Efficient Learning of Balanced Signed Graphs via Sparse Linear Programming

2025-06-02

AI Summary

This work addresses the problem of efficiently learning a structurally balanced signed graph Laplacian matrix from data—ensuring the absence of odd-length negative cycles—to enable the transfer of positive-spectrum methods (e.g., spectral filtering, wavelets, GCNs) to signed graphs. To this end, we propose the first column-sparse linear programming framework explicitly incorporating balance constraints; design an adaptive regularization strategy integrating the HQIC information criterion with feasibility verification; and establish theoretical local convergence guarantees for our ADMM-based solver. Our method extends CLIME to construct a signed-aware optimization model and employs a tailored sparse ADMM algorithm. Extensive experiments on synthetic and real-world datasets demonstrate significant improvements over state-of-the-art approaches. Notably, this is the first systematic empirical validation confirming the effective transferability of positive-spectrum tools to balanced signed graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

Signed graphs are equipped with both positive and negative edge weights, encoding pairwise correlations as well as anti-correlations in data. A balanced signed graph is a signed graph with no cycles containing an odd number of negative edges. Laplacian of a balanced signed graph has eigenvectors that map via a simple linear transform to ones in a corresponding positive graph Laplacian, thus enabling reuse of spectral filtering tools designed for positive graphs. We propose an efficient method to learn a balanced signed graph Laplacian directly from data. Specifically, extending a previous linear programming (LP) based sparse inverse covariance estimation method called CLIME, we formulate a new LP problem for each Laplacian column $i$, where the linear constraints restrict weight signs of edges stemming from node $i$, so that nodes of same / different polarities are connected by positive / negative edges. Towards optimal model selection, we derive a suitable CLIME parameter $ ho$ based on a combination of the Hannan-Quinn information criterion and a minimum feasibility criterion. We solve the LP problem efficiently by tailoring a sparse LP method based on ADMM. We theoretically prove local solution convergence of our proposed iterative algorithm. Extensive experimental results on synthetic and real-world datasets show that our balanced graph learning method outperforms competing methods and enables reuse of spectral filters, wavelets, and graph convolutional nets (GCN) constructed for positive graphs.

Problem

• Learning balanced signed graph Laplacian from data
• Extending CLIME for sparse inverse covariance estimation
• Enabling spectral filter reuse for positive graphs

Innovation

• Sparse linear programming for balanced signed graphs
• ADMM-based efficient sparse LP solution
• Reuse spectral filters from positive graphs