Graph Guided Diffusion: Unified Guidance for Conditional Graph Generation

TL;DR

This work addresses the challenge of conditionally generating graphs under arbitrary rewards by reframing conditional graph diffusion as a stochastic optimal control problem and introducing GGDiff, a gradient-free, plug-and-play guidance framework. It derives an optimal control via the Feynman-Kac formalism and provides practical greedy, zeroth-order estimators (one-point, best-of-N, and multi-point) to handle non-differentiable rewards, unifying gradient-based and gradient-free strategies. The authors demonstrate GGDiff on constrained, fair, and incomplete graph generation, showing superior reward alignment while preserving graph distribution quality and diversity compared to prior methods like PRODIGY. The approach broadens the applicability of diffusion-based graph generation to complex, black-box constraints and paves the way for flexible, user-defined conditional graph synthesis in real-world domains. Key contributions include the SOC formulation for graphs, the gradient-free ZO estimators, and extensive empirical validation across diverse tasks.

Abstract

Diffusion models have emerged as powerful generative models for graph generation, yet their use for conditional graph generation remains a fundamental challenge. In particular, guiding diffusion models on graphs under arbitrary reward signals is difficult: gradient-based methods, while powerful, are often unsuitable due to the discrete and combinatorial nature of graphs, and non-differentiable rewards further complicate gradient-based guidance. We propose Graph Guided Diffusion (GGDiff), a novel guidance framework that interprets conditional diffusion on graphs as a stochastic control problem to address this challenge. GGDiff unifies multiple guidance strategies, including gradient-based guidance (for differentiable rewards), control-based guidance (using control signals from forward reward evaluations), and zero-order approximations (bridging gradient-based and gradient-free optimization). This comprehensive, plug-and-play framework enables zero-shot guidance of pre-trained diffusion models under both differentiable and non-differentiable reward functions, adapting well-established guidance techniques to graph generation--a direction largely unexplored. Our formulation balances computational efficiency, reward alignment, and sample quality, enabling practical conditional generation across diverse reward types. We demonstrate the efficacy of GGDiff in various tasks, including constraints on graph motifs, fairness, and link prediction, achieving superior alignment with target rewards while maintaining diversity and fidelity.

Figures (4)

• Figure 1: Illustration of GGDiff, a method that guides the generation of graphs to satisfy a set of constraints (in this case, the constraint is fairness). The guidance ${\mathbf U}_t$ is a local direction obtained via SOC, and approximated using ZO techniques, like the multi-point estimate shown here.
• Figure 2: Samples for the force stars constraint in the Ego small dataset.
• Figure 3: Samples from the fair graph generation experiment.
• Figure 4: Samples generated for the incomplete graph generation experiment. The graph on the left is the test graph from which we observe the entries in its adjacency matrix. The generated graphs are represented in the rows, one for each of the methods. In the generated graphs, the solid green (red) lines are observed edges that were (not) preserved in the generated graphs, while dotted green (red) lines are observed entries not corresponding to an edge that were (not) preserved in the generated graphs.