The Polynomial Counting Capabilities of Message Passing Neural Networks

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.

We study the expressive power of message-passing aggregate-combine-readout graph neural networks (ACR-GNNs). Particularly, we focus on the first-order (FO) properties expressible by this formalism. While a tight logical characterisation remains a difficult open question, we make two contributions towards answering it. First, we show that sum aggregation and readout suffice for GNNs to capture FO properties that cannot be expressed in the logic C2 on both directed and undirected graphs. This strengthens known results by Hauke and Wał{\k e}ga (2026) where aggregation and readout functions are specially crafted for the task. Second, we identify two natural ways of restoring characterisability (with regard to C2) for ACR-GNNs. One option is to limit local aggregation (without imposing restrictions on global readout), whilst the second is to run ACR-GNNs over graphs of bounded degree (but unbounded size). In both cases, the FO properties captured by GNNs are exactly those definable by a formula in graded modal logic with global counting modalities. Our results thus establish an innate lower- and upper-bound in terms of how far (fragments of) C2 can be taken to characterise GNNs, and imply that is indeed the unbounded interaction of aggregation and readout that pushes the logical expressive power of GNNs above C2.

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.