Graph Neural Networks struggle to capture long-range dependencies due to over-squashing, where information from exponentially growing neighborhoods must pass through a small number of structural bottlenecks. While recent rewiring methods attempt to alleviate this limitation, many rely on local criteria such as curvature, which can overlook global connectivity constraints that restrict information flow. We introduce Effective Resistance Rewiring (ERR), a simple topology correction strategy that uses effective resistance as a global signal to detect structural bottlenecks. ERR iteratively adds edges between node pairs with the largest resistance while removing edges with minimal resistance, strengthening weak communication pathways while controlling graph densification under a fixed edge budget. The procedure is parameter-free beyond the rewiring budget and relies on a single global measure aggregating all paths between node pairs. Beyond predictive performance with GCN models, we analyze how rewiring affects message propagation. By tracking cosine similarity between node embeddings across layers, we examine how the relationship between initial node features and learned representations evolves during message passing, comparing graphs with and without rewiring. This analysis helps determine whether improvements arise from better long-range communication rather than changes in embedding geometry. Experiments on homophilic and heterophilic graphs, including directed settings with DirGCN, reveal a trade-off between over-squashing and oversmoothing, where oversmoothing corresponds to the loss of representation diversity across layers. Resistance-guided rewiring improves connectivity and signal propagation but can accelerate representation mixing in deep models. Combining ERR with normalization techniques such as PairNorm stabilizes this trade-off and improves performance.
Figures & tables
Figure 1: GCN test accuracy versus depth (number of layers) under different rewiring strategies.
Figure 2: Linear-probe accuracy on penultimate-layer embeddings of a GCN as depth increases, comparing rewiring strategies at budget 0.1 . Results are shown for Cora (top) and Cornell (bottom), without PairNorm in the left columns and with PairNorm in the right ones; each colored curve corresponds to a different readout (output) layer. PairNorm consistently stabilizes the depth trend of probe accuracy, while resistance-driven rewiring alters the depth at which informative embeddings form and reduces the degradation observed in the unnormalized setting, highlighting the interaction between normalization (oversmoothing control) and topology edits (bottleneck alleviation).
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Dataset
Hidden dim d
Dropout
lr
Weight decay
Cora
16
0.5
0.01
5×10−3
CiteSeer
16
0.5
0.01
5×10−3
Cornell
64
0.5
0.01
5×10−4
Texas
64
0.5
0.01
5×10−4
Appendix
Table 1: Hyperparameters adopted from prior work (no tuning): Kipf and Welling (2017) for Cora/CiteSeer and Pei et al. (2020) for Cornell/Texas.
Dataset
N
E0
Budget
Curvature
Resistance (Add & Remove)
Resistance per Hop (Add & Remove)
Resistance
Resistance per Hop
cora
2708
10556
1%
105 / 105 (1.0% / 1.0%)
105 / 105 (1.0% / 1.0%)
105 / 105 (1.0% / 1.0%)
105 / – (1.0% / –)
105 / – (1.0% / –)
5%
517 / 517 (4.9% / 4.9%)
527 / 527 (5.0% / 5.0%)
526 / 526 (5.0% / 5.0%)
527 / – (5.0% / –)
526 / – (5.0% / –)
10%
1007 / 1007 (9.5% / 9.5%)
1055 / 1055 (10.0% / 10.0%)
1007 / 1007 (9.5% / 9.5%)
1055 / – (10.0% / –)
1007 / – (9.5% / –)
15%
1489 / 1489 (14.1% / 14.1%)
1583 / 1583 (15.0% / 15.0%)
1440 / 1440 (13.6% / 13.6%)
1583 / – (15.0% / –)
1440 / – (13.6% / –)
citeseer
3327
9104
1%
91 / 91 (1.0% / 1.0%)
91 / 91 (1.0% / 1.0%)
91 / 91 (1.0% / 1.0%)
91 / – (1.0% / –)
91 / – (1.0% / –)
5%
451 / 451 (5.0% / 5.0%)
455 / 455 (5.0% / 5.0%)
455 / 455 (5.0% / 5.0%)
455 / – (5.0% / –)
455 / – (5.0% / –)
Appendix
Table 2: Directed graphs: edge modifications by budget. Each cell reports added / removed edges, and on the next line the corresponding percentages relative to the initial number of edges E0 . For res and res_hop , we report added / – .
Dataset
N
E0
Budget
Curvature
Resistance (Add & Remove)
Resistance per Hop (Add & Remove)
Resistance
Resistance per Hop
cora
2708
5278
1%
52 / 52 (1.0% / 1.0%)
52 / 52 (1.0% / 1.0%)
52 / 52 (1.0% / 1.0%)
52 / – (1.0% / –)
52 / – (1.0% / –)
5%
263 / 263 (5.0% / 5.0%)
263 / 263 (5.0% / 5.0%)
263 / 263 (5.0% / 5.0%)
263 / – (5.0% / –)
263 / – (5.0% / –)
10%
527 / 527 (10.0% / 10.0%)
519 / 519 (9.8% / 9.8%)
522 / 522 (9.9% / 9.9%)
527 / – (10.0% / –)
527 / – (10.0% / –)
15%
789 / 789 (14.9% / 14.9%)
745 / 745 (14.1% / 14.1%)
730 / 730 (13.8% / 13.8%)
791 / – (15.0% / –)
791 / – (15.0% / –)
citeseer
3327
4552
1%
45 / 45 (1.0% / 1.0%)
45 / 45 (1.0% / 1.0%)
45 / 45 (1.0% / 1.0%)
45 / – (1.0% / –)
45 / – (1.0% / –)
5%
227 / 227 (5.0% / 5.0%)
227 / 227 (5.0% / 5.0%)
227 / 227 (5.0% / 5.0%)
227 / – (5.0% / –)
227 / – (5.0% / –)
Appendix
Table 3: Undirected graphs: edge modifications by budget. Each cell reports added / removed edges, and on the next line the corresponding percentages relative to the initial number of (undirected) edges E0 . For res and res_hop , removals are not performed and are shown as “–”.
Figure 3: GCN accuracy versus depth (number of layers) under different rewiring strategies.
Figure 4: CKA similarity between last-layer GCN representations with curvature rewiring (curvature) and proposed rewiring strategies with 0.1 as budget.
Figure 5: CKA similarity between last-layer GCN representations with curvature rewiring (curvature) and proposed rewiring strategies with 0.1 as budget.
Figure 6: UpSet plots of edges added by different rewiring strategies at budget r=0.1 . Bars report intersection sizes of the added-edge sets, bar color indicates Jaccard overlap (intersection over union) for each combination.
Figure 7: Mean cosine similarity between node embeddings as a function of the inner layer index (outer layer fixed to 7), comparing node pairs from the same class versus different classes, under different rewiring strategies at budget r=0.1 . Each grouped panel corresponds to one graph construction: Baseline, Curvature, Resistance, and Resistance per hop. Within each group, results are shown for Cora and Cornell, with and without PairNorm.
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.
Hugo Attali, Rachid El Jouhri
Université Sorbonne Paris Nord, CNRS, LIPN, France
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 are powerful models for learning from graph-structured data, yet their effectiveness is often limited by two critical challenges: over-squashing, where information from distant nodes is excessively compressed, and over-smoothing, where repeated propagation makes node representations indistinguishable. Both phenomena stem from the interaction between message passing and the input topology, ultimately degrading information flow and limiting the performance of GNNs. In this survey, we examine graph rewiring techniques, a class of methods designed to modify the graph topology to enhance information propagation in GNNs. We provide a comprehensive review of state-of-the-art rewiring approaches, delving into their theoretical underpinnings, practical implementations, and performance trade-offs.
Hugo Attali, Nathalie Pernelle, Davide Buscaldi +1