CoPHo: Classifier-guided Conditional Topology Generation with Persistent Homology

2025-12-17

AI Summary

Existing graph diffusion models struggle to simultaneously achieve topological controllability and generation efficiency: retraining-based strategies lack generalizability, while classifier-guided methods neglect topological scale awareness and practical constraints. This paper addresses sparse topological data by proposing a retraining-free discrete graph diffusion generative framework. We introduce the first approach that directly injects gradients from a pretrained graph classifier into the discrete reverse diffusion posterior. Moreover, we pioneer the dynamic incorporation—within each denoising step—of multi-scale topological features extracted via persistent homology (PH) filtering as differentiable conditional signals. Our method unifies global structural modeling with local-scale constraints. Experiments on four network datasets demonstrate significant improvements in target topological metric fidelity. On the QM9 molecular dataset, our framework validates cross-domain transferability, reducing Betti number control error by 32%.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Diffusion Models for Vision
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The structure of topology underpins much of the research on performance and robustness, yet available topology data are typically scarce, necessitating the generation of synthetic graphs with desired properties for testing or release. Prior diffusion-based approaches either embed conditions into the diffusion model, requiring retraining for each attribute and hindering real-time applicability, or use classifier-based guidance post-training, which does not account for topology scale and practical constraints. In this paper, we show from a discrete perspective that gradients from a pre-trained graph-level classifier can be incorporated into the discrete reverse diffusion posterior to steer generation toward specified structural properties. Based on this insight, we propose Classifier-guided Conditional Topology Generation with Persistent Homology (CoPHo), which builds a persistent homology filtration over intermediate graphs and interprets features as guidance signals that steer generation toward the desired properties at each denoising step. Experiments on four generic/network datasets demonstrate that CoPHo outperforms existing methods at matching target metrics, and we further validate its transferability on the QM9 molecular dataset.

Problem

• Generates synthetic graphs with desired topological properties
• Incorporates classifier guidance into discrete reverse diffusion process
• Steers generation toward specified structural features using persistent homology

Innovation

• Classifier-guided gradients steer diffusion generation
• Persistent homology filtration interprets intermediate graph features
• Discrete reverse diffusion integrates topology constraints without retraining