Dynamic Graph Structure Learning via Resistance Curvature Flow

TL;DR

Dynamic Graph Structure Learning via Resistance Curvature Flow introduces Resistance Curvature Flow (RCF) as a scalable alternative to Ollivier-Ricci curvature for evolving graph topology. By leveraging effective resistance, RCF converts curvature optimization into efficient matrix operations, enabling end-to-end differentiable integration in deep models through DGSL-RCF with preprocessing, regularization, or post-processing paradigms. Theoretical analysis demonstrates computational efficiency and manifold-enhancement capabilities, while extensive experiments across deep metric learning, manifold learning, and graph structure learning show improved representation quality and task performance with substantially reduced runtime. This work offers a practical, geometry-informed approach to dynamic graph learning suitable for large-scale, real-time deep learning applications.

Abstract

Geometric Representation Learning (GRL) aims to approximate the non-Euclidean topology of high-dimensional data through discrete graph structures, grounded in the manifold hypothesis. However, traditional static graph construction methods based on Euclidean distance often fail to capture the intrinsic curvature characteristics of the data manifold. Although Ollivier-Ricci Curvature Flow (OCF) has proven to be a powerful tool for dynamic topological optimization, its core reliance on Optimal Transport (Wasserstein distance) leads to prohibitive computational complexity, severely limiting its application in large-scale datasets and deep learning frameworks. To break this bottleneck, this paper proposes a novel geometric evolution framework: Resistance Curvature Flow (RCF). Leveraging the concept of effective resistance from circuit physics, RCF transforms expensive curvature optimization into efficient matrix operations. This approach achieves over 100x computational acceleration while maintaining geometric optimization capabilities comparable to OCF. We provide an in-depth exploration of the theoretical foundations and dynamical principles of RCF, elucidating how it guides the redistribution of edge weights via curvature gradients to eliminate topological noise and strengthen local cluster structures. Furthermore, we provide a mechanistic explanation of RCF's role in manifold enhancement and noise suppression, as well as its compatibility with deep learning models. We design a graph optimization algorithm, DGSL-RCF, based on this framework. Experimental results across deep metric learning, manifold learning, and graph structure learning demonstrate that DGSL-RCF significantly improves representation quality and downstream task performance.

Figures (13)

• Figure 1: The analogy between resistance in electrical circuits and paths in graphs.
• Figure 2: Cost comparison for RCF and OCF under different $k$ (top-left: S Curve, top-right: Swiss Roll, bottom-left: Truncated Sphere, bottom-right: Gaussian Surface).
• Figure 3: Comparison of convergence on NMI, F1 and Recall metric between the +DGSL-RCF method and the baseline method on the CUB-200-2011 (Top) and Cars-196 (Bottom) datasets for DML task.
• Figure 4: Impact of hyperparameter $k$ on the ACC metric of LEM, LEM+OCF, and LEM+RCF in ML task (top-left: S Curve, top-right: Swiss Roll, bottom-left: Truncated Sphere, bottom-right: Gaussian Surface).
• Figure 5: Impact of hyperparameter $k$ (density) on the NMI (left), F1(middle) and Recall (right) under different batch size in DML task (top: CUB-200-2011, bottom: Cars-196).