Signed Graph Learning: Algorithms and Theory

2025-07-13

AI Summary

This paper addresses the critical problem of signed graph structure learning. We propose the first signed graph learning framework based on the net Laplacian operator. Unlike conventional unsigned-graph methods, our approach integrates low-pass filtering intuition with the signed-graph signal smoothness assumption, formulating a nonconvex optimization model that minimizes total variation. Theoretically, we establish the first convergence guarantee and derive an explicit estimation error bound, characterizing the interplay among sample size, number of nodes, and graph topology. Computationally, we design an efficient ADMM-based algorithm that reduces per-iteration complexity from $O(n^2)$ to $O(n)$, substantially improving scalability. Extensive experiments—including synthetic benchmarks and gene regulatory network inference—demonstrate that our method consistently outperforms state-of-the-art approaches in both structural recovery accuracy and computational efficiency.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Graphical Models

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
• Economics, Online Markets and Human Computation: Social networks and social learning
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Real-world data is often represented through the relationships between data samples, forming a graph structure. In many applications, it is necessary to learn this graph structure from the observed data. Current graph learning research has primarily focused on unsigned graphs, which consist only of positive edges. However, many biological and social systems are better described by signed graphs that account for both positive and negative interactions, capturing similarity and dissimilarity between samples. In this paper, we develop a method for learning signed graphs from a set of smooth signed graph signals. Specifically, we employ the net Laplacian as a graph shift operator (GSO) to define smooth signed graph signals as the outputs of a low-pass signed graph filter defined by the net Laplacian. The signed graph is then learned by formulating a non-convex optimization problem where the total variation of the observed signals is minimized with respect to the net Laplacian. The proposed problem is solved using alternating direction method of multipliers (ADMM) and a fast algorithm reducing the per-ADMM iteration complexity from quadratic to linear in the number of nodes is introduced. Furthermore, theoretical proofs of convergence for the algorithm and a bound on the estimation error of the learned net Laplacian as a function of sample size, number of nodes, and graph topology are provided. Finally, the proposed method is evaluated on simulated data and gene regulatory network inference problem and compared to existing signed graph learning methods.

Problem

• Learning signed graphs from smooth graph signals
• Developing a non-convex optimization method for graph learning
• Providing theoretical guarantees for algorithm convergence

Innovation

• Uses net Laplacian as graph shift operator
• Solves non-convex optimization via ADMM
• Reduces iteration complexity to linear