标题： 可重整化图嵌入用于多尺度网络重构 显示英文标题

标题： Renormalizable Graph Embeddings For Multi-Scale Network Reconstruction

摘要： 在机器学习中，图嵌入算法旨在对输入的网络数据进行低维表示，从而使得在压缩编码上的下游任务成为可能。 最近，在网络重整化框架内，提出了多尺度嵌入方法，这些方法在将节点任意聚合到块节点下保持一致，并因此在输入网络数据的任意分辨率变化下保持一致。 在这里，我们研究了在修改后的背景下这样的多尺度图嵌入，其中由于数据限制或隐私约束，输入网络并非完全可观测。 这种情况在金融和经济网络中很常见，由于保密性，个别银行或公司之间的连接被隐藏，必须从汇总信息中概率性地重建底层网络。 我们首先考虑基于最大熵原理的最先进的网络重构技术，该技术设计用于在固定分辨率水平下最优运行。 然后我们讨论当这些方法用作图嵌入以在不同分辨率水平上进行预测时的局限性。 最后，我们提出了从尺度不变性不同原理导出的自然“可重整化”对应方法，从而为多尺度网络重构提供一致的图嵌入。 我们在国家经济投入产出网络和国际贸易网络上展示了这些方法，这些网络可以分别在多个工业和地理分辨率层次上自然表示。 显示英文摘要

摘要： In machine learning, graph embedding algorithms seek low-dimensional representations of the input network data, thereby allowing for downstream tasks on compressed encodings. Recently, within the framework of network renormalization, multi-scale embeddings that remain consistent under an arbitrary aggregation of nodes onto block-nodes, and consequently under an arbitrary change of resolution of the input network data, have been proposed. Here we investigate such multi-scale graph embeddings in the modified context where the input network is not entirely observable, due to data limitations or privacy constraints. This situation is typical for financial and economic networks, where connections between individual banks or firms are hidden due to confidentiality, and one has to probabilistically reconstruct the underlying network from aggregate information. We first consider state-of-the-art network reconstruction techniques based on the maximum-entropy principle, which is designed to operate optimally at a fixed resolution level. We then discuss the limitations of these methods when they are used as graph embeddings to yield predictions across different resolution levels. Finally, we propose their natural 'renormalizable' counterparts derived from the distinct principle of scale invariance, yielding consistent graph embeddings for multi-scale network reconstruction. We illustrate these methods on national economic input-output networks and on international trade networks, which can be naturally represented at multiple levels of industrial and geographic resolution, respectively.