How Low Can You Go? Searching for the Intrinsic Dimensionality of Complex Networks using Metric Node Embeddings

2025-03-03

AI Summary

This paper addresses the problem of determining the intrinsic dimension of complex networks—the minimum Euclidean embedding dimension required to preserve pairwise distances exactly. Conventional vector-based embeddings (e.g., LPCA) suffer from dimensional redundancy and lack theoretically grounded lower bounds. To overcome this, we propose a metric-embedding-based logarithmic search framework that combines low-rank approximation of the distance matrix with triangle inequality verification to identify the minimal feasible dimension in logarithmic time. We provide the first rigorous proof that Euclidean metric embeddings substantially reduce dimensionality requirements. Empirically, our method achieves significantly lower embedding dimensions than state-of-the-art approaches on small-scale networks and, for the first time, enables lossless reconstruction of million-node networks in 2–10-dimensional Euclidean space. The resulting embeddings robustly support downstream tasks including community detection, node classification, and fidelity-preserving visualization.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Low-dimensional embeddings are essential for machine learning tasks involving graphs, such as node classification, link prediction, community detection, network visualization, and network compression. Although recent studies have identified exact low-dimensional embeddings, the limits of the required embedding dimensions remain unclear. We presently prove that lower dimensional embeddings are possible when using Euclidean metric embeddings as opposed to vector-based Logistic PCA (LPCA) embeddings. In particular, we provide an efficient logarithmic search procedure for identifying the exact embedding dimension and demonstrate how metric embeddings enable inference of the exact embedding dimensions of large-scale networks by exploiting that the metric properties can be used to provide linearithmic scaling. Empirically, we show that our approach extracts substantially lower dimensional representations of networks than previously reported for small-sized networks. For the first time, we demonstrate that even large-scale networks can be effectively embedded in very low-dimensional spaces, and provide examples of scalable, exact reconstruction for graphs with up to a million nodes. Our approach highlights that the intrinsic dimensionality of networks is substantially lower than previously reported and provides a computationally efficient assessment of the exact embedding dimension also of large-scale networks. The surprisingly low dimensional representations achieved demonstrate that networks in general can be losslessly represented using very low dimensional feature spaces, which can be used to guide existing network analysis tasks from community detection and node classification to structure revealing exact network visualizations.

Problem

• Determining the minimum embedding dimensions for complex networks.
• Comparing Euclidean metric embeddings with Logistic PCA embeddings.
• Providing scalable, exact reconstruction for large-scale networks.

Innovation

• Euclidean metric embeddings reduce dimensionality.
• Logarithmic search identifies exact embedding dimensions.
• Scalable exact reconstruction for large-scale networks.