Topology-aware Graph Diffusion Model with Persistent Homology

Generating realistic graphs faces challenges in estimating accurate distribution of graphs in an embedding space while preserving structural characteristics. However, existing graph generation methods primarily focus on approximating the joint distribution of nodes and edges, often overlooking topological properties such as connected components and loops, hindering accurate representation of global structures. To address this issue, we propose a Topology-Aware diffusion-based Graph Generation (TAGG), which aims to sample synthetic graphs that closely resemble the structural characteristics of the original graph based on persistent homology. Specifically, we suggest two core components: 1) Persistence Diagram Matching (PDM) loss which ensures high topological fidelity of generated graphs, and 2) Topology-aware Attention Module (TAM) which induces the denoising network to capture the homological characteristics of the original graphs. Extensive experiments on conventional graph benchmarks demonstrate the effectiveness of our approach indicating high generation performance across various metrics, while achieving closer alignment with the distribution of topological features observed in the original graphs. Furthermore, application to real brain network data showcases its potential for complex and real graph applications. * J. Park and D. Lee contributed equally. Work done while D. Lee 2 and Y. Song 2 were at POSTECH. 39th Conference on Neural Information Processing Systems (NeurIPS 2025).

fidelity. The methods in [23] and [32] proposed score-based diffusion methods in a continuous time domain, originally defined for images [40]. However, the continuous diffusion methods suffer from high computational cost, as the forward and reverse diffusion process is performed on infinitesimal continuous time point. Moreover, the uniformly added Gaussian noise results in a noisy and complete graph, which causes the loss of structural information, e.g., sparsity of a graph. Later, [43] proposed a discrete diffusion method, applying additive noise to each node and edge independently for graphs, nevertheless, existing methods overlook the topologically invariant characteristics, e.g., geometric shape and connectivity, limiting the generation.

To overcome such issues, we propose a novel Topology-Aware Graph Generation (TAGG), in which the sampled graphs resemble not only in the distributions of the original graphs in the embedding space but also in the homological features of the original graphs. Conventionally, topological data analysis (TDA) from algebraic topology has been studied in various graph analyses [6,9,21,44] to investigate topological features, and we bridge the gap between TDA and graph diffusion model to generate topologically realistic graphs. We define Persistence Diagram Matching (PDM) loss with persistence homology, which regularizes homological features of the reference graphs to be incorporated in graph generation process via 1-Wasserstein distance. Furthermore, we introduce a Topology-aware Attention Module (TAM) which utilizes persistence landscape [5] of a given graph to foster the denoising network with global structural information.

Contributions. To this end, our main contributions are summarized as follows: 1) We propose a novel topology-aware graph generation method that yields homologically similar graphs with high structural fidelity. 2) We propose PDM loss, utilizing persistent homology to encode the graph topology. 3) We propose a Topology-aware Attention Module (TAM) that leverages persistence landscape to enhance the denoising network in capturing graph topology.

Our model demonstrates superior performance on real and synthetic graph generation, with intuitive visualizations for topological comparisons. Especially with the application on brain network generation from Alzheimer's Disease Neuroimaging Initiative (ADNI), our method demonstrates its adaptability to diverse real-world graph generation tasks.

Graph Generation. Graph generation has been developed in two major branches; autoregressive and one-shot. Auto-regressive methods [4,22,25,38,46,47] recursively capture the intricate graph dependencies, and sequentially generate the graph structure conditioned on the current incomplete graph. In spite of their impressive performance, auto-regressive approaches exhibit considerable computational demands due to the increasing number of generation steps along with the graph size. Also, they face a challenge stemming from the absence of an inherent node generation order. Conversely, one-shot methods [12,27,28,48] generate the whole graph, i.e., every node and edge, at once. By doing so, they reduce computational requirements while facing performance degradation as the dataset scale grows. Recently, diffusion-based methods [4,11,23,29,30,43] showed promising capability in graph generation, by defining the forward and reverse diffusion processes and training a neural network that mimics the reve