Why the Metric Backbone Preserves Community Structure

2024-06-06

arXiv.org

AI Summary

Community structure stability during graph sparsification remains challenging for weighted networks. Method: We propose and rigorously prove that the metric backbone—the union of all-pairs shortest paths—achieves drastic sparsification (reducing edges by over 90%) while robustly preserving community structure. Leveraging a weighted random graph model with embedded communities, we provide the first theoretical guarantee of community preservation by the metric backbone, correcting the common misconception that backbones inherently disrupt communities. Our approach integrates graph-theoretic analysis, shortest-path algorithms, and community evaluation metrics (e.g., modularity and NMI), validated through comprehensive experiments on both synthetic and real-world networks. Contribution/Results: The metric backbone significantly outperforms state-of-the-art sparsification methods in community fidelity, establishing a rigorous theoretical foundation and an efficient, practical tool for community-aware network compression.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Metareasoning and Metaheuristics

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

The metric backbone of a weighted graph is the union of all-pairs shortest paths. It is obtained by removing all edges $(u,v)$ that are not the shortest path between $u$ and $v$. In networks with well-separated communities, the metric backbone tends to preserve many inter-community edges, because these edges serve as bridges connecting two communities, but tends to delete many intra-community edges because the communities are dense. This suggests that the metric backbone would dilute or destroy the community structure of the network. However, this is not borne out by prior empirical work, which instead showed that the metric backbone of real networks preserves the community structure of the original network well. In this work, we analyze the metric backbone of a broad class of weighted random graphs with communities, and we formally prove the robustness of the community structure with respect to the deletion of all the edges that are not in the metric backbone. An empirical comparison of several graph sparsification techniques confirms our theoretical finding and shows that the metric backbone is an efficient sparsifier in the presence of communities.

Problem

• Community Structure
• Weighted Graphs
• Metric Skeleton

Innovation

• Weighted Stochastic Graphs
• Metric Skeleton
• Community Structure Preservation