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.

Generating novel graph structures that optimize given objectives while obeying some given underlying rules is fundamental for chemistry, biology and social science research. This is especially important in the task of molecular graph generation, whose goal is to discover novel molecules with desired properties such as drug-likeness and synthetic accessibility, while obeying physical laws such as chemical valency. However, designing models that finds molecules that optimize desired properties while incorporating highly complex and non-differentiable rules remains to be a challenging task. Here we propose Graph Convolutional Policy Network (GCPN), a general graph convolutional network based model for goal-directed graph generation through reinforcement learning. The model is trained to optimize domain-specific rewards and adversarial loss through policy gradient, and acts in an environment that incorporates domain-specific rules. Experimental results show that GCPN can achieve 61% improvement on chemical property optimization over state-of-the-art baselines while resembling known molecules, and achieve 184% improvement on the constrained property optimization task.

We introduce a novel score-based diffusion framework named Twigs that incorporates multiple co-evolving flows for enriching conditional generation tasks. Specifically, a central or trunk diffusion process is associated with a primary variable (e.g., graph structure), and additional offshoot or stem processes are dedicated

Diffusion generative models, as an emerging research focus, have been applied to graph generation tasks.