加权网络的多尺度重整化

中文摘要 AI 辅助

网络重整化的近期进展已经确定了多种在不同分辨率水平上一致地转换网络表示的方法。特别是，描述随机图模型在节点聚合下如何变换的重整化流，已经识别出一个对应于聚合不变模型的不动点，该模型在任意尺度上保持连接概率的形式。尽管这种概率性多尺度方法已成功用于建模真实世界网络、构建潜在空间节点嵌入，并以原则性的分辨率不变方式重建聚合网络数据的更细粒度版本，但迄今为止它仅限于二值图。在这里，我们为加权随机图重新表述了多尺度网络重整化。通过生成函数方法，我们识别出一个加权多尺度模型，其完整概率律（编码链接创建和权重分配）在任意节点聚合下具有不变复合泊松形式。跨聚合级别变换参数的重整化规则允许该集成在一个级别上校准并在另一个级别上应用，从而能够在任意尺度上对加权网络进行一致的建模和重建。

英文摘要

Recent progress in network renormalization has identified various ways to transform network representations consistently across resolution levels. In particular, the renormalization flow describing how random graph models transform under node aggregation has identified a fixed point corresponding to an aggregation-invariant model that preserves the form of the connection probability across arbitrary scales. While this probabilistic multiscale approach has successfully been used to model real-world networks, construct latent-space node embeddings, and reconstruct finer-grained versions of aggregate network data in a principled resolution-invariant way, it has so far been restricted to binary graphs. Here we reformulate multiscale network renormalization for weighted random graphs. Via a generating-function approach, we identify a weighted multiscale model whose full probability law, encoding both link creation and weight assignment, has an invariant compound-Poisson form under arbitrary node aggregation. The renormalization rules that transform parameters across aggregation levels allow the ensemble to be calibrated at one level and applied at another, enabling consistent modeling and reconstruction of weighted networks across arbitrary scales.

发表机构

• IMT School for Advanced Studies（意大利高等研究学院）
• Lorentz Institute for Theoretical Physics, University of Leiden（莱顿大学洛伦兹理论物理研究所）
• Statistics Netherlands（荷兰统计局）
• INdAM-GNAMPA Istituto Nazionale di Alta Matematica ‘Francesco Severi’（意大利高等数学国家研究所·伽尔冈帕）
• Korteweg - de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）

机构由 AI 辅助整理，请以论文原文为准。

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12 pages, 2 figures