Message-passing neural networks (MPNNs) often struggle when task-relevant information is distributed across distant regions of a graph, since local propagation must compress remote signals through limited structural interfaces. Graph rewiring provides a structural response to over-squashing. Most existing methods rely on edge-level bottleneck scores or graph-level connectivity surrogates. With a limited rewiring budget, the key question is which pairwise communications most need structural support. This paper proposes PairAlign, a pair-centric graph rewiring framework that makes this question explicit through demand-support shortage. Specifically, PairAlign combines original-graph structural demand with current-graph finite-hop propagation support; their ratio highlights interactions whose communication demand is poorly supported by topology, and our theory shows that this score provides a computable proxy for the corresponding Jacobian-based shortage with a pair-level interpretation of over-squashing. Our theory reveals a two-sided effect of edge insertion: a new edge can create useful walks and simultaneously dilute existing normalized transition mass. Guided by this observation, PairAlign optimizes shortage to favor edge additions that alleviate over-squashing. Beyond selecting useful additions, PairAlign further introduces an Optimal Transport-guided rewiring mechanism to coordinate the finite edge budget for pair-level structural compatibility and shortage-target coverage. It formulates communication alignment between the candidate edge budget and the shortage targets, and the theory shows that this allocation covers shortage targets more broadly and effectively than a greedy-local assignment. Experiments on standard graph benchmarks show PairAlign's improvement across message-passing backbones, validating pair-level repair as an effective route for alleviating over-squashing.
Graph Neural Networks (GNNs) suffer from two fundamental limitations: over-smoothing, where node representations become indistinguishable with depth, and over-squashing, where long-range information is compressed through limited message-passing channels. Existing metrics such as Dirichlet energy provide global characterizations of over-smoothing but lack the resolution to analyze node-level behavior and guide architectural improvements. In this paper, we propose LEED (Local Embedding Evolution Distance), a novel local metric that quantifies over-smoothing by tracking the evolution of individual node embeddings across layers. By operating at the node level, LEED enables fine-grained analysis of representation dynamics during training, revealing heterogeneous over-smoothing patterns that are invisible to global energy-based measures. This locality induces informative node importance scores, interpreted as embedding-driven centrality measures. We leverage LEED to design a more efficient strategy for virtual node selection. Unlike existing approaches that depend on multiple heuristic centrality measures, our method uses LEED as a unique criterion to guide the construction of Local Virtual Nodes to mitigate over-squashing. Experiments show that LEED provides more informative diagnostics than Dirichlet energy while preserving global evaluation, and enables more effective virtual node integration, improving GNN performance across datasets.
The information flow in the graph neural networks (GNNs) is fundamentally constrained by over-squashing, where structural bottlenecks impede long range information propagation. Graph-rewiring methods, which modify graph topology, have been extensively used to alleviate this. However, existing approaches often introduce prohibitive structural and computational bottlenecks, fail to preserve the critical properties of original graphs, and increase the edge counts massively. We introduce a novel method Schreier-Coset Graph Rewiring , a group-theoretic rewiring method that augments the input graph with a Schreier-Coset graph derived from a special linear group. Our method provides theoretical guarantees, a graph that exhibits spectral gap and a bounded effective resistance, creating a low-resistance bypass for long-range communication. Empirical evaluations demonstrate that SCGR reduces effective resistance by 5-40% across various learning tasks, effectively mitigating connectivity bottlenecks while maintaining competitive accuracy.
Graph Neural Networks (GNNs) have emerged as a powerful paradigm for learning on graph-structured data by iteratively propagating and aggregating information across edges. However, conventional message passing schemes often suffer from over-squashing, whereby exponentially large neighborhoods are compressed into fixed-dimensional embeddings, impeding effective long-range dependency learning. In this work, we introduce Ramanujan Propagation, a graph rewiring strategy that leverages Ramanujan graphs to alleviate topological bottlenecks in GNNs. We first establish that suitably chosen Ramanujan graphs guarantee non-negative resistance curvature, which mitigates over-squashing and facilitates efficient information flow. We then propose an algorithmic framework to construct a Ramanujan rewired graph that preserves the local connectivity of the original graph. Our experiments demonstrate that our method outperforms nine state-of-the-art rewiring techniques. These results establish Ramanujan graphs as a rigorous structural prior for scalable, topology-aware message passing in GNNs.