Graph Energy Matching: Transport-Aligned Energy-Based Modeling for Graph Generation

TL;DR

Graph Energy Matching (GEM) is a novel energy-based framework for graph generation that improves sampling quality and enables flexible, constrained, and compositional graph inference tasks.

Contribution

GEM introduces a transport-aligned energy-based model for graphs, bridging the fidelity gap with diffusion models and enabling efficient, high-quality graph sampling and inference.

Findings

GEM matches or exceeds diffusion baselines on molecular graph benchmarks.

Explicit likelihood modeling allows targeted, property-constrained graph sampling.

The sampling protocol effectively balances rapid transport and exploration.

Abstract

Energy-based models for discrete domains, such as graphs, explicitly capture relative likelihoods, naturally enabling composable probabilistic inference tasks like conditional generation or enforcing constraints at test-time. However, discrete energy-based models typically struggle with efficient and high-quality sampling, as off-support regions often contain spurious local minima, trapping samplers and causing training instabilities. This has historically resulted in a fidelity gap relative to discrete diffusion models. We introduce Graph Energy Matching (GEM), a generative framework for graphs that closes this fidelity gap. Motivated by the transport map optimization perspective of the Jordan-Kinderlehrer-Otto (JKO) scheme, GEM learns a permutation-invariant potential energy that simultaneously provides transport-aligned guidance from noise toward data and refines samples within regions of high data likelihood. Further, we introduce a sampling protocol that leverages an energy-based switch to seamlessly bridge: (i) rapid, gradient-guided transport toward high-probability regions to (ii) a mixing regime for exploration of the learned graph distribution. On molecular graph benchmarks, GEM matches or exceeds strong discrete diffusion baselines. Beyond sample quality, explicit modeling of relative likelihood enables targeted exploration at inference time, facilitating compositional generation, property-constrained sampling, and geodesic interpolation between graphs.