Transferable Graph Metanetworks

Graph neural networks (GNNs) can operate on large graphs but become infrastructure-sensitive at the scale of millions of nodes and typically require scalable training techniques for even larger graphs. This raises a central question: when can a model trained on a smaller, scaled-down replica of a graph be deployed on the full-resolution graph without retraining? We introduce a zero-shot transfer protocol in which a GNN is trained on a graph coarse-grained by geometric renormalization (GR), and the resulting weights are transferred directly to the original network. Across synthetic and real-world networks, training on GR scaled-down replicas preserves much of the original-scale predictive performance while significantly reducing training cost. We further find that learned representations and predictive trajectories remain aligned across scales. These findings suggest that structural similarity may be more important than network size in determining GNN transferability, opening a path toward scale-equivariant graph architectures.

Metanetworks are neural architectures designed to operate directly on pretrained weights to perform downstream tasks. However, the parameter space serves only as a proxy for the underlying function class, and the parameter-function mapping is inherently non-injective: distinct parameter configurations may yield identical input-output behaviors. As a result, metanetworks that rely solely on raw parameters risk overlooking the intrinsic symmetries of the architecture. Reasoning about functional identity is therefore essential for effective metanetwork design, motivating the development of equivariant metanetworks, which incorporate equivariance principles to respect architectural symmetries. Existing approaches, however, typically enforce strict equivariance, which imposes rigid constraints and often leads to sparse and less expressive models. To address this limitation, we introduce the novel concept of quasi-equivariance, which allows metanetworks to move beyond the rigidity of strict equivariance while still preserving functional identity. We lay down a principled basis for this framework and demonstrate its broad applicability across diverse neural architectures, including feedforward, convolutional, and transformer networks. Through empirical evaluation, we show that quasi-equivariant metanetworks achieve good trade-offs between symmetry preservation and representational expressivity. These findings advance the theoretical understanding of weight-space learning and provide a principled foundation for the design of more expressive and functionally robust metanetworks.

Self-organization is an emergent property of life, driven by the collective behavior of individual components acting on local information. Biological neurons, through local interactions transmitted through synapses, are able to learn efficiently and can adapt their connections over an organism's lifespan. Motivated by these desirable properties of adaptability and local interaction, neural cellular automata (NCA) models have been successful at learning morphogenesis solely through local update rules, demonstrating stability over many updates and robustness to perturbations. In this work, we introduce Meta Neural Cellular Automata (MetaNCA), a framework that learns local rules which self-organize the weights of artificial neural networks. A learned rule network iteratively updates the weights of a task network using only local interactions on the computation graph. We propose a novel Weight Transformer architecture for the local rule network, which uses linear attention to aggregate signals from neighboring weights and hidden states. Once trained, the rule network generates task networks of diverse architectures without backpropagation. We show that MetaNCA generates weights for feedforward MLPs, CNNs, and ResNets on MNIST and CIFAR-100, scaling to networks of 2 million parameters. We further show that MetaNCA generalizes to architectures not seen during meta-training, and that architectural diversity in the training phase strengthens this generalization.