Learn When and Where to Connect: Adaptive Virtual Nodes for Dynamic Message Passing on Graphs

Message-passing neural networks (MPNNs) are a powerful framework for learning representations of graph-structured domains. However, weights in MPNNs act on features only, limiting their ability to capture structural patterns. We introduce a novel structure-aware weight sharing principle that explicitly incorporates information inherent to the graph structure. Weights are indexed directly by user-chosen graph invariants, i.e., functions preserved under node permutations, enabling systematic reuse across structurally equivalent subgraphs. We present ShareGNNs, which instantiate this principle within a simple encoder-decoder architecture, resulting in an MPNN with learnable adjacency and transformer-like connectivity. We show that their expressivity is at least as strong as the discriminative power of the chosen invariants, providing explicit control over the model complexity. Experiments on synthetic and real-world data, as well as subgraph counting tasks, demonstrate consistent improvements over standard MPNNs, competitive expressivity beyond the 1-WL test, and scalability to large datasets.

We define an information-complexity property for aggregation functions, capturing a vast range of practical aggregations, and prove that any Message-Passing Graph Neural Network (MP-GNN) model with such aggregations induces only a polynomial number of equivalence classes on all graphs - while the number of non-isomorphic graphs is super-exponential (in number of vertices). Adding a familiar perspective, we observe that merely 2 iterations of Color Refinement (CR) induce at least an exponential number of equivalence classes, making the aforementioned MP-GNNs relatively infinitely weaker. Previous studies state that sum-aggregation MP-GNNs match full CR however they consider a weak, 'non-uniform', notion of distinguishing-power where each graph size may require a different MP-GNN to distinguish graphs up to that size. Our results concern both distinguishing between non-equivariant vertices and distinguishing between non-isomorphic graphs.

Pre-propagation graph neural networks (PPGNNs) push all graph-dependent computation into a preprocessing step and train only on the resulting dense hop features, which makes them highly scalable. A puzzle in this regime is that more complex hop aggregators do not reliably outperform simpler ones: on many benchmarks, a plain MLP-based aggregator matches or beats hop-attention variants. We revisit this behavior from a graph-filter perspective. Over a precomputed diffusion basis, existing PPGNNs differ mainly in how filter coefficients are shared across nodes and feature channels, rather than simply in raw aggregator capacity. MLP-based architectures learn channel-dependent filters that are largely shared across nodes, while hop-attention-based architectures learn node-dependent mixtures that are largely shared across channels. This reveals a missing regime in standard PPGNN designs: joint node- and channel-adaptive filtering under the pre-propagation computational contract. We propose FilterMoE, a mixture-of-experts PPGNN in which a small bank of learnable Chebyshev filter experts is routed jointly over nodes and channels by a 3D gating tensor. Across eleven homophilic and heterophilic benchmarks, FilterMoE outperforms strong PPGNN baselines on nine datasets and ranks first on all three large-scale benchmarks, improving the average test score by 1.53 points. These results establish joint node-channel filter routing as a robust alternative to dataset-specific hop-aggregator selection.