Oversquashing in Graph Neural Networks

• Oversquashing is a failure mode in graph neural networks where exponentially many signals are compressed into fixed-size node representations, impairing long-range dependencies.
• It is analyzed through formal lenses like information contraction, Jacobian sensitivity, and effective resistance, which quantify how bottlenecks limit signal propagation.
• Mitigation strategies—including graph rewiring, architectural modifications, and selective propagation—aim to adjust topology and dynamics to preserve vital information.

Oversquashing is a failure mode of message-passing graph neural networks in which information from many nodes, often but not only at long range, is compressed into fixed-size intermediate representations as it traverses a graph. In the standard formulation, a node representation after $L$ layers depends on its $L$-hop computation tree, so exponentially many signals may be forced through a sparse set of graph pathways and summarized in a constant-width vector; the resulting loss of influence is the canonical oversquashing picture (Banerjee et al., 2022). Subsequent work has broadened this picture: oversquashing has been tied to information contraction, graph expansion, effective resistance, commute time, conductance, and Jacobian sensitivity; it has also been shown to admit a short-range bottleneck form that is not captured by long-range gradient-decay explanations alone (Black et al., 2023, Southern et al., 2024, Chehreghani, 27 Mar 2026, Mishayev et al., 25 Nov 2025).

1. Conceptual definition and scope

At the level of a standard message-passing GNN, node representations evolve as

$$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$

so $h_v^{(L)}$ can only depend on nodes within $L$ hops (Banerjee et al., 2022). In this setting, oversquashing is the failure mode in which exponentially many long-range signals are squashed into fixed-size node vectors, causing the model to lose information needed for long-range tasks (Banerjee et al., 2022).

This description is distinct from, but often entangled with, two other limitations. Underreaching refers to the fact that a depth-$K$ message-passing GNN cannot propagate information beyond $K$ hops; oversmoothing refers to the convergence of node embeddings toward increasingly similar representations under repeated propagation (Errica et al., 2023, Chehreghani, 27 Mar 2026). Much of the oversquashing literature studies the regime in which depth is sufficient to reach distant nodes but insufficient to preserve their influence.

A central refinement is that oversquashing is not limited to long-range tasks. “Short-Range Oversquashing” constructs tasks with radius $2$ in which any exact MPNN solution requires hidden dimension growing with graph size, and separates two mechanisms: a bottleneck phenomenon, which can arise even in low-range settings, and a vanishing gradient phenomenon, which is closely associated with long-range tasks (Mishayev et al., 25 Nov 2025). This suggests that “long-range dependency” is not a complete synonym for oversquashing.

2. Formal lenses and quantitative proxies

One influential lens is information contraction. For a Markov chain $U \rightarrow X \rightarrow Y$, the data processing inequality gives

$I(U;Y) \le I(U;X),$

and the strong data processing inequality quantifies strict loss through contraction coefficients (Banerjee et al., 2022). In this view, a deep message-passing computation graph resembles a noisy computation graph in which repeated local transformations quench distant signals.

A second lens is Jacobian sensitivity. A common operational quantity is

$L$0

or its normalized variant, which measures how much source node $L$1 can influence target node $L$2 after $L$3 layers (Black et al., 2023, Saber et al., 12 Aug 2025). One topology-focused approximation defines

$L$4

and models pairwise oversquashing through an exponential decay rate $L$5 fitted from $L$6 across depth (Saber et al., 12 Aug 2025). This produces four graph-level statistics: prevalence, intensity, variability, and extremity (Saber et al., 12 Aug 2025).

A third lens is graph connectivity via effective resistance. For a connected graph with Laplacian $L$7,

$L$8

and high effective resistance indicates weak communication through the graph (Black et al., 2023). In the same line of work, total effective resistance $L$9 is proposed as a graph-level proxy for total oversquashing, with rewiring aimed at minimizing it (Black et al., 2023).

A related spectral-topological view uses commute time and conductance. In the virtual-node analysis, commute time is proportional to effective resistance,

$$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$0

and bounded-depth MPNNs require depth at least $$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$1 to realize graph functions with sufficiently strong mixing between nodes $$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$2 and $$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$3 (Southern et al., 2024). In a more cut-theoretic formulation, conductance

$$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$4

serves as a proxy for the worst bottleneck in the graph (Chehreghani, 27 Mar 2026).

3. Structural explanations and unsettled interpretations

The classical structural explanation ties oversquashing to sparse cuts, weak expansion, and bottlenecks. The Cheeger constant, spectral gap, and expander-graph intuition all encode the idea that too much information may have to cross too few edges (Banerjee et al., 2022, Karhadkar et al., 2022). In this framing, poor expansion creates bottlenecks; bottlenecks induce information contraction; repeated contraction causes signal quenching (Banerjee et al., 2022).

Curvature-based explanations refine this by localizing bottlenecks. Ollivier–Ricci, Forman, balanced Forman, and related curvatures are used as edge-local indicators of poor transport geometry, and several rewiring methods target highly negatively curved edges (Attali et al., 2024). Yet this interpretation is not universal. “Expander Graph Propagation” argues that propagation over sparse expander templates can alleviate bottlenecks while the propagation graph remains negatively curved everywhere; it further shows that for large sparse graphs, bounded degree, uniform expansion, and non-negative Ollivier curvature cannot all coexist (Deac et al., 2022). A plausible implication is that negative curvature is not, by itself, a sufficient diagnosis of harmful oversquashing.

Another unsettled point is whether sensitivity alone is an adequate metric. “Adaptive Message Passing” argues that rewiring may increase

$$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$5

while also sending more information into the same bottleneck, so sensitivity can improve even when the compression burden worsens (Errica et al., 2023). “Beyond Oversquashing” pushes this further by reframing the problem as one of signal routing rather than all-pairs communication: a model should move signal from task-relevant sources to task-relevant destinations while keeping it concentrated, rather than merely increasing generic diffusion (Nagar et al., 13 May 2026).

4. Rewiring and structural transformation

Graph rewiring is the dominant mitigation family. A central idea is to alter topology so that long-range information need not cross narrow cuts. “Oversquashing in GNNs through the lens of information contraction and graph expansion” proposes Random Local Edge Flip and its greedy variant G-RLEF, a local degree-preserving rewiring scheme inspired by expander constructions; it preserves node degrees exactly, never disconnects the graph, and improves expansion while targeting bottleneck-like edges with few triangles (Banerjee et al., 2022).

Spectral rewiring methods pursue the same goal through global expansion proxies. FoSR adds the edge predicted to most improve the graph’s spectral gap using a first-order perturbation approximation, and combines this with a relational architecture that distinguishes original edges from added edges to control oversmoothing (Karhadkar et al., 2022). Effective-resistance rewiring instead selects edges that maximally decrease total effective resistance, thereby lowering a global oversquashing proxy (Black et al., 2023). Conductance-based optimal rewiring, however, is computationally intractable in the worst case: exact optimization of conductance under bounded edge edits is NP-hard, and the decision version is NP-complete under the paper’s formulation (Chehreghani, 27 Mar 2026).

Other structural transformations depart from ordinary graph rewiring. One line lifts graphs to hypergraphs using Forman–Ricci curvature and turns curvature-identified backbones or bridge structures into hyperedges, thereby “virtually shortening the distance between potentially relevant nodes” (Banf et al., 15 Aug 2025). Another uses fixed sparse expander templates as propagation graphs rather than graph-specific rewiring (Deac et al., 2022). Both are motivated by the same anti-bottleneck principle, but they modify the communication domain more radically than edge addition alone.

5. Architectural remedies beyond explicit rewiring

Several approaches attempt to mitigate oversquashing without changing the original graph globally. Virtual nodes add a hub connected to every node, so the augmented graph $$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$6 changes commute times and effective resistances in a topology-dependent way rather than uniformly helping every graph (Southern et al., 2024). The same work shows that classical virtual nodes often induce uniform sensitivity to distant nodes, whereas a reordered variant, $$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$7, can produce graph-structured heterogeneous sensitivity at the same computational complexity (Southern et al., 2024).

A different line modifies the propagation operator itself. “Graph Unitary Message Passing” proposes applying a unitary adjacency matrix for message passing; the abstract explicitly frames this as an alternative to rewiring techniques that improve graph spectrums but disrupt structural bias and have limited improvement on oversquashing in terms of oversquashing measure (Qiu et al., 2024). “SWAN” uses antisymmetry in both feature and spatial operators to obtain local and global non-dissipativity, with the stated consequence that the information propagation rate among graph nodes is constant, independent of time $$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$8 (Gravina et al., 2024).

Selective or asynchronous propagation is another remedy. “Adaptive Message Passing” learns both the effective depth and a soft layer-dependent filtering tensor $$h_{v}^{(l)}=f_{l}\bigl(h_{v}^{(l-1)},\{h_{u}^{(l-1)} : u \in N_{G}(v)\}\bigr),$$9, so the model can suppress irrelevant outgoing messages rather than broadcasting everything synchronously (Errica et al., 2023). “Asynchronous Message Passing” updates only centrality-ordered node batches at each layer; the framework is explicitly motivated by the claim that synchronous message passing compresses too much information at once through fixed-capacity channels (Bose et al., 8 Sep 2025).

Hierarchical and higher-order methods address oversquashing by changing the unit of communication. “Cross-attentive Cohesive Subgraph Embedding” builds $h_v^{(L)}$0-core-derived cohesive subgraphs, pools them, lets them communicate via cross-attention, and fuses the resulting subgraph representations back into node embeddings; this creates representation-level alternative pathways rather than explicit new edges (Hossain et al., 29 Mar 2026). In topological deep learning, oversquashing has been generalized to simplicial and cellular message passing by representing such models as relational structures and analyzing them through an induced influence graph (Taha et al., 6 Jun 2025).

Redundancy control offers yet another angle. “On the Two Sides of Redundancy in Graph Neural Networks” argues that repetitive exchange and re-encoding of identical information amplifies oversquashing; it replaces unfolding trees with pruned neighborhood trees and computes embeddings on a merged DAG, yielding the ordering

$h_v^{(L)}$1

which the paper interprets as progressively less oversquashing under stronger pruning (Bause et al., 2023).

6. Empirical patterns, diagnostics, and open questions

Across the literature, rewiring and alternative architectures frequently improve long-range synthetic tasks and several real benchmarks, but the relationship between oversquashing reduction and downstream accuracy is not monotone. The causal analysis of rewiring strategies reports that most graph-classification datasets suffer from over-squashing, that rewiring often mitigates it, and that the degree of mitigation varies by dataset and method; it also reports that over-squashing is less notable in node-classification datasets, where rewiring often increases over-squashing and performance variations are uncorrelated with over-squashing changes (Saber et al., 12 Aug 2025). This suggests that rewiring is most useful when over-squashing is substantial and corrected with restraint.

A recurrent empirical theme is that aggressive global densification and restrained topological surgery behave differently. DIGL can reduce topology-based oversquashing metrics strongly while failing to translate that reduction into better accuracy, whereas more restrained methods such as FoSR or BORF can align mitigation more closely with performance (Saber et al., 12 Aug 2025). A plausible implication is that anti-oversquashing interventions must be evaluated jointly with their effects on oversmoothing, locality, and inductive bias.

The field also lacks a single accepted definition. Some papers privilege sensitivity decay, others effective resistance or commute time, others conductance or curvature, and still others selective routing or dynamical non-dissipativity (Black et al., 2023, Gravina et al., 2024, Nagar et al., 13 May 2026). The short-range results further show that existing long-range diagnostics do not capture every bottleneck mechanism (Mishayev et al., 25 Nov 2025). An objective synthesis is therefore that oversquashing is best understood not as one invariant scalar, but as a family of related failures in which graph structure, propagation dynamics, representational width, and aggregation geometry jointly determine whether distant information remains influential.