Graph neural networks (GNN) based on message passing are provably no more powerful than the one-dimensional Weisfeiler--Leman colour-refinement test (1-WL): two graphs it cannot tell apart receive identical representations, however deep or wide the network. A common remedy augments node or edge features with precomputed structural descriptors, most often counts of a fixed small subgraph such as triangles or longer cycles, but such counts require committing in advance to the size of the substructure counted, a choice usually made blind to the data. We study a descriptor that avoids this choice. The edge-girth of an edge is the length of a shortest cycle through it, and its multiplicity is the number of such shortest cycles; together they form a per-edge invariant that reports cycles of arbitrary length, computable exactly by a single breadth-first search per edge. Injected into a gated message-passing architecture, EGAGNN, it reaches a test MAE a factor three below the closest gated comparator on the ZINC-12k regression benchmark at 104k parameters; against bounded cycle-counting descriptors under the same architecture, it matches only a dictionary counting cycles up to length eight, using twice as many channels, while a dictionary capped at length four performs no better than no structural information at all. On graph discrimination we prove a matching limitation: on graphs where every edge sees the same number of shortest cycles of the same length, the descriptor becomes constant and any model built on it collapses back to the 1-WL bound. This holds without exception across all 400 pairs of the BREC benchmark: not one of the 90 such pairs is distinguished.
Edge-attributed multigraphs, in which multiple edges with distinct attributes connect the same pair of nodes, arise naturally in many real-world systems. In these graphs, effective learning requires preserving information from repeated interactions while distinguishing contributions from different neighbors. Existing neural network solutions for edge-attributed multigraphs remain limited: some lose information from repeated interactions, while others break permutation equivariance. To address this, we introduce \emph{neighbor-aware aggregation}, an operator that first combines multi-edge features for each neighbor and then aggregates across neighbors. This operator captures per-neighbor statistics that standard single-stage aggregation cannot represent. Building on this operator, we present MEGA-GNN, a model-agnostic message-passing framework for edge-attributed multigraphs. We show that MEGA-GNN is permutation equivariant and has the same asymptotic complexity as standard GNNs with edge updates. We evaluate our approach on datasets from social networks and financial transaction networks. Neighbor-aware aggregation consistently improves GNN performance and matches or surpasses state-of-the-art methods.
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 introduce a constrained two-view framework for node prediction that aligns structure-conditioned GNN embeddings with a structure-free feature prior learned by an anchor model. Conventional Graph Neural Networks (GNNs) couple feature transformation and neighborhood aggregation, which renders them vulnerable to topology noise and heterophilous connections. To decouple this dependency, our framework utilizes an independent anchor network to capture intrinsic attribute features via a self-supervised reconstruction objective. Furthermore, we propose a Channel-Split Adaptive Gated GNN (CSAG-GNN) that dynamically routes representations between global spectral smoothing and local spatial discrimination through a node-wise gating mechanism. We propose a stable cyclic alternating optimization strategy to solve the resulting coupled bi-level objective, preventing mutual representation drift during training. Empirical results on both homophilous and heterophilous benchmarks show balanced performance gains and structural robustness over competitive baselines.