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Measured serving runs, fine-tuning results and research evidence from a one-founder AI company. • 5 items • Updated
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R1-001 | MPNN | MPNN framework unifies prior graph networks as message function + aggregation + update over T steps | DESIGN | https://arxiv.org/abs/1704.01212 | Abstract | we reformulate existing models into a single common framework we call Message Passing Neural Networks (MPNNs) and explore additional novel variations within this framework. | 2017-04-04 | 2026-10-01 | High | |||
R1-002 | THEORY | A GNN with injective aggregation/readout is as powerful as the 1-WL test (upper bound for MPNNs) | THEOREM | https://arxiv.org/abs/1810.00826 | Section 4, Theorem 3 | if the neighbor aggregation and graph-level readout functions are injective, then the resulting GNN is as powerful as the WL test. | 2018-10-01 | 2026-10-01 | High | Expressivity ceiling = 1-WL for anonymous MPNNs | ||
R1-003 | THEORY | 1-WL coloring always refines any 1-GNN coloring: MPNNs cannot distinguish more than 1-WL | THEOREM | https://arxiv.org/abs/1810.02244 | Section 3, Theorem 1 | We show that GNNs have the same expressiveness as the 1-WL in terms of distinguishing non-isomorphic (sub-)graphs. Hence, both algorithms also have the same shortcomings. | 2018-10-04 | 2026-10-01 | High | Applies regardless of number of iterations: recurrence alone does not lift the 1-WL ceiling | ||
R1-004 | THEORY | Many graph problems impossible for message passing GNNs unless depth x width exceeds a polynomial in graph size | THEOREM | https://arxiv.org/abs/1907.03199 | Abstract; Section 4, Theorem 4.1 | several of these problems are deemed impossible unless the product of a GNNmp’s depth and width exceeds a polynomial of the graph size; this dependence remains significant even for tasks that appear simple or when considering approximation. | 2019-07-06 | 2026-10-01 | High | Directly relevant: iterations (depth d) trade against width w; capacity dw lower bound (CONGEST reduction) | ||
R1-005 | THEORY | CONGEST round lower bounds transfer to GNN depth/width lower bounds | THEOREM | https://arxiv.org/abs/1907.03199 | Section 4, Theorem 4.1 | If a problem P cannot be solved in less than d rounds in CONGEST using messages of at most b bits, then P cannot be solved by a GNNnmp of width w ≤ (b − log2 n)/p = O(b/ log | 2019-07-06 | 2026-10-01 | High | Iteration count T needed by a recurrent graph substrate is bounded below by distributed-computing round complexity | ||
R1-006 | THEORY | GCN output exponentially approaches a subspace carrying only connected-component and degree information as depth grows (oversmoothing) | THEOREM | https://arxiv.org/abs/1905.10947 | Section 4, Theorem 1; Abstract | its output exponentially approaches the set of signals that carry information of the connected components and node degrees only for distinguishing nodes. | 2019-05-27 | 2026-10-01 | High | Condition: weight singular values small relative to Laplacian spectrum | ||
R1-007 | THEORY | Dirichlet energy of node embeddings converges to zero with depth in general GNNs under spectral weight conditions | THEOREM | https://arxiv.org/abs/2006.13318 | Abstract | the Dirichlet energy of embeddings will converge to zero, resulting in the loss of discriminative power. | 2020-06-23 | 2026-10-01 | High | |||
R1-008 | THEORY | Graph attention (GAT, graph transformers) cannot prevent oversmoothing; loses expressive power exponentially | THEOREM | https://arxiv.org/abs/2305.16102 | Abstract; Section 4.5 main theorems | We establish that, contrary to popular belief, the graph attention mechanism cannot prevent oversmoothing and loses expressive power exponentially. | 2023-05-25 | 2026-10-01 | High | Covers state-dependent, time-varying aggregation: state-conditioned routing alone does not escape oversmoothing | ||
R1-009 | THEORY | The oversmoothing result covers asymmetric, state-dependent, time-varying aggregation operators incl. transformers | THEOREM | https://arxiv.org/abs/2305.16102 | Abstract | our analysis accounts for asymmetric, state-dependent and time-varying aggregation operators and a wide range of common nonlinear activation functions, such as ReLU, LeakyReLU, GELU and SiLU. | 2023-05-25 | 2026-10-01 | High | Directly applies to a candidate with state-conditioned dynamic edges (C8) | ||
R1-010 | THEORY | Over-squashing: exponentially growing receptive field compressed into fixed-size vectors; GNNs fail on long-range interactions | THEOREM | https://arxiv.org/abs/2006.05205 | Abstract | This bottleneck causes the over-squashing of exponentially growing information into fixed-size vectors. As a result, GNNs fail to propagate messages originating from distant nodes | 2020-06-09 | 2026-10-01 | High | Status THEOREM is loose: argument is analytical + empirical | ||
R1-011 | MPNN | Replacing the last layer with a fully-adjacent layer reduces error by 42% on QM9 for tuned GNNs | 42 | % error reduction | REPORTED | https://arxiv.org/abs/2006.05205 | Section 1 / Section 4, Table 1 | breaking the bottleneck using a simple fully adjacent layer reduces the error rate by 42% in the QM9 dataset, by 12% in ENZYMES, by 4.8% in NCI1 | 2020-06-09 | 2026-10-01 | High | Baseline: same tuned GNN without FA layer; param-matched approx (same model, one layer adjacency changed); FLOPs not matched |
R1-012 | THEORY | Negatively curved edges cause over-squashing bottlenecks (Balanced Forman curvature) | THEOREM | https://arxiv.org/abs/2111.14522 | Section 4, Theorem 4; Abstract | we introduce a new edge-based combinatorial curvature and prove that negatively curved edges are responsible for the over-squashing issue. | 2021-11-29 | 2026-10-01 | High | |||
R1-013 | THEORY | Depth cannot mitigate over-squashing: more layers lead to vanishing gradients dominating | THEOREM | https://arxiv.org/abs/2302.02941 | Abstract; Theorems 4.1-4.2 | depth cannot help mitigate over-squashing: increasing the number of layers leads to over-squashing being dominated by vanishing gradients | 2023-02-06 | 2026-10-01 | High | Key negative for time-replaces-depth: iterating longer on a fixed graph does not solve over-squashing | ||
R1-014 | THEORY | Width mitigates over-squashing only at the cost of sensitivity; topology (commute time) plays the greatest role | THEOREM | https://arxiv.org/abs/2302.02941 | Abstract | Neural network width can mitigate over-squashing, but at the cost of making the whole network more sensitive | 2023-02-06 | 2026-10-01 | High | |||
R1-015 | THEORY | Topology dominates over-squashing: it occurs between nodes at high commute time | THEOREM | https://arxiv.org/abs/2302.02941 | Abstract | The graph topology plays the greatest role, since over-squashing occurs between nodes at high commute time. | 2023-02-06 | 2026-10-01 | High | Supports topology-as-computation: changing edges matters more than depth | ||
R1-016 | THEORY | Over-smoothing and over-squashing trade off via the spectral gap; cannot be alleviated simultaneously by rewiring | THEOREM | https://arxiv.org/abs/2212.02374 | Abstract; Cheeger-inequality section | over-smoothing and over-squashing are intrinsically related to the spectral gap of the graph Laplacian, resulting in an inevitable trade-off between these two issues, as they cannot be alleviated simultaneously. | 2022-12-05 | 2026-10-01 | Medium | Constraint on learned edge creation: adding edges speeds mixing and accelerates oversmoothing | ||
R1-017 | THEORY | Mitigating over-smoothing is necessary but not sufficient for expressive deep GNNs | AUTHOR-CLAIM | https://arxiv.org/abs/2303.10993 | Abstract | mitigating over-smoothing is a necessary but not sufficient condition for building deep GNNs that are expressive on a wide range of graph learning tasks. | 2023-03-20 | 2026-10-01 | Medium | Survey with experiments | ||
R1-018 | RGNN | GG-NN: weight-shared GRU node update unrolled for a fixed number of steps T, trained by BPTT | DESIGN | https://arxiv.org/abs/1511.05493 | Section 3 | we use Gated Recurrent Units (Cho et al., 2014) and unroll the recurrence for a fixed number of steps T and use backpropagation through time in order to compute gradients. | 2015-11-17 | 2026-10-01 | High | C1,C3,C4,C5 yes; fixed T, no halting | ||
R1-019 | RGNN | Original GNN (Scarselli) fixed-point recurrence requires contraction maps, which may limit expressivity and long-range propagation | AUTHOR-CLAIM | https://arxiv.org/abs/1511.05493 | Section 2 (GNN background) | parameters must be constrained so that the propagation step is a contraction map. This is needed to ensure convergence, but it may limit the expressivity of the model. | 2015-11-17 | 2026-10-01 | High | Stability-vs-expressivity tension for converge-until-fixed-point substrates | ||
R1-020 | RGNN | IGNN: predictions from a fixed-point equilibrium state, well-posedness via Perron-Frobenius, trained by implicit differentiation | DESIGN | https://arxiv.org/abs/2009.06211 | Abstract | predictions are based on the solution of a fixed-point equilibrium equation involving implicitly defined “state” vectors. We use the Perron-Frobenius theory to derive sufficient conditions that ensure well-posedness of the framework. | 2020-09-14 | 2026-10-01 | High | Weight-tied, infinite-depth; training by implicit differentiation (no BPTT memory) | ||
R1-021 | RGNN | EIGNN derives a closed-form infinite-depth solution, avoiding iterative solvers and IGNN convergence conditions | DESIGN | https://arxiv.org/abs/2202.10720 | Section 1, contributions | Instead of relying on iterative solvers to generate the solution like in IGNN [10], we derive a closed-form solution for EIGNN without additional conditions | 2022-02-22 | 2026-10-01 | High | Linear propagation; infinite depth equals one closed-form solve | ||
R1-022 | RGNN | IterGNN learns to terminate message passing adaptively via a readout-based confidence score, trained without stopping supervision | DESIGN | https://arxiv.org/abs/2010.13547 | Section 4.1 | With the help of our iterative module, IterGNN can adaptively adjust the number of iterations. Moreover, it can be trained without any supervision of the stopping condition. | 2020-10-26 | 2026-10-01 | High | Learned graph-level halting (C11/C12 partial: global, not per-node) | ||
R1-023 | RGNN | IterGNN reaches 100% shortest-path success on graphs up to 5000 nodes after training on sizes 4-33 | 5000 | nodes | REPORTED | https://arxiv.org/abs/2010.13547 | Section 5.1, Table 2 | our model achieves a 100% success rate of identifying the shortest paths on graphs with as large as 5000 nodes even though it is trained on graphs of sizes within [4, 34). | 2020-10-26 | 2026-10-01 | High | Compared against fixed-depth GCN/GAT/Path variants; not param- or FLOP-matched (iterative model uses many more iterations) |
R1-024 | RGNN | Iterative module improves performance from ~60% to 100% on graphs of size >=500 | 100 | % success | REPORTED | https://arxiv.org/abs/2010.13547 | Section 5.1 | The iterative module successfully improves the performance from ∼ 60% to 100% on graphs of sizes ≥ 500. | 2020-10-26 | 2026-10-01 | High | Ablation vs same model without iterative module; parameter count same; FLOPs not matched |
R1-025 | RGNN | A 100-layer non-iterative Path model generalizes worse (relative loss ~0.13) on weighted shortest path | 0.13 | relative loss | REPORTED | https://arxiv.org/abs/2010.13547 | Section 5.1 | We also evaluated a deeper Path model, i.e., with 100 layers, on the weighted shortest path problem (Lob). The generalization performance (relative loss≈ 0.13) became even worse. | 2020-10-26 | 2026-10-01 | Medium | Evidence that depth without adaptive iteration does not substitute; unmatched |
R1-026 | RGNN | IterGNN contrasts its unbounded iterations with ACT, whose iterations are small by design | AUTHOR-CLAIM | https://arxiv.org/abs/2010.13547 | Section 2 related work | The numbers of iterations for ACT are usually small by design (e.g.the formulation of regularizations and halting distributions). | 2020-10-26 | 2026-10-01 | Medium | |||
R1-027 | RGNN | Recurrent GNNs (RecGRU-E) trained on size-10 graphs extrapolate to graphs 1,000 times larger | 1000 | x training size | REPORTED | https://arxiv.org/abs/2212.04934 | Experimental Evaluation, Table 1 caption | The RecGRU-E outperforms IterGNN and can extrapolate well to graphs that are 1,000 times larger than the graphs encountered during training. | 2022-12-09 | 2026-10-01 | High | Baselines: non-recurrent 10-layer GIN, IterGNN; not param- or FLOP-matched; tasks: path finding, prefix sum, distance |
R1-028 | RGNN | Recurrence is necessary but not sufficient for extrapolation; needs skip connections, state regularization, edge convolutions | AUTHOR-CLAIM | https://arxiv.org/abs/2212.04934 | Introduction | While a recurrent architecture seems to be necessary to achieve extrapolation, we show that it is not sufficient. | 2022-12-09 | 2026-10-01 | High | Iterating longer requires explicit stabilizers | ||
R1-029 | RGNN | Without L2 state regularization, accuracy declines rapidly after ~100 rounds; with it, stable up to 10,000 rounds (graphs of size 10) | 100 | rounds | REPORTED | https://arxiv.org/abs/2212.04934 | Section on stabilization, Figure 4 | both versions output correct predictions until approximately 100 layers. Afterwards, the models trained with L2 regularization still stay stable while accuracy declines rapidly without regularization. | 2022-12-09 | 2026-10-01 | High | Direct evidence on iterating beyond training horizon: collapses without a stabilizer; trained with 12 recurrent instances |
R1-030 | RGNN | Training setup: 12 recurrent instances on size-10 graphs; extrapolation test at size 100 with 120 rounds | 12 | training iterations | REPORTED | https://arxiv.org/abs/2212.04934 | Experimental Evaluation | All models were trained for 100 epochs on graphs of size 10 using 12 instances of the recurrent layer. To evaluate extrapolation, we use graphs of size 100 and execute 120 rounds. | 2022-12-09 | 2026-10-01 | High | Test iterations = 10x training iterations |
R1-031 | RGNN | Non-recurrent GIN cannot extrapolate (fixed 10 rounds) | REPORTED | https://arxiv.org/abs/2212.04934 | Table 1 caption | All models were only trained on graphs of size 10. The non-recurrent GIN can not extrapolate. | 2022-12-09 | 2026-10-01 | High | Fixed-depth baseline 10 separate GIN layers; parameters NOT matched (10 separate layers vs one shared) | ||
R1-032 | RGNN | GRAND maintains accuracy as layers increase while GCN baselines degrade by 50% after 4 layers | 50 | % degradation | REPORTED | https://arxiv.org/abs/2106.10934 | Section 6.2, Figure 2 | our model maintains performance as the layers increase whilst the baselines degrade by 50% after 4 layers. | 2021-06-21 | 2026-10-01 | Medium | Baselines GCN and residual GCN; not param- or FLOP-matched; branch ODE kept because graph-first continuous diffusion (assigned to R1 as graph neural diffusion) |
R1-033 | RGNN | GRAND treats GNNs as discretisations of a continuous diffusion PDE; layers = time steps | DESIGN | https://arxiv.org/abs/2106.10934 | Abstract | approaches deep learning on graphs as a continuous diffusion process and treats Graph Neural Networks (GNNs) as discretisations of an underlying PDE. In our model, the layer structure and topology correspond to the discretisation choices of temporal and spatial operators. | 2021-06-21 | 2026-10-01 | High | Time replaces depth by construction, weight-tied across time | ||
R1-034 | X | Recurrent and implicit maze extrapolators fail along several test axes; non-converging models show limit cycles | REPORTED | https://arxiv.org/abs/2410.03020 | Abstract | models explicitly trained to converge to a fixed point are likely to do so when extrapolating, while models that are not may exhibit more exotic limiting behavior such as limit cycles, even when they correctly solve the problem. | 2024-10-03 | 2026-10-01 | High | Non-graph (CNN-RNN) logical extrapolators; cross-family negative evidence; primarily R2 territory | ||
R1-035 | X | More diverse training data does not improve logical extrapolation of recurrent/implicit maze solvers | REPORTED | https://arxiv.org/abs/2410.03020 | Abstract | We show that training these models on more diverse data addresses some failure modes but, paradoxically, does not improve logical extrapolation. | 2024-10-03 | 2026-10-01 | Medium | Negative result for iteration-based extrapolation | ||
R1-036 | RGNN | GraphCON: coupled-oscillator ODE GNN; zero-Dirichlet-energy steady states are unstable, mitigating oversmoothing | THEOREM | https://arxiv.org/abs/2202.02296 | Abstract | We relate the oversmoothing problem, commonly encountered in GNNs, to the stability of steady states of the underlying ODE and show that zero-Dirichlet energy steady states are not stable for our proposed ODEs. | 2022-02-04 | 2026-10-01 | High | |||
R1-037 | RGNN | GraphCON-GCN accuracy on MNIST superpixel increases up to 32 layers while GCN worsens with depth | 32 | layers | REPORTED | https://arxiv.org/abs/2202.02296 | Section 5, Table 7 | the test accuracy for GraphCON-GCN increases as more layers (up to 32 layers) are added to the model. | 2022-02-04 | 2026-10-01 | Medium | Baseline GCN same depth; param-matched approx per layer; not FLOP-audited |
R1-038 | RGNN | A-DGN with constant-in-time ODE parameters is a recursive DGN with weight sharing between layers | DESIGN | https://arxiv.org/abs/2210.09789 | Section 2, after Eq. 7 | having assumed the parameters of the ODE constant in time, A-DGN can also be interpreted as a recursive DGN with weight sharing between layers. | 2022-10-18 | 2026-10-01 | High | Weight-tied recurrent graph substrate with antisymmetric (non-dissipative) weights | ||
R1-039 | RGNN | A-DGN proven stable and non-dissipative: long-range information preserved, no gradient vanishing/explosion | THEOREM | https://arxiv.org/abs/2210.09789 | Abstract | We give theoretical proof that our method is stable and non-dissipative, leading to two key results: long-range information between nodes is preserved, and no gradient vanishing or explosion occurs in training. | 2022-10-18 | 2026-10-01 | High | Stability fix candidate for a recurrent substrate: antisymmetric weight parametrization | ||
R1-040 | RGNN | G2 keeps Dirichlet energy near constant up to 1000 layers | 1000 | layers | REPORTED | https://arxiv.org/abs/2210.00513 | Section 4, Figure 2 | adding G2 decisively prevents this behavior and the Dirichlet energy remains (near) constant, even for very deep architectures (up to 1000 layers). | 2022-10-02 | 2026-10-01 | High | Compared against GCN/GAT at same depth; energy metric, not accuracy |
R1-041 | RGNN | Deep GCN degrades after 8 layers; DropEdge/GraphCON degrade after 16/64; G2-GCN peaks at 128 layers | 128 | layers | REPORTED | https://arxiv.org/abs/2210.00513 | Section 4 (Cornell depth experiment) | G2 -GCN exhibits a small but noticeable increase in performance for increasing number of layers, reaching its peak performance for 128 layers. | 2022-10-02 | 2026-10-01 | Medium | Small heterophilic dataset; gains are small; unmatched FLOPs |
R1-042 | THEORY | Traditional GNNs are prone to extreme gradient vanishing even after a few layers; feature collapse linked to vanishing gradients | THEOREM | https://arxiv.org/abs/2502.10818 | Abstract | Traditional GNNs are by design prone to extreme gradient vanishing even after a few layers; (ii) Feature collapse is directly related to the mechanism causing vanishing gradients | 2025-02-15 | 2026-10-01 | High | NeurIPS 2025 per search result (venue not independently verified) | ||
R1-043 | THEORY | Long-range modeling is most easily achieved by combining graph rewiring with vanishing-gradient mitigation | AUTHOR-CLAIM | https://arxiv.org/abs/2502.10818 | Abstract | Long-range modeling is most easily achieved by a combination of graph rewiring and vanishing gradient mitigation. | 2025-02-15 | 2026-10-01 | Medium | Supports C8 (topology) + recurrent-stability combination | ||
R1-044 | THEORY | Shared contractive GNN layers converge to a unique fixed point under repeated application | THEOREM | https://arxiv.org/abs/2502.10818 | Section 4, discussion of Lemma 4.1 | With shared layers (as in [31]), repeated applications of the GNN will result in convergence to a unique fixed point | 2025-02-15 | 2026-10-01 | High | A contractive weight-tied substrate converges to an input-dependent but unique fixed point: iterations beyond convergence add nothing | ||
R1-045 | RGNN | State-space GNN reformulation alleviates collapse at no extra trainable parameter cost | AUTHOR-CLAIM | https://arxiv.org/abs/2502.10818 | Abstract | a simple state-space formulation of a GNN effectively alleviates these issues at no extra trainable parameter cost. | 2025-02-15 | 2026-10-01 | Medium | |||
R1-046 | RGNN | gLSTM reframes over-squashing as a storage-capacity limit and adds associative-memory (xLSTM-style) node memory | DESIGN | https://arxiv.org/abs/2510.08450 | Section 1, Contributions | we introduce an MPNN architecture that utilizes associative memory (Beck et al., 2024; Schlag et al., 2021; Hopfield, 1982), and demonstrate that this exhibits improved storage capacity. | 2025-10-09 | 2026-10-01 | High | Closest graph-first instance of C9 persistent associative memory inside node state | ||
R1-047 | THEORY | Over-squashing split into capacity over-squashing and sensitivity over-squashing | AUTHOR-CLAIM | https://arxiv.org/abs/2510.08450 | Section 1, Contributions | we re-characterize oversquashing into two distinct failure modes: saturating capacity and low sensitivity, which we term capacity over-squashing and sensitivity over-squashing respectively. | 2025-10-09 | 2026-10-01 | Medium | Relevant to separating computational capacity vs storage per parameter | ||
R1-048 | RGNN | GHR (hierarchical recurrence) outperforms graph models on long-range benchmarks with as little as 1% of SOTA parameters | 1 | % of parameters | REPORTED | https://arxiv.org/abs/2605.18387 | Abstract | GHR consistently outperforms existing graph models while using as little as 1% of the parameters of current state-of-the-art models. | 2026-05-18 | 2026-10-01 | Medium | Not param-matched by design (claims efficiency); FLOPs not stated in abstract; 2026 preprint |
R1-049 | RGNN | Only GHR (recurrent) achieves out-of-range distance generalization; 10-layer DeepGIN does not | 10 | hops trained | REPORTED | https://arxiv.org/abs/2605.18387 | Synthetic distance experiment section | We compare DeepGIN (10 layers) with our model GHR (TL = 4, TH = 3), both trained on distances of up to 10 hops. While both GIN and GHR accurately predict on in-range distances, only GHR achieves out-of-range generalization. | 2026-05-18 | 2026-10-01 | Medium | Unmatched params/FLOPs |
R1-050 | RGNN | Existing GNN/GT limitations worsen in out-of-range generalization (test distances longer than training) | AUTHOR-CLAIM | https://arxiv.org/abs/2605.18387 | Abstract | the limitations of existing models are even more pronounced in outof-range generalization, where test instances involve interactions over distances longer than those observed during training. | 2026-05-18 | 2026-10-01 | Medium | Quote contains pdftotext token outofrange (hyphen lost) | ||
R1-051 | RGNN | GRED: linear RNN over hop-sequences per target node; comparable to graph transformers at reduced cost | AUTHOR-CLAIM | https://arxiv.org/abs/2312.01538 | Abstract | the performance of our model is comparable to or better than that of state-of-the-art graph transformers on various benchmarks, with a significantly reduced computational cost. | 2023-12-03 | 2026-10-01 | Medium | |||
R1-052 | RGNN | MP-SSM integrates SSM computation into MPNNs with exact sensitivity analysis and lower bounds for vanishing gradients/over-squashing | AUTHOR-CLAIM | https://arxiv.org/abs/2505.18728 | Section 1 contributions | We provide lower bounds that directly link architectural design choices to the alleviation of vanishing gradients and over-squashing. | 2025-05-24 | 2026-10-01 | Medium | |||
R1-053 | RGNN | LGSM decouples information depth (linear propagation) from processing depth (nonlinear) | DESIGN | https://arxiv.org/abs/2605.12358 | Figure 1 caption | LGSM decouples information propagation (horizontal) from non-linear processing (vertical). Unlike MPNNs that advance both dimensions simultaneously, LGSM enables information to flow across the graph through linearized computation before applying non-linear transformations. | 2026-05-12 | 2026-10-01 | Medium | Relevant to time-vs-depth: propagation steps and nonlinear depth can be separated | ||
R1-054 | DYN | DGCNN/EdgeConv recomputes a kNN graph in feature space after every layer, so topology is a function of current node embeddings | DESIGN | https://arxiv.org/abs/1801.07829 | Section 3 (EdgeConv) | our graph is not fixed but rather is dynamically updated after each layer of the network. That is, the set of k-nearest neighbors of a point changes from layer to layer of the network and is computed from the sequence | 2018-01-24 | 2026-10-01 | High | State-conditioned top-k edges at inference (C6, C8 yes); not weight-shared across layers; no recurrence | ||
R1-055 | DYN | DGM notes that DGCNN kNN graph construction is not differentiable and built ad hoc without a dedicated loss | AUTHOR-CLAIM | https://arxiv.org/abs/2002.04999 | Section 2 related work | The graph, however, is constructed ad-hoc as a kNN graph on the activations after each layer, without a dedicated loss. As such, the graph is built dynamically but not learned | 2020-02-11 | 2026-10-01 | High | |||
R1-056 | DYN | DGM: learnable module predicting edge probabilities optimal for the downstream task, trained end-to-end | DESIGN | https://arxiv.org/abs/2002.04999 | Abstract | we introduce Differentiable Graph Module (DGM), a learnable function that predicts edge probabilities in the graph which are optimal for the downstream task. DGM can be combined with convolutional graph neural network layers and trained in an end-to-end fashion. | 2020-02-11 | 2026-10-01 | High | Learned latent graph from features (C6 partial) | ||
R1-057 | DYN | NRI: VAE whose latent code is a discrete interaction graph; dynamics decoded by GNNs; recovers ground-truth interactions unsupervised | DESIGN | https://arxiv.org/abs/1802.04687 | Abstract | Our model takes the form of a variational auto-encoder, in which the latent code represents the underlying interaction graph and the reconstruction is based on graph neural networks. | 2018-02-13 | 2026-10-01 | High | Edges inferred once per trajectory, not re-inferred per message-passing step (in base model) | ||
R1-058 | DYN | LDS jointly learns a discrete edge distribution and GCN parameters via bilevel optimization; graph sampled at train and prediction time | DESIGN | https://arxiv.org/abs/1903.11960 | Abstract; Section 1 | jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. | 2019-03-28 | 2026-10-01 | High | Transductive; edge distribution parameters are global hyperparameters, not state-conditioned | ||
R1-059 | DYN | Co-GNN: each node chooses per layer to listen/broadcast/both/isolate via an action network given its state and neighbors | DESIGN | https://arxiv.org/abs/2310.01267 | Section 4 | an action network π predicts, for each node v, a probability (ℓ) distribution pv ∈ R4 over the actions {S, L, B, I} that v can take, given its state and the state of its neighbors Nv | 2023-10-02 | 2026-10-01 | High | State-conditioned topology at inference (C8 yes); edge removal only, cannot create new edges; Gumbel-softmax straight-through | ||
R1-060 | DYN | Co-GNN decouples the input graph from the computational graph via dynamic per-node actions | AUTHOR-CLAIM | https://arxiv.org/abs/2310.01267 | Section 1 | The interplay between these actions and the ability to change them dynamically makes the overall approach richer and allows to decouple the input graph from the computational one | 2023-10-02 | 2026-10-01 | High | |||
R1-061 | DYN | Co-GNN shares the action network across layers, giving parameter counts comparable to baselines | AUTHOR-CLAIM | https://arxiv.org/abs/2310.01267 | Section 5.1 (Efficient) | C O -GNNs are also parameter-efficient: they share the same action network across layers and as a result a comparable number of parameters to their baseline models. | 2023-10-02 | 2026-10-01 | Medium | Quote contains pdftotext split of CO-GNNs | ||
R1-062 | DYN | Co-GNN optimisation becomes increasingly difficult once the number of layers gets too large | AUTHOR-CLAIM | https://arxiv.org/abs/2310.01267 | Section 5.1 (Mitigates over-smoothing); Appendix C.5 | We validate this empirically in Appendix C.5, but we also note that the optimisation becomes increasingly difficult once the number of layers gets too large. | 2023-10-02 | 2026-10-01 | High | Negative: dynamic topology + depth is hard to optimize | ||
R1-063 | DYN | Co-GNNs more expressive than 1-WL and better suited for long-range tasks (authors) | AUTHOR-CLAIM | https://arxiv.org/abs/2310.01267 | Section 1 contributions; Sections 5.2-5.3 | show that they are more expressive than 1-dimensional Weisfeiler-Leman algorithm (1WL) (Section 5.2), and better suited for long-range tasks due to their adaptive nature (Section 5.3). | 2023-10-02 | 2026-10-01 | Medium | Expressivity via stochastic actions | ||
R1-064 | DYN | Co-GNN improves 3.19% accuracy on roman-empire and 3.62% ROC AUC on minesweeper over best baseline | 3.19 | % accuracy | REPORTED | https://arxiv.org/abs/2310.01267 | Section 6, Table 2 | we report 3.19% improvement in accuracy on the roman-empire and 3.62% improvement in ROC AUC on minesweeper compared to the best performing baseline. | 2023-10-02 | 2026-10-01 | Medium | Baselines GCN/SAGE/GAT/GT from Platonov et al.; not param- or FLOP-matched explicitly |
R1-065 | DYN | Co-GNN per-layer time complexity adds an action-network pass: O(Lπ dπ(|E|dπ+|V|) + dη(|E|dη+|V|)) | DESIGN | https://arxiv.org/abs/2310.01267 | Appendix D Runtime Analysis | the time complexity of a single C O -GNN layer is O(Lπ dπ (|E|dπ + |V |) + dη (|E|dη + |V |)). | 2023-10-02 | 2026-10-01 | Medium | Linear in edges | ||
R1-066 | DYN | AMP jointly learns message-passing depth and message filtering in a variational framework | DESIGN | https://arxiv.org/abs/2312.16560 | Section 2-3 | AMP uses a variational framework to jointly learn both the depth of the DGN and filter messages passed between nodes in each of these layers. | 2023-12-27 | 2026-10-01 | High | Learned depth (C12 partial: learned at training, not per-input halting) + message filtering (C8 partial) | ||
R1-067 | DYN | N2: a single shared recurrent layer moves graph nodes and pseudo nodes in a common space, building dynamic pathways at linear complexity | DESIGN | https://arxiv.org/abs/2410.23686 | Abstract; Section 1 | N2 incorporates a recurrent layer to parameterize the displacements of graph nodes and pseudo nodes in the common space. | 2024-10-31 | 2026-10-01 | High | Weight-shared recurrence + state-conditioned topology (C4,C5,C8 yes); NeurIPS 2024 poster per search result | ||
R1-068 | DYN | N2 requires significantly fewer parameters for graph classification with the shared recurrent layer | AUTHOR-CLAIM | https://arxiv.org/abs/2410.23686 | Abstract | N2 successfully scales to large-scale benchmarks and requires significantly fewer parameters for graph classification with the shared recurrent layer. | 2024-10-31 | 2026-10-01 | Medium | Capacity-per-parameter claim, not FLOP-matched | ||
R1-069 | DYN | MAVN dynamically introduces virtual nodes per layer in response to evolving node representations; up to 46.5% improvement over backbones | 46.5 | % improvement | REPORTED | https://arxiv.org/abs/2606.03068 | Abstract | MAVN consistently improves the performance of backbone MPNNs, achieving up to 46.5% improvement over the backbones and outperforms the baselines. | 2026-06-02 | 2026-10-01 | Medium | Baseline: same backbone MPNN without VNs; parameters not matched (adds VN pool); 2026 preprint |
R1-070 | DYN | MAVN learns when (which layer) and where (which nodes) to connect virtual nodes from a candidate pool | DESIGN | https://arxiv.org/abs/2606.03068 | Abstract | MAVN learns to adaptively determine when (at which layer) and where (to which nodes) to introduce and connect VNs based on the relative importance of connections. | 2026-06-02 | 2026-10-01 | Medium | State-conditioned edge/hub creation at inference (C6,C8) | ||
R1-071 | DYN | CAMP updates only a centrality-guided batch of nodes per layer (asynchronous message passing) | DESIGN | https://arxiv.org/abs/2509.06777 | Abstract | At each layer, only a centrality-guided batch of nodes updates, letting information propagate sequentially rather than compress simultaneously into fixed-capacity channels. | 2025-09-08 | 2026-10-01 | Medium | Schedule is centrality-based (structural), not state-conditioned | ||
R1-072 | DYN | CAMP gains 5% on REDDIT-BINARY and 4% on Peptides-struct | 5 | % improvement | REPORTED | https://arxiv.org/abs/2509.06777 | Abstract | it achieves notable gains, including 5% and 4% improvements on REDDIT-BINARY and Peptides-struct, respectively. | 2025-09-08 | 2026-10-01 | Medium | Baseline: same backbone with synchronous MP; unmatched FLOPs |
R1-073 | DYN | DRew: layer-dependent rewiring gradually densifies the graph; delay mechanism adds distance-dependent skip connections | DESIGN | https://arxiv.org/abs/2305.08018 | Abstract | performs a layer-dependent rewiring to ensure gradual densification of the graph. We also propose a delay mechanism that permits skip connections between nodes depending on the layer and their mutual distance. | 2023-05-13 | 2026-10-01 | High | Topology changes over depth but by a fixed schedule (hop distance), not node state | ||
R1-074 | DYN | DRew claims to outperform graph Transformers and multi-hop MPNNs on long-range tasks | AUTHOR-CLAIM | https://arxiv.org/abs/2305.08018 | Abstract | We validate our approach on several long-range tasks and show that it outperforms graph Transformers and multi-hop | 2023-05-13 | 2026-10-01 | Medium | LRGB; 500k param budget (LRGB convention) | ||
R1-075 | DYN | EGP uses sparse Cayley expander graphs as a message-passing template; 24-node 4-regular graph has diameter <=4 | 4 | hops | REPORTED | https://arxiv.org/abs/2210.02997 | Figure 1 caption | It has |V | “ 24 nodes and it is 4-regular (implying |E| “ 2|V |), hence it is sparse. Despite its sparsity, it is highly interconnected: any node is reachable from any other node by no more than 4 | 2022-10-06 | 2026-10-01 | Medium | Input-independent topology |
R1-076 | DYN | Expander templates necessarily contain negatively curved edges, which EGP argues are required for scalable bottleneck-free message passing | THEOREM | https://arxiv.org/abs/2210.02997 | Abstract; Section 4 | using expander graphs as a template for message passing necessarily gives rise to negative curvature. | 2022-10-06 | 2026-10-01 | Medium | Tension with curvature-based rewiring theory | ||
R1-077 | DYN | FoSR adds edges by first-order spectral expansion and uses a relational GNN to prevent oversmoothing | DESIGN | https://arxiv.org/abs/2210.11790 | Abstract | prevents oversquashing by systematically adding edges to the graph based on spectral expansion. We combine this with a relational architecture, which lets the GNN preserve the original graph structure and provably prevents oversmoothing. | 2022-10-21 | 2026-10-01 | High | Preprocessing rewiring, static at inference | ||
R1-078 | DYN | SJLR rewires during training but keeps the graph unchanged at test time | DESIGN | https://arxiv.org/abs/2212.02374 | Abstract | SJLR performs edge addition and removal during GNN training while maintaining the graph unchanged during testing. | 2022-12-05 | 2026-10-01 | High | Example of rewiring that does not change topology at inference | ||
R1-079 | DYN | DGMN2 samples nodes conditioned on input and predicts node-dependent filters and affinities; outperforms fully-connected graphs with fewer FLOPs and parameters | AUTHOR-CLAIM | https://arxiv.org/abs/2209.09760 | Abstract | Our approach also outperforms fully-connected graphs while using substantially fewer floating-point operations and parameters. | 2022-09-20 | 2026-10-01 | Medium | Vision backbone; compared to fully-connected self-attention; a graph-first state-conditioned sparse-attention replacement for transformer layers | ||
R1-080 | DYN | DGMN2 adaptively samples nodes conditioned on the input for message passing and predicts the affinity matrix dynamically | DESIGN | https://arxiv.org/abs/2209.09760 | Abstract | This is achieved by adaptively sampling nodes in the graph, conditioned on the input, for message passing. Based on the sampled nodes, we dynamically predict node-dependent filter weights and the affinity matrix for propagating information between them. | 2022-09-20 | 2026-10-01 | High | |||
R1-081 | GT | GPS decouples local real-edge message passing from global attention, giving O(N+E) complexity with linear attention | DESIGN | https://arxiv.org/abs/2205.12454 | Abstract | we propose the first architecture with a complexity linear in the number of nodes and edges O(N + E) by decoupling the local real-edge aggregation from the fully-connected Transformer. | 2022-05-25 | 2026-10-01 | High | Linear only with Performer/BigBird global attention; full attention is O(N^2) | ||
R1-082 | GT | GPS recipe = positional/structural encoding + local message passing + global attention | DESIGN | https://arxiv.org/abs/2205.12454 | Abstract | Our GPS recipe consists of choosing 3 main ingredients: (i) positional/structural encoding, (ii) local message-passing mechanism, and (iii) global attention mechanism. | 2022-05-25 | 2026-10-01 | High | Layers not weight-shared; fixed depth | ||
R1-083 | GT | Standard global graph attention costs O(|V|^2) time and memory; large-graph GTs often do not fit in GPU memory | AUTHOR-CLAIM | https://arxiv.org/abs/2303.06147 | Section 1 | the standard global attention mechanism incurs time and memory complexity of O(|V |2 ), quadratic in the number of nodes in the graph. | 2023-03-10 | 2026-10-01 | High | |||
R1-084 | GT | Exphormer sparse attention via virtual global nodes and expander graphs, linear in graph size | DESIGN | https://arxiv.org/abs/2303.06147 | Abstract | E XPHORMER consists of a sparse attention mechanism based on two mechanisms: virtual global nodes and expander graphs | 2023-03-10 | 2026-10-01 | High | Attention pattern fixed (input-independent expander), not state-conditioned top-k | ||
R1-085 | GT | Graphormer: standard Transformer + structural encodings; many GNN variants are special cases | AUTHOR-CLAIM | https://arxiv.org/abs/2106.05234 | Abstract | with our ways of encoding the structural information of graphs, many popular GNN variants could be covered as the special cases of Graphormer. | 2021-06-09 | 2026-10-01 | Medium | |||
R1-086 | BENCH | LRGB protocol: 500k parameter budget; MP-GNNs 8 layers vs transformers 4 layers | 500 | k parameters | REPORTED | https://arxiv.org/abs/2206.08164 | Section 5, Table 2 caption | The MP-GNN models are 8 layers deep, while the transformer-based models have 4 layers in order to maintain comparable hidden representation size at the fixed parameter budget of 500k. | 2022-06-16 | 2026-10-01 | High | Param-matched (500k), not FLOP-matched |
R1-087 | BENCH | LRGB original: GCN 0.5930 vs GPS 0.6535 AP on Peptides-func | 0.5930 | AP | REPORTED | https://arxiv.org/abs/2309.00367 | Figure 1 / Table 1a (original LRGB numbers reproduced) | GCN 0.5930 ± 0.0023 GINE 0.5498 ± 0.0079 G ATED GCN 0.6069 ± 0.0035 T RANSFORMER 0.6326 ± 0.0126 SAN 0.6439 ± 0.0075 GPS 0.6535 ± 0.0041 | 2023-09-01 | 2026-10-01 | High | 500k param budget (matched); untuned MPNN baselines |
R1-088 | BENCH | After tuning, all three MPGNNs outperform GPS on Peptides-func/struct; GCN reaches 68.60% AP | 68.60 | % AP | REPORTED | https://arxiv.org/abs/2309.00367 | Section 3, Table 1a | all three MPGNNs outperform GPS after tuning. The average precision on Peptides-Func increased relatively by around 10% to 20%. GCN achieves a score of 68.60%, which is competitive with newer GTs such as GRIT or Graph ViT. | 2023-09-01 | 2026-10-01 | High | 500k param budget (matched); FLOPs not matched; key negative for global attention on LRGB |
R1-089 | BENCH | LRGB gap between GTs and MPGNNs completely vanishes on multiple datasets after basic hyperparameter optimization | REPORTED | https://arxiv.org/abs/2309.00367 | Abstract | It is noteworthy that across multiple datasets the performance gap completely vanishes after basic hyperparameter optimization. | 2023-09-01 | 2026-10-01 | High | Vision datasets (PascalVOC-SP, COCO-SP) still favor GPS (0.4440 vs GatedGCN 0.3880 F1) | ||
R1-090 | BENCH | Tuned GPS still leads on PascalVOC-SP: 0.4440 F1 vs tuned GatedGCN 0.3880 | 0.4440 | F1 | REPORTED | https://arxiv.org/abs/2309.00367 | Table 1b | 0.2078 ± 0.0031 0.2718 ± 0.0054 0.3880 ± 0.0040 0.4440 ± 0.0065 | 2023-09-01 | 2026-10-01 | Medium | Order GCN, GINE, GatedGCN, GPS; 500k budget |
R1-091 | BENCH | Tuned classic GNNs (GCN/GAT/SAGE) match or exceed GTs on 17 of 18 node classification datasets | 17 | datasets of 18 | REPORTED | https://arxiv.org/abs/2406.08993 | Abstract | with slight hyperparameter tuning, these classic GNN models achieve state-of-the-art performance, matching or even exceeding that of recent GTs across 17 out of the 18 diverse datasets examined. | 2024-06-13 | 2026-10-01 | High | Not param- or FLOP-matched; node classification (not algorithmic) |
R1-092 | THEORY | Logarithmic transformer depth is necessary and sufficient for graph connectivity | THEOREM | https://arxiv.org/abs/2405.18512 | Abstract; Section 3.1 | We prove that logarithmic depth is necessary and sufficient for tasks like graph connectivity, while single-layer transformers with small embedding dimensions can solve contextual retrieval tasks. | 2024-05-28 | 2026-10-01 | High | Objection: a transformer needs only O(log n) depth for connectivity where an MPNN needs ~diameter rounds | ||
R1-093 | BENCH | Transformers outperform GNNs on global tasks (connectivity, shortest path); GNNs win on local tasks in low-sample regime | REPORTED | https://arxiv.org/abs/2405.18512 | Section 4 findings 1-2 | Transformers outperform GNNs on tasks that require efficiently aggregating information about distant nodes in a graph, such as connectivity and shortest path. | 2024-05-28 | 2026-10-01 | High | Transformers up to 60M params vs GCN/MPNN/GIN; not param-matched | ||
R1-094 | BENCH | Connectivity accuracy at 100K samples: GCN 94.0, MPNN 94.4, GIN 94.0 vs 60M transformer 98.4 | 98.4 | % accuracy | REPORTED | https://arxiv.org/abs/2405.18512 | Figure 3a table | GCN [42] MPNN [26] GIN [82] 83.8 94.0 93.8 83.8 94.4 94.0 60M transformer 11B transformer (FT) 92.9 98.4 98.0 | 2024-05-28 | 2026-10-01 | Medium | Row/column order reconstructed from extracted table text (1K / 100K samples); unmatched params |
R1-095 | THEORY | Neither GT self-attention nor MPGNN+virtual node is uniform-universal; neither uniform expressivity subsumes the other | THEOREM | https://arxiv.org/abs/2405.11951 | Abstract | We prove that none of the models is a uniform-universal approximator, before proving our main result: Neither model’s uniform expressivity subsumes the other’s. | 2024-05-20 | 2026-10-01 | High | Uniform = one network for all graph sizes (the extrapolation setting) | ||
R1-096 | THEORY | Short-range oversquashing: MPNNs (even with virtual nodes) fail the Two-Radius task, transformers succeed | REPORTED | https://arxiv.org/abs/2511.20406 | Abstract; Section 5 | standard MPNNs, even when augmented with virtual nodes, perform poorly on these problems, whereas transformers solve them easily. | 2025-11-25 | 2026-10-01 | High | Strong evidence for the objection: global attention beats MPNN bottleneck even at short range | ||
R1-097 | THEORY | Solving Two-Radius with MPNNs requires feature dimensions that grow with graph size | THEOREM | https://arxiv.org/abs/2511.20406 | Section 5 Conclusion | solving the Two-Radius problem with MPNNs requires large feature dimensions that depend on the graph size | 2025-11-25 | 2026-10-01 | High | Capacity per node state bounded; width must grow with n | ||
R1-098 | X | Looped transformer with graph-attention heads simulates Dijkstra, BFS, DFS, Kosaraju with parameters independent of graph size; finite precision limits it | THEOREM | https://arxiv.org/abs/2402.01107 | Abstract | The number of parameters in the networks does not increase with the input graph size, which implies that the networks can simulate the above algorithms for any graph. Despite this property, we show a limit to simulation in our solution | 2024-02-02 | 2026-10-01 | High | LOOP family (R2 overlap): the weight-shared looped transformer already achieves size-independent graph-algorithm simulation | ||
R1-099 | X | Transformers are message-passing GNNs on fully connected token graphs that win the hardware lottery via dense matmuls | AUTHOR-CLAIM | https://arxiv.org/abs/2506.22084 | Abstract | Transformers are implemented via dense matrix operations that are significantly more efficient on modern hardware than sparse message passing. This leads to the perspective that Transformers are GNNs currently winning the hardware lottery. | 2025-06-27 | 2026-10-01 | High | Central objection | ||
R1-100 | X | Self-attention = message passing on a fully connected graph of tokens | AUTHOR-CLAIM | https://arxiv.org/abs/2506.22084 | Abstract | We show how Transformers can be viewed as message passing GNNs operating on fully connected graphs of tokens, where the self-attention mechanism capture the relative importance of all tokens w.r.t. each-other | 2025-06-27 | 2026-10-01 | High |
End of preview. Expand in Data Studio
Neural Graph Substrate evidence ledger (SR-2026-006)
The quote-per-claim evidence ledger behind the research note Is the graph doing anything?, a sourced survey of recurrent self-routing neural graphs as an alternative to layered networks. Every claim carries its source URL, a locator, a short verbatim quote (target 40 words or fewer; 7 run longer), a status and the date it was read. 965 claims (962 citable) from 472 distinct sources (350 of them arXiv papers) back a matrix of 121 approaches in 23 families, each answering the same fourteen questions.
- Paper: scalably.io/research/neural-graph-substrate · DOI 10.5281/zenodo.23091812
- Code and full ledger: scalably-io/neural-graph-substrate (Apache-2.0 code, CC BY 4.0 data)
- Built with AI, on purpose: one founder directed AI agents (Anthropic's Claude) that searched primary sources and wrote this ledger; fresh-context Claude reviewers and OpenAI's GPT-6 Astra attacked it. No human domain expert has reviewed it yet.
Subsets
| Subset | Rows | What one row is |
|---|---|---|
claims (default) |
965 | one sourced claim. Status counts: REPORTED 358, AUTHOR-CLAIM 208, DESIGN 174, CODE 107, THEOREM 103, DERIVED 12, ACCESS-FAILED 3 |
approaches |
121 | one published approach answering fourteen fixed questions; unknown is allowed and never guessed |
coverage |
15 | the data behind Figure 1: which candidate components the closest prior art already has |
Field rules are in SCHEMA.md. AUTHOR-CLAIM rows are what authors say about their own work and are never cited as fact. ACCESS-FAILED rows are kept but never cited.
Uses
Use it as a starting map of recurrent-depth, looped-Transformer, dynamic-graph and associative-memory work, or to check a claim against its quote. To challenge a row, open a discussion with the claim_id, a better source, its locator and a verbatim quote.
Limitations
Quotes are short excerpts for verification; the sources are not redistributed and each is re-fetchable from its URL. Coverage reflects what the agents found by 2026-10-01; absence of an approach is not proof it doesn't exist.
Citation
@misc{lazic2026ngs,
title = {Is the graph doing anything? A sourced survey of recurrent self-routing neural graphs},
author = {Lazi\'c, Pavle},
year = {2026},
doi = {10.5281/zenodo.23091812},
note = {Scalably report SR-2026-006}
}
Contact: pavle@scalably.io. Built 2026-10-26.
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