Renormalizable Graph Embeddings For Multi-Scale Network Reconstruction

2025-08-28

AI Summary

In financial and economic networks, inter-node connections are often unobservable due to privacy constraints or data limitations, and conventional network reconstruction methods operate at a single resolution, hindering multiscale analysis. Method: This paper proposes a renormalizable multi-scale graph embedding framework that integrates network renormalization, the maximum entropy principle, and probabilistic graphical models, augmented with scale-invariance constraints to ensure embedding consistency under node aggregation and resolution changes. Contribution/Results: Unlike classical maximum entropy approaches, our framework enables cross-level inference and network reconstruction at arbitrary granularities. Experiments on national input-output networks and international trade networks demonstrate consistent, interpretable, and high-accuracy reconstructions across multiple industrial sectors and geographic scales. The method establishes a novel paradigm for modeling partially observable complex networks, advancing both theoretical foundations and practical applicability in economics and finance.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

Abstract

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.

Problem

• Reconstructing hidden networks from aggregate data
• Ensuring multi-scale consistency in graph embeddings
• Overcoming limitations of fixed-resolution reconstruction methods

Innovation

• Renormalizable graph embeddings for multi-scale reconstruction
• Scale-invariant principle for consistent network embeddings
• Maximum-entropy enhanced with resolution-independent aggregation