SS-GUMAP, SL-GUMAP, SSSL-GUMAP: Fast UMAP Algorithms for Large Graph Drawing

2025-09-23

AI Summary

UMAP suffers from an $O(nm)$ time complexity bottleneck when applied to large-graph visualization, as it requires computing all-pairs shortest paths, severely limiting scalability. This work presents the first systematic evaluation of UMAP’s effectiveness for graph drawing and proposes SSSL-GUMAP—a novel acceleration framework integrating spectral sparsification (SS), partial BFS sampling, and linear-time $k$-nearest-neighbor graph construction. The framework yields three variants—SS-GUMAP, SL-GUMAP, and SSSL-GUMAP—that achieve near-linear time complexity. Empirically, all variants maintain high layout quality (metric degradation <15%) while accelerating GUMAP by 28%–80%+ and outperforming tsNET by 90% in speed and superior average quality. These results significantly advance the state of the art by breaking longstanding performance barriers in scalable graph embedding and visualization.

Technology Category

• Machine Learning: Graph-based Machine Learning
• 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
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

UMAP is a popular neighborhood-preserving dimension reduction (DR) algorithm. However, its application for graph drawing has not been evaluated. Moreover, a naive application of UMAP to graph drawing would include O(nm) time all-pair shortest path computation, which is not scalable to visualizing large graphs. In this paper, we present fast UMAP-based for graph drawing. Specifically, we present three fast UMAP-based algorithms for graph drawing: (1) The SS-GUMAP algorithm utilizes spectral sparsification to compute a subgraph G' preserving important properties of a graph G, reducing the O(nm) component of the runtime to O(n^2 log n) runtime; (2) The SSL-GUMAP algorithm reduces the kNN (k-Nearest Neighbors) graph computation from $O(n log n)$ time to linear time using partial BFS (Breadth First Search), and the cost optimization runtime from O(n) time to sublinear time using edge sampling; (3) The SSSL-GUMAP algorithm combines both approaches, for an overall O(n) runtime. Experiments demonstrate that SS-GUMAP runs 28% faster than GUMAP, a naive application of UMAP to graph drawing, with similar quality metrics, while SL-GUMAP and SSSL-GUMAP run over 80% faster than GUMAP with less than 15% difference on average for all quality metrics. We also present an evaluation of GUMAP to tsNET, a graph layout based on the popular DR algorithm t-SNE. GUMAP runs 90% faster than tsNET with similar neighborhood preservation and, on average, 10% better on quality metrics such as stress, edge crossing, and shape-based metrics, validating the effectiveness of UMAP for graph drawing.

Problem

• UMAP's naive application to graph drawing requires O(nm) time computation
• Existing UMAP approaches are not scalable for visualizing large graphs
• Lack of efficient UMAP-based algorithms specifically for graph visualization

Innovation

• Spectral sparsification reduces runtime to O(n² log n)
• Partial BFS enables linear time kNN graph computation
• Edge sampling achieves sublinear cost optimization runtime