间的关系尚未被理解。本文基于网络重归一化理论的最新成果，同时解决了这两个限制，定义了一种多尺度节点嵌入方法。该方法在任意粗粒度化条件下，能确保块节点的嵌入向量与其组成节点的嵌入向量之和保持统计一致性。我们通过两个可在多分辨率层级自然表示的经济网络（即国家（集合）间的国际贸易网络与荷兰产业（集合）间的投入产出流网络）展示了该方法的优势。我们证实了从粗粒度节点向量重建的网络与从细粒度节点向量之和重建的网络之间存在统计一致性，这是其他方法无法实现的结果。即使使用极低维度的嵌入，也能成功复现包括大量三角形在内的多个关键网络特性，从而能够在任意分辨率层级生成原始网络的忠实副本。