CoPHo: Classifier-guided Conditional Topology Generation with Persistent Homology

TL;DR

CoPHo introduces a classifier-guided, discrete diffusion framework that integrates persistent homology filtrations to condition topology generation without retraining the diffusion backbone. By constructing PH-based multi-scale filtrations and using classifier gradients as guidance signals at each denoising step, CoPHo achieves precise control over both global and fine-grained graph properties and mitigates sparse-gradient collapse. Empirical results across diverse graph datasets and transfer to QM9 molecular graphs demonstrate superior conditional fidelity while maintaining sample quality, with favorable training-time trade-offs. This topology-aware diffusion approach enables practical, scalable generation of synthetic graphs that adhere to complex structural constraints for networking and molecular design applications.

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.

Figures (5)

• Figure 1: Comparison between the current methods and CoPHo. (a), (c) provide a high-level explanation of how CoPHo differs from existing methods, while (b) and (d) show that sparse gradients on continuous graphs cause collapse; CoPHo instead discretely edits the graph to avoid this.
• Figure 2: Overview of CoPHo. At each denoising step, we build a decreasing filtration via persistent homology to capture multi-scale topological features. A local subgraph around the conditioned nodes and the full graph are fed into a lightweight GNN $\Phi$, which outputs node-level and edge-level gradient signals. These signals drive monotonic node and edge removals, producing a sequence of subgraphs that converges to a final graph $\widehat{G}$ meeting the desired global and fine-grained properties without retraining the diffusion backbone.
• Figure 3: Visualization of Community-small. Top: real topologies from Community-small. Middle: The corresponding conditional samples without CoPHo. Bottom: Conditional samples with CoPHo.
• Figure 4: A case study on shortest-path conditioning. Red: negative gradient. Blue: positive gradient.
• Figure 5: Generate sample resulting from posterior and gradient guidance.