LRIM: a Physics-Based Benchmark for Provably Evaluating Long-Range Capabilities in Graph Learning
Authors: Joël Mathys
Henrik Christiansen
Federico Errica
Takashi Maruyama
Francesco Alesiani
Venue: ICLR 2026 Poster (2026)
Keywords: Graph Learning
Long-Range
Benchmark
Primary area: learning on graphs and other geometries & topologies

Abstract: Accurately modeling long-range dependencies in graph-structured data is critical for many real-world applications. However, incorporating long-range interactions beyond the nodes' immediate neighborhood in a $\textit{scalable}$ manner remains an open challenge for graph machine learning models. Existing benchmarks for evaluating long-range capabilities either cannot $\textit{guarantee}$ that their tasks actually depend on long-range information or are rather limited.
Therefore, claims of long-range modeling improvements based on said performance remain questionable.
We introduce the Long-Range Ising Model Graph Benchmark, a physics-based benchmark utilizing the well-studied Ising model whose ground truth $\textit{provably}$ depends on long-range dependencies. Our benchmark consists of ten datasets that scale from 256 to 65k nodes per graph, and provide controllable long-range dependencies through tunable parameters, allowing precise control over the hardness and ``long-rangedness". We provide model-agnostic evidence that local information is insufficient, further validating the design choices of our benchmark. Via experiments on classical message-passing architectures and graph transformers, we show that both perform far from the optimum, especially those with scalable complexity. Our goal is that our benchmark will foster the development of scalable methodologies that effectively model long-range interactions in graphs.

Published as a conference paper at ICLR 2026
LRIM:
A PHYSICS-BASED BENCHMARK FOR PROVABLY EVALUATING LONG-RANGE CAPABILITIES IN GRAPH LEARNING
Joë Mathys
ETH Zurich jmathys@etha.chh
Henrik Christiansen', Fecdeeio Errica', Takashi Maruyama* & Francesco Alesiani
NEC Laboratories Europe
[firstname. Lastnamme@@ecce @neclab eu,
hag.calaaabbeaaaaaaaaaaaaaaaa #49takashi@nec.com
ABSTRACT
Accurately modeling long-range dependencies in graph-structured data is critical for many real-world applications. However, incorporating long-range interactions beyond the nodes' immediate neighborhood in a scalable manner remains an open challenge for graph machine learning models. Existing benchmarks for evaluat- ing long-range capabilities either cannot guarantee that their tasks actually depend on long-range information or are rather limited. Therefore, claims of long aangrange modeling improvements based on said performance remain questionable. We in- bench-
troduce the Long-Range Ising Model Graph Benchmark, physics- mark utilizing the well-studied Ising model whose ground truth provably depends on long-range dependencies. Our benchmark consists of ten datasets that scale from 256 to 65k nodes per graph, and provide controllable long-range dependen- cies through tunable parameters, allowing precise control over the hardness and ronggraangedees".. We provide modellaggostic evidence that local information is insufficient, further validating the design choices of our benchmark. Via ex- periments on classical message-passing architectures and graph transformers, we show that both perform far from the optimum, especially those with scalable com- plexity. Our goal is that our benchmark will foster the development of scalable methodologies that effectively model long-range interactions in graphs.
INTRODUCTION
Since the early days of deep learning on graphs (Sperduti & Starita 1997 Micheli & Sestito 2005
Gori et al. 22055 Scarselli al., 2009 Micheli| (2099), researchers have tried to automatically learn task-defined mapping given graph-structure data. At the very heart of the most popular architecture, namely Graph Neural Networks (GNNs), is the idea that repeated aggregation of local information expands the "receptive field" of each node (Bacciu et al. 2020) in a way similar to convolutional neural networks for images (LeCun et al.) 1995). Such an expansion is crucial, for instance, in tasks where the true mapping requires non-local processing of information among nodes in the graph. In this case, researchers typically talk about capturing long-range dependencies.
Long-range dependencies or interactions are an important component of physics and chemistry, manifesting, for example, quantum systems (Defenu et al.) (023). protein folding (Gromiha & Sclvaraj 1999) or astronomy (Carroll & Ostlie, 2017). An example from biology is mRNA splicing, a fundamental part of the gene expression process: splicing is inhibited if we "disable" long-range dependencies between distant regions of mRNA (Rüegsegger et al., 2001). The protein binding mechanism, whose understanding is crucial for the development of vaccines, also depends on long- range interactions between proteins (Ferber et al. 2012).
As of today, most deep learning models on graphs struggle to incorporate long-range interactions. For GNNs, the reason is tied to their efficient but limited message-passing scheme: expanding the

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receptive field to capture long-range information rapidly increases the amount of information that every node has to process and store, generating a computational bottleneck (Amaiz-Rodrighee Rodriguez & Emrica 2025). When addressing the long-range limitations of existing models, the first step should be to benchmark novel methods on tasks that provably depend on long-range information. Unfor- tunately, while popular existing benchmarks focus on real-world data (Dwivedi al. 2022), they cannot guarantee that the task to solve hinges on long-range dependencies (Tönshoff' et al.) 2023).
The reason is that the definition of the task is unknown, which is almost always the case machine learning.
We introduce the Long- Range Ising Model (LRIM) Graph Benchmark provable and controllable long-range benchmark based on the well-studied and fundamental Ising model for graph learning.
The Ising model was originally introduced in statistical physics to study magnetic materials (Peierls) 1936) and fundamental properties related to phase transitions [Wilson] 1983, Stanley 1987] Yeo- mans 1992). Over the decades, the influence of the model has spread to aid understanding and analyzing complex phenomena far beyond the original intention, with applications in protein fold- ing (Bryngelson Wolynes 1987), percolation (Balogh et al.) (012), the theory of disordered sys- tems (Parisi 2023), social systems such as stock markets (Durlauf 1999) to name only few. The tunable parameters of the LRIM control the dependency of each spin's 's energy on distant spins, thereby allowing to control the impact of long-range dependencies; in other words, through the underlying physics we can easily control the Hlong-rangedness of the task.
In the following, we introduce ten novel LRIM datasets scaling from 256 to 65k nodes, each pro- viding provable and controllable long-range dependencies through the underlying physical model. Besides the inherent long-range dependency in the task formulation, we ensure that we sample configurations having large spatial correlations between node features by simulating at the pseudo- critical temperature. addition, we demonstrate the validity of our approach by providing baseline- agnostic evidence. We analyze the behavior of oracle predictors restricted to local neighborhoods, which systematically degrade performance. Moreover, this consistent behavior provides a valuable continuous feedback signal for approaches, both for performance evaluation and during training. The analysis of our proposed benchmark through the lens of recently proposed long-range met- rics (Bamberger et al. 2025) further supports our claims. Finally, also empirically quantify the performance of message- mnssagg-ppassng architectures and graph transformers on our benchmark, revealing substantial gaps, especially when factoring in the computational cost of the methods. As such, our aim is that the LRIM Graph Benchmark provides the graph learning community with a powerful tool for evaluating and advancing long-range capabilities in provable, controllable manner.
RELATED WORK
The literature on long-range benchmarks for graph learning can be roughly divided into two main directions. The first focuses on real-world datasets in which long-range interactions should occur, often related to phenomena in biology or chemistry. The most popular one is perhaps the Long- Range Graph Benchmark (LRGB) by Dwivedi et al. (2022), which consists of image segmentation tasks - adapted to graphs together with peptides' function classification and property regression tasks. While this work identifies properties that should be required for long-range candidates, such as sufficiently large graphs, there appears to be no conclusive evidence that the proposed tasks require LR interactions. Even if they did, it remains unclear how to assess their impact and necessity for the empirical performance on these datasets. The benchmark served as a common test-bed to compare graph transformers (Müller et al. (2024), which seemed to achieve superior performance compared to classical GNNs. However, such empirical claims were later revised (Tönshoff et al. 2023 [echler- Speicher et al. 2025). showing that with proper syper-parameter tuning, both techniques achieve similar performance on the peptide tasks. As such, LRGB still provides a valuable test-bed on real-world data, but any claim about long-range capabilities cannot be guaranteed.
On the other hand, there exist a variety of synthetic datasets to evaluate long-range capabilities of machine learning models. These often focus on gathering or copying information (Gu et al., 2020 Di Giovanni et al. 2023) beyond a certain distance or predicting non-local graph properties such as connectivity, distances or eccentricity (Corso et al. 2020; Rampášek & Wolf 2021 Liang et al.,
2025). Similarly, the GLoRa benchmark (Zhou et al. (0255) provides a detailed analysis of these
Project available https://giihaahccm/lllo/ttttm graph benchmark
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datasets and proposes new task that provably depends on specific lengths. While this provides a promising direction towards a more principled evaluation, these datasets still exhibit significant limitations: most prediction tasks exhibit binary-like feedback signal. This results in models either perfectly solving the task or achieving null performance. Contrarily to these works, we propose prediction task that: i) is central historically played a crucial role in advancing research and understanding of many fields (Budrikis) 2024]; ii) provides evidence for the required long-range interaction; tii) allows for a more fine- grained and controllable continuous feedback signal both during training and evaluation.
the simulation of physical systems, as the Ising model has
Lastly, we mention that heterophilic datasets have often been believed to require long long-range capa- bilities of GNNs. However, this viewpoint was recently criticized in Armaiz-Rodriguu Rodriguez & Erricaa (2025) by providing clear counterexamples that the task might induce heterophily regardless of the nature of the problem.
3 BACKGROUND
The description of systems with many interacting components is ubiquitous in the natural and social sciences. For example, in nature, a colony of ants evolves based on interactions among the indi- vidual ants and based their genetic inclinations, in a transportation network, vehicles move, combining individual objectives and traffic rules. Using domain-specific modeling approaches, these case-by-case basis using convoluted system prototypes that are of-
systems can be investigated ten difficult to understand and interpret. Conversely, spin models, such as the Ising model, have proven powerful in describing relevant features of real systems while retaining simplicity. Based on simple microscopic interaction laws, they show rich emergent behavior, non-trivial phase- transiiiansss and spin-spin correlations. Formally, the LRIM defined by the graph topology G (excluding self- interactions) and the Hamiltonian
H(({:))
Σ J18818,, with power-law potential Jig tjEG
d+o Fij
where ray |r, rıl is the distance between two spins s; +1 located at coordinate r,. The exponent a controls how long-ranges the interactions are. In our case, we consider the spins to be located regular d = 2 dimensional lattice. In general, the choice of Jij determines the model class: If instead one chooses nearest-neighbor interactions, one recovers the standard short-range lattice models (Jlz 1 and G (ijlry 1)), and random Jej define spin glasses such as the Sherrington- -irkkpatkk model (Sherrington & Kirkpatrick] 1975). We consider the system to be in contact with its thermal environment, that is, canonical setting in which microscopic configurations follow the Boltzmann distribution
1 [.(((s))) P((s:))
z exp
(-x,,
(2)
K2T
where the Boltzmann constant is unity (ks, 1), T is the temperature, and Z the (unknown) normalization constant.
This model has three distinct phases a function of T, distinguished by the critical temperature To: (i) for T > To the system is disordered, (ii) for T T. it is critical, and (iii) for T' < T. it ordered. For finite systems as considered in our benchmark, the relevant temperature is not T. directly, but rather the pseudo-critical temperature characterizing the transition for the given system size L (see Appendix B). It is known that at T. the correlations in the system diverge, i.e., spins are correlated across all distances, and no length scale dominates. As consequence, self-similar structures emerge, and the spin clusters become fructal. More formally, our setting, the connected correlation function is defined as
1 C(r) (8185) - (8)))8)) 7"
for T = Tc and r Do,
where (.) [1L-] symbolized expectations under the Boltzmann distribution of Eq. and includes average over the system. At T. and in the limit of large distances has that (r) decays algebraically with the universal critical exponent and the corresponding correlation length diverges. This implies that spins are correlated over large distances in non-trivial way and,
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Input Spins
Target AE
Input Spins
Target A E
Figure 1: A 4 Ising spin configuration (left) represented as an attributed graph (middle) in the LRIM Graph Benchmark. (Right) Four representative 32x32 spin configurations from the LRIM- 32 dataset and their corresponding Á energy landscapes, demonstrating the non-trivial energetic landscape which provably depends on long-range interactions.
in addition, are interacting via a long-range potential by construction, posing an ideal way to construct appropriate configurations for our benchmark.
The value of a Equation ① controls how long-range the spins interact, and as a consequence, changes the equilibrium behavior of the model. For < 1 one has mean-field behavior and hence 7 = 1 (Kadanofi) 2009). for 1 < a < 0x, the critical exponents depends on 0 as 7) = a, and for 0x < σ, is in the short-range (nearest-neighhor) universality class with vj 1/4. The value of 0x is discussed in the literature (Picco) 2012; Angelini et al.] 2014 Shiratani & Todo 2024 Liu al. 2025), with 0x 2 (Fisher al.) 1972) 0x 1.75 (Sak) 1973) being the most likely candidates. To simulate the system, a Markov Chain Monte Carlo simulation can be set up by proposing random single spin flip f...........
f.....,,....... and accepting the new
configuration according to the Metropolis criterion with probability
p min exp (-2AE:/kg7)), with AE, = Σs
Using this stochastic approach to generate states results in a chain of (correlated) configurations that, after equilibration period, become samples from the Boltzmann distribution defined in Eq, [ Although there exist cluster algorithms that decorrelate quickly at criticality in equilibrium (Luijten) Flores-Sola et al. 2017). and recent advances for single spin flip simulations (Müller et al. 2023 that avoid exact calculations of A E, by exploiting the way Monte Carlo simulations constructed, the calculation of AE, is central for simulations of many models. We therefore use the prediction of A E, as our long-range task.
& Blöte 1995 Fukui & Todo 2009
To obtain samples from the target distribution for the dataset, implement the single cluster variant for the LRIM as presented by FFloresSSlla et al.] (207)). We make sure to equilibrate the simulation before measuring and to decorrelate subsequent samples by sufficient number of cluster updates. For more details, see Appendix
4 LRIM GRAPH BENCHMARK
The long-range graph learning task Our goal is to directly translate the Ising model into an appropriate graph-based task formulation with controllable long-range interactions while preserving its simplicity. We focus on the d = LRIM on a grid lattice and want to predict energy changes throughout the system. Each LRIM instance corresponds arranged in a 2D periodic grid. Note that the topology is shared among all instances and that each node connected its nearest neighbors. Moreover, each node has a single feature, representing its physical spin (-1, +1}. We formulate the energy prediction as a node regression task, where each node v; has to predict its energy change AE, E R. A visual illustration of how the graph task constructed is shown in Figure I An important remark is that learning on grid graph is a non- trivial challenge for most message passsing architectures. The reason that a grid graph exhibits an exponential computational bottleneck due to its computation tree, detailed in Armaiz- Rodriguez & Errical (025)), which can deteriorate performances as shown in previous work (Alon & Yahav]
graph G with L x L nodes that are
2021
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Table 1: Overview of all 10 datasets in the LRIM Graph Benchmark. Our benchmark can system- atically vary the complexity across 5 graph sizes (256 to 65,536 nodes) and 2 difficulty levels per size controlled by the interaction parameter o. Each dataset contains 10,000 spin configurations represented as 2D periodic grid graphs with 4-regular connectivity. The proposed task considers node-level regression to predict energy changes A E,, with performance measured using 1og MSE.
Avg. Eff. Resistance
Avg. Short. Path
Dataset
Graphs
Nodes
Edges
Diameter
casy
hard
LRIM-16 16
1.5
0.6
10,000
256
512
0.49
8.03
16
LRIM-32
1.5
0.6
10,000
1,024
2,048
0.60
16.02
32
LRIM-64
1.5
0,6
10,000
4,096
8,192
0.71
32,01
64
LRIM-128
1.5
0.6
10,000
16,384
32,768
0.82
64.00
128
LRIM-256
1.5
0.6
10,000
65,536
131,072
0.93
128.00
256
Dataset control and diversity To thoroughly evaluate the long-range capabilities with the LRIM benchmark across scale and difficulty, we provide five datasets of increasing size and two difficulty levels (easy and hard) for each size configuration based on the values of a. In particular, the datasets range from LRIM-16 (256 nodes) to LRIM-256 (65, 536 nodes). Complete graph and dataset statistics for all variants are provided in Table ! with further details on the calculation in Appendix C Each dataset variant contains 10, 000 distinct graph instances. For each size, the long-range interaction strength is varied between a "hard" variant (a 0.6) and an "easy" variant (o 1.5) (see Section 3 for the justification of these values). Across settings, our task formulation has the advantage of yielding continuous feedback as the true energy can be more closely approximated by integrating more long-range information. Simultaneously, the harder tasks require models to aggregate information from more distant nodes, as we validate later.
Data generation Data generation follows the outlined Monte Carlo sampling protocol of Sec- tion 3 For each system size I. and a value, we first determine the appropriate pseudo-critical tem- perature To L) where the system exhibits longest correlation lengths, thereby creating the most interesting interactions. Each configuration is sampled from simulation which is first equilibrated followed by a decorrelation phase between two subsequent samples to ensure statistical indepen- dence between them. Then, we split the dataset according to 80/ 80/10/10 into dedicated splits for training. validation, and testing. For more details we refer to Appendix
Task evaluation metries We want to highlight the computational efficiency aspect when evalu- ating methods for their long-range capabilities, known improvements often come with increased computational costs. To ensure rigorous and fair comparisons, we require that methods report their runtime complexity for their computational budget (e.g. O(L E) for standard MPNNs with T. layers and edges) and any precomputation costs, including creation of additional struc- ture and feature preprocessing. We put little restriction what can be used the benchmark on purpose, for instance extra features, encouraging novel methods. However, we ask that all modifications and their computational overhead must be transparently documented. To report prediction performance, we use the base-10 10 logarithm of the mean square error (LogMSE).
EVALUATION
5.1 LONG-RANGE ANALYSIS
In this section, we empirically and theoretically demonstrate that our proposed LRIM benchmark is suitable for testing long-range interactions. We remind the reader that the dataset consists of syn- thetic simulation data over which we have full knowledge and control of the underlying generation process. This allows us observe the (in)ability of current models to capture long-range dependen- cies and encourage the design and development of new approaches that solve our benchmark under computational budgets mentioned above. In analysis, we consider three different perspec- tives: i) how the simulation accuracy degrades when an oracle is restricted to local neighborhoods, including worst-case error bounds; ii) the trade-off between overfitting and generalization based
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-1
-2
LRIM-128- 128-hard
-4
AF log MSE
5
LRIM 128-easy
LRIM-16 16-hard
6
LRIM-16 10--166eeass
LRIM-256-haB 256-hard
8
-9
-10
LRIM-32-hard
-11
LRIM-32-easH
LRIM-64-hard
-12
-13
LRIM-64-easy
log2 Number of Hops
Figure 2: LogMSE (1) performance of the oracle predictor degrades when restricted to consider local r-hop neighborhoods only. The oracle uses the true underlying energy function, but only con- siders spins within hop-distance from each target node. Results demonstrate that smaller a values (harder tasks) require larger neighborhoods to achieve the same accuracy, confirming stronger long- range dependencies. Second, larger system sizes increase task difficulty, even within the same a. Moreover, the performance decays smoothly, providing continuous feedback both during eval- uation and training. Therefore, achieving low prediction error requires information from neigh- borhoods spanning significant fractions of the graph, providing model-agnostic evidence that local information insufficient.
the Weisfeiler- Lemann (WL) test of graph isomorphism; iii) and theoretical results on recent long- rangedness metrics (Bamberger et al. (2025). Taken together, these analyses provide strong evidence that our benehmark requires long-rang long-range reasoning capabilities grounded in well-knows physics lit- erature, contrary to previous works.
First, we analyze how prediction accuracy degrades when the information of oracle restricted to local r-hop neighborhoods only. The oracle predicts AÆ, based on the correct contributions within the r-hop neighborhood; it follows that when the oracle has global access to the graph for each node, its prediction matches the simulation ground truth. Note that this is not strict lower bound on achievable error for a given r-hop neighborhood.
We vary the parameter from I to the diameter of the graph across datasets, and plot the oracle's 's score Figure 2 These results show that task difficulty can be precisely controlled by both the parameter and the size of the system L.. We observe that lower values consistently require larger neighborhoods to achieve the same prediction accuracy, which is indicative of stronger long-range interactions required to solve the harder task (o 0.6). Furthermore, for the same a, a larger system size L also contributes to increase the task's difficulty. The prediction error decays smoothly as the neighborhood size increases from local global, showing that incorporating information from more distant nodes consistently improves prediction accuracy. This provides both continuous feedback signal for both model performance evaluation and during training. Crucially. this analysis reveals that achieving low error requires considering interactions across substantial fractions of the entire graph.
Further, we provide theoretical arguments that underscore the fundamental necessity of long-range information for the best performance of the task. We first establish lower bound on the worst-case emoor that any method restricted to local neighborhoods must exhibit by not considering the rest of the graph. As part of the system remains unknown, there exists a variety possible configurations compatible with the observed neighborhood which exhibit wide range of possible energies. There- fore, no predictor that only considers the local information can have a maximum error significantly below that range. For the detailed derivation refer to Appendix

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0=0.3
0-0.6
0=0.9
10
1.1.
log ((AE))))
0=1.8
50
150 200
Figure 3: Normalized range measure (AEE) increases with system size L and decreases with The measure quantifies the relative contribution of distant nodes to the energy prediction task, computed analytically using the oracle's s gradient. Smaller a values lead to higher measures across all system sizes, with the growth rate accelerating for I 1. This validates both the dependence and controllability of long-range dependencies throughout our proposed benchmark.
Lemma 5.1. Let fo be model that predicts AE for given node v on spin configuration X with a periodic grid graph of size N n x n with diameter D n. If fe only considers the spins of nodes within radius & D of then there exists spin configuration X' where fo("=
Y - So(X") )ol 9
Following Leman & Weisfeiler 1968), analyze limitations of MPNNs and find that WL equiv- alence classes initially differ energy targets, requiring larger receptive fields to be distinguished. However, this effect diminishes faster than oracle analysis due finite data (detailed analysis in Appendix D). This indicates that expressiveness does not fundamentally restrict fitting the data, however proper generalization likely requires receptive fields closer the oracle predictor.
We connect our proposed datasets to the long-rangedness metric for graphs defined BBabberger et al. (2025). The range measure P=(F") of node-level differentiable function F for node u and its normalized measure (F) are defined as
Long-rangedness Measure
(F(X),
Pa(F)
Pa(F)
Σ DEV
da(u,
(n(F))
Or,
(F(X),
Ox.
VEV
Here, T. is the node feature of and de V X + a distance function, graph G E) with nodes V and edges E. We adapt the measure to the oracle predictor, which is given by the data generation process, by setting AE as F (assuming spins are embedded R) and consider +1 spins as node features X. We first present one of our main results on the expression of the adapted long- range measure in the following proposition. The proof and further derivation of the proposition are provided in Appendix G
Ising model defined over a grid of size L, with F AE. Assume that the binary spins s; relaxed continuous spin values s, E R. Then, when de V V > IR is the shortest path distance, the range measures (a(EE) and p; (AE) are represented
Proposition 5.2. Let be a periodic
Pa AE
P (AE) Σe
Jik: ISEST kEN,
P(GE)
Σ
Σ
Jik
REN,
where Ne(1) denotes the set of spins at E-hop distant from i.
In addition, the following proposition, which states the effect of changing the parameter the metric proposed in (Bamberger et al. 2023), supports the controllability of long-rangsd in terms
of pa(A (A) and 7s( (AFE) in LRIM. Its proof provided in Appendix G for Proposition G.4
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Proposition 5.3. Given the same assumption Proposition 5.2 with r L, both range measures Pa(EE) and (aE diverge as L - oC when a S 1, and converge when a >> 1.
We plot the normalized range measure Pi [) obtained in Proposition 5.2 across sizes and for var- ious in order to demonstrate the impact on the long-range measure. The computation of p. (A)) relies on an analytical expression of its intermediate expression Σ Ne(i) Jik, whose formal state- ment and proof are provided in Theorems G.2 and G.3 We observe in Figure a clear upward trend that smaller a values lead to higher corresponding node value in the metric across different grid sizes.
We also observe that the growth rate of the measure significantly increases for smaller o, which suggests that smaller a could increase the likelihood that machine learning models face more diffi- culty in capturing long-range dependency. We believe the presented result further supplements the legitimacy of using the LRIM as a benchmark for evaluating the performance of models in graph learning focusing on long- range interactions.
Table 2: Baseline performance on LRIM-16 and LRIM-32 datasets. The number of edges E corre-
sponds to 4N in our datasets. We emphasize the importance of reporting computational complexity alongside performance results, as scalability is a crucial aspect in long-range modeling.
Preprocessing
Computation
16-M-16-hard
LRIM-32-hard
LRIM-I6-casy 4
LRIM-32-ensy J
GIN
O E)
-2.533 +am
-2.249 6.s
-3.564 +
-3.446 + nan
GaledGCN
OIL E)
-3.844 + uns
-4.087 022
-,817 + tem
4.940 + use
GaledGCN VNg O(N)
OIL E N-N
-4,068 Las
-3.243
-4.612 a
-4.322 asse
GPS-Base
OIL N2)
-4,211 tass
-.044
-5.296 s
-5.134 + 0054
GPS RWSE
O(k-N") N²)
OIL N
4,011 a1s
4.134 ouns
-5.133 e
-5,103 1 0091
GPS-LapPE
OlL- N*)
..334 was
4.082
-5.154 4m
5.2 EMPIRICAL EVALUATION
The last part of the section presents our quantitative results. We evaluate two of the most common and popular message-passingg architectures, namely GIN (Xu et al. 2019) and GatedGCN (Bresson & <Laurent] 2018). In addition, we include a MPNN variation using a virtual node (Southern et al. 2025) and three versions of GPS (Rampášek et al. 2022) as representatives of graph transformer models: one that augments node features with random- walk structural encodings (RWSE); one that relies on Laplacian positional encodings (LapPE) instead; and another that has no additional in- formation (Base). To provide some initial results, that practitioners can compare their methods against, we carefully train and evaluate these models on the easy and hard variants of LRIM- 16 and LRIM-32. To ensure rigorous evaluation, we perform model selection on the most important hy- perparameters: the model readout, hidden dimension, and number of layers. The number layers proved to be the most impactful parameter, intuitively expected, which we chose centage of the system size. For more details on the model selection and training of the baselines we refer Appendix E. We report the mean test LogMSE in Table with standard deviations computed over 3 runs. On LRIM-16, we observe a consistent difference between the easy and hard variants. The message-passing based models, which require substantially less computation than graph trans- formers, perform worse. This highlights the importance of comparing models with respect to their computational complexity, as this can significantly impact the results and incentivize future work on the trade-offs between computational requirements and performance.
On the larger datasets, we expect the situation be worse. In this setup, due computational constraints, we do not directly train models but rather evaluate the transfer ability of LRIM- - hard baselines on the larger hard versions of the benchmark: LRIM-32, LRIM-64, LRIM-128 and LRIM- 256. These transferability results are reported in Table 3 We note that there is smooth increase in the test error as we evaluate all models on systems of larger sizes, as expected. The LogMSE of the message-passing architectures tends to saturate, possibly because the relative error is quite high. Using a naive implementation of the attention, which requires quadratic memory, one cannot perform inference would require around 160GB of VRAM. If we compare these results with the oracle predictions of Figure it is clear that the graph community still has long path ahead in designing novel models
the largest instances
with a batch size on an A100 80GB GPU, as it
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Table 3: We evaluate how well models trained on LRIM-16-hard transfer larger systems with- out additional training. The results show that performance generally degrades as system size in- creases, GPS variants encounter out-of mumoof- (OOM) errors on LRIM-256 even for inference- only, demonstrating the importance of considering the scalability of long-range methods.
Preprocesing
Computation
LRIM-16-bard
LRIM-52 hand 1
LRIM-64-harl 1
LRIM- 128 hand
LRIM-256- hand
GIN
OZL Ey
-2,406 +
-1.043 =
.0.774 + cun
-0.703 +
-0.03
GateGGCN
O
-3,919
-1.050
-0.781
40.708 +:
.0.952
GaeIOCN-VNN
OIN)
-3.756
-1.054
-0.968
GPS-Base Base
OVL-Na
-4.340 1: am
-1.057 0.0
-0.790 com
-0.719 nox
OOM
GPS RWSE
Olk N²)
-4.345 4: usa
-1.057
00.90 10
-0.716 4: uno
OOM 00M
GPS-LapPE
Oue-
OVL-N9)
-4,218 1
-1.053
-0.785
-0.716 1
that efficiently and effectively capture long-range interactions, even relatively regular graphs such as the grid.
To investigate the impact of the receptive field, we perform ablation study on the number of layers across all architectures. In Figure 4 we reuse the best configurations from Table 2 and retrain the models by using between to 32 layers. Note that the datasets have diameter of 16 and 32 spectively, therefore, we would expect improvement up to that point. We observe that performance initially consistently improves with increased depth. However, the performance flattens after roughly 10 and 16 layers, respectively, on the different system sizes. While GT baselines seem to exhibit a slight edge compared to the MPNN basclines. They see the whole graph immediately due to positional encodings and attention mechanisms. For the message-passingg architectures, this may be partly due to well-known problems such as computational bottlenecks and vanishing gradients, which hamper the ability of these architectures to properly learn a good representation. For the graph transformers, it may tied to a too unrestricted attention mechanism. We provide an addi- tional ablation investigating scaling the number of parameters used for the GatedGCN baseline in Appendix E.4
Notwithstanding these considerations, what we want to stress here is how our LRIM benchmark has uncovered, in a rather explicit way, that both gessage-paasiigg and graph transformer architectures suffer from limitations when modeling this kind of long-range problems, and that our contribution stands as a novel, theoretically, and empirically grounded tool towards developing and evaluating more capable long-range techniques for graph learning.
LIMITATIONS
Our proposed LRIM benchmark a synthetic dataset with a well-understood and well-defined task, and as such it cannot be directly tied to real-world problem. Its primary purpose and main advan- tage, as argued throughout the paper, to provide an understandable and controllable framework to assess long-range capabilities rather than to solve open problem. As such, we do not intend LRIM as a replacement for real-world benchmarks, but rather as a complementary tool for advancing the study of long-range interactions for graph learning. Furthermore, our current benchmark is limited to regular lattice structures, which may not capture the diverse topological patterns encountered in general graphs. Future work may incorporate other structured graph types, but this would require careful consideration of how to properly obtain appropriate simulated data (particularly, finding the critical temperature for such systems) and the accompanying long-range analysis.
CONCLUSION
With current graph learning benchmarks, it is difficult to properly isolate and assess the ability of models to capture long-range dependencies. To address these ambiguities, we introduced the Long- Range Ising Model (LRIM) Graph Benchmark, a physics-based benchmark based on the power-law interacting Ising model that relies on controllable and provable long-range dependencies. We can precisely vary the hardness of the task across datasets, which scale from 256 to 65k nodes. In addi- tion, we provide model-agnostic evidence that LRIM tasks genuinely require long- range reasoning, with oracle prediction degrading when information is restricted to local neighborhoods. This is further supported through theoretical insights using recent long-rangsdness measures. Our empir-

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GPS-LapPE
-2
AE log MSE 4
GPS-Base
GPS-RWSE
GatedGCN-VNG
GatedGCN
-6
GIN
Sim. Oracle
-8
-10
2
6
10
12
14
16
Layers
0
GPS-LapPE
-2
GPS-RWSE
AE log MSE
GPS-Base
GatedGCN
GatedGCN-VNG
-6
Sim. Oracle
-8-
-10
20
32
Layers
Figure 4: LogMSE (4) of the layer ablation study plotting performance function of number of layers. Oracle performance is clamped at -10 for visualization purposes. Note, each combination of model and layer a separate trained model instance. On the top, models are trained and evaluated on LRIM- 14MM-16-hard 16-dard, while on the bottom they are trained and evaluated on LRIM-32- hard. All models consistently improve with increased depth but plateau with significant gap remaining to the oracle predictor.
ical evaluation reveals large gaps between current methods and oracle performance, highlighting limitations in existing graph learning approaches when confronted with provably long-range tasks. Our aim is that the benchmark will contribute establishing foundation for developing, properly evaluating, and advancing our understanding what architectural innovations are needed to tackle long-range dependency modeling graph-structured data.
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A
ORACLE PREDICTOR
A.1
ALL
LNN 128 hard
LIIN-128-ewyy
hord
casy
1Hg MASE
-1.
LSN-128 hard
LIN-122-4ady
LRHH 54 hard
LRIN 30-essy
-2.0
12M-15-brr
-2.5
-3.0
eayy
3.5
casy
4.0
10
50 130
150
250
250
Number Hops
Number of Hops
Figure 5: LogMSE (4) performance of the oracle predictor across all LRIM datasets when restricted to r-hop neighborhoods. (Left) indicates the first 10 hops, while (right) shows performance across the graph diameter. Performance degrades smoothly neighborhood size decreases. Further, harder variants consistently require larger neighborhoods than easier variants for the same accuracy. Fi- nally, larger sizes also increase task difficulty even for fixed a. The oracle uses the true underlying Ising energy function but only considers spins within hop-distance providing model model-agnostic evi- dence that our benchmark tasks genuinely require long-range information across substantial parts of the graph.
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A.2 DOUBLE
1.0
LaR-1b-bard
LRR--B-oard
LRN-16-aay
LAM-32hhrdd
LRR 32 hard
LAN-32-00
-1.0
91g VISE
-1.
2.0
-2.5
3.0
-3.5
-33
-4.0
Number Hops
Number of Hops
haard
LRN 32 hord
LRW--6-haml
LRM casy
LRB 64 casy
AE 1og NSE
-11
-32
-12
Hops
Number Hops
1.0
120--120-wwrl
LRIM 128 casy
LRIM hard
LIIM sa-h-----arrd
18114-55-eeay
LRIM
-3.5
AF MSE
ogg MSE
-1.0
3.5
-2.0
-11
-12
-33
3.0
20
120
Numbor Hops
Number of Hops
1.0
URIM hard
UNN 128 hard
0.5
128--aay
LRIM 256-hamd
URM 256 hard
3.0
URIH 256 casy
3.5
1E11g MSE
-3.0
-1.5
-2.0
-30
-11
-2.
20 -13
-3.0
50 100
150
200
250
Number of Hops
Number of Hops
Figure 6: LogMSE (J) performance of the oracle predictor across size adjacent LRIM datasets when restricted to h-o neighborhoods. (Left) indicates the first 10 hops, while (right) shows perfor- mance across the graph diameter. Performance degrades smoothly neighborhood size decreases. Further, harder variants consistently require larger neighborhoods than easier variants for the same accuracy. Finally, larger sizes also increase task difficulty even for fixed a. The oracle uses the true underlying Ising energy function but only considers spins within hop-distance r, providing aonel-agnostic evidence that our benchmark tasks genuinely require long-range information across substantial parts of the graph.
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A.3 SINGLE
1.0
LaM-1b-hard LRM-16-ear
LRR--1-oard LRM--66-00
-1.0
91g MISE
-1.
2.0
-2.5
3.0
-3.5
-12
-33
-4.0
Number Hops
Number of
32
LRN 32 hard
LRB--2-44y
AE 1og NSE
-11
-32
-12
Number Hops
Number Hops
1.0
LRN-64-eary
LRM 64 casy
-3.5
AF MSE
ogg MSE
-1.0
3.5
-2.0
-11
-12
-13
3.0
50
Numbor Hops
Number Hops
1.0
URIM hard LI0M-1266-eam
UNI 128 hard
0.5
LIIN- 128-e-yy
3.0
3.5
RISE
1E1ig MSE
-3.0
-1.5
-2.0
-30
-11
-2.
-13
-3.0
20
20
60
30 100
120
Number of Hops
Number of Hops
Figure 7: LogMSE (1) performance of the oracle predictor across a single LRIM dataset when restricted to r-hop neighborhoods. (Left) indicates the first 10 hops, while (right) shows performance across the graph diameter. Performance degrades smoothly as neighborhood size decreases. Further, harder variants consistently require larger neighborhoods than easier variants for the same accuracy. Finally, larger sizes also increase task difficulty even for fixed a. The oracle uses the true underlying Ising energy function but only considers spins within hop-distance r, providing model-agnostic evidence that our benchmark tasks genuinely require long-rang information across substantial parts of the graph.
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B
ISING MODEL: BACKGROUND AND SIMULATION DETAILS
1.0
- 16
L 32
0.8
L 128
256
T-11
0.6
[m]
T=5
0.4
T=7
0.2
T-13
0.0
4
10
12
14
T
Figure 8: Absolute of the magnetic order parameter per spin [m] = 1/L2 versus temperature T' for the LRIM with = 0.6 for different system sizes L as noted in the figure. The snapshots show respective configurations from simulations for L 256 at the mentioned temperatures. The plot shows the transition from ordered configurations at small T to disordered configurations at large and the associated shift transition temperature for different system sizes. The pseudo-critical temperature T, marks the temperature at which the spins are highly correlated.
The (long-range) Ising model has three distinct phases depending the temperature T;; it is dis- ordered above Te, critical at Te, and ordered below T., The transition from disordered to ordered phase is evident from the behavior of the magnetic order parameter age spin orientation the system. In Figure we plot the absolute magnetization |m] as function of temperature T for the LRIM with the system has close to unit magnetization and is ordered apart from very few thermal excitations. This is reconfirmed by the snapshots of the system configuration for T 5 and T 7 in the same figure. For large temperatures, |m| is approaching zero, corresponding to random or disordered configuration.
E: & s;NN, defined as aver-
0.6 for various system sizes L. At low temperatures,
Between these two phases there is a transition, characterized by so-called critical behavior with diverging correlations the system. For our graph benchmark, we determine this point of transition by determining the pseudo-crite critical temperature for each system size. As discussed in the main text, this temperature used for benchmark, guarantees correlation between spins, as discussed in the main text.
B. DETERMINATION OF THE PSEUDO CRITICAL TEMPERATURE
We determine the pseudo-critical temperature from the location maxima of the magnetic suscep- tibility, defined as
Xm = 2m = T ((m²)r (m)
03
where N Ixx the number of spins, m = & is the magnetization per spin, and (...) symbolizes the expectation under the Boltzmann distribution at given temperature T. The magnetic susceptibility is related to the correlation function as
Xm KoT Σ C(r),
(4)
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0.6
- 16
80
-X-
16
10
64
128
128
40
2-
5.4
5.8 6.0
Figure 9: Magnetic susceptibility X = KrTN (mm² [lm]))) as function of temperature T for the long-range Ising model foro 0.6 (left) and a 1.5 (right). The positions of the maxima of x mark the position of pseudo Te. and are used to extract our simulation temperature as discussed in Appendix [..]] For a 0.6, the slower decaying system, the system size influences the pseudo critical temperature much more strongly than for the more short-range model with 1.5.
which means that the maximum of X also marks the maximum the system-wide (summed) con- nected correlation function over all distance vectors r. In practice, as usual for finite systems, we consider subtracting the expectation of the absolute magnetization [m], resulting in
x K,TN ((m²)r (mll)7)
(5)
This avoids symmetry cancellations due to the simulation of finite systems with a cluster algorithm. The peaks of Xm and X coincide in the absence of external magnetic field, thus the maximum of x corresponds to maximizing the correlations in the system.
To locate the maximum of X. we perform a fit of a quadratic function to the maximum and sur- rounding data points. Standard errors of pseudo T, are estimated by leave one out jackknife resam- pling (Efron 1992).
AE
P(AE))
0.020
0.015
0,010
0.000
-10
10
AF)
Figure 10: Visualization of the mapping from spin configuration (left) to the corresponding A AE; (middle); (right) the resulting histogram P(AF.) F.) of A F.. The presented representative snapshot was generated for o 0.6 and L 256 at the pseudo critical point.
B.2 MARKOV CHAIN MONTE CARLO SIMULATIONS
A default approach to generate samples from the target distribution for spin systems in statistical physics is the use of Markov chain Monte Carlo simulations. For this, one (randomly) proposes a change to the system and accepts the proposal in such way that the stationary distribution of the samples follows the target distribution. Specifically, for the LRIM, one proposes a spin flip at random location i, and accepts the proposal with Metropolis probability given by
= min (1, exp (- 2kEs/kgT)) with AE, = S; Σ J.y..
Calculating AE; involves N terms, where is the number of spins. Thus, a sweep consisting of N updates, has complexity O(N2). This is in contrast to short-range interacting spin models,
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for which a sweep has complexity O(N). Pseudo code implementation of standard Metropolis MCMC simulation for the LRIM is shown in Algorithms and 2 Figure I0 demonstrates the mapping from spin configuration 8; AE, for single snapshot of the system. The last plot shows the normalized histogram of P(A F2²) of energy difference values at the critical point, which has a clearly non-trivial ton-Gaussian shape.
Algorithm 1: DeltaEnergy x, y)
Input: Interaction strength J RLXL, spins s (-1, +1) site (ax, v)
4 0:
for 0 L do
for v = 0 to L 1 do
Ar min( | 2.. x ul), Ay min(ly u/. 11- ol):
h += Jr, Ay] u,,
m):
AE t 2- s|z yl h; return A E;
Algorithm 2: MetropolissonteCartoo
Input: J. 3. number of samples N, RNG seed R Initialize RNG with ft; initialize s| = +1 randomly; for k 1 to N do
for t to I do
Draw (x, y) uniformly from
11:
AE + DeltaEnergy AtSite(J, y):
& + min(1, exp( -8 AE));
Draw r Uni 1);
if < a then set s|x, v) + ssxx,
Such local updates for the LRIM are thus very costly to perform since all spins interact with each other. In addition, the critical point, one has critical slowing down, i.e., the autocorrelation times grow strongly, often exponentially. This is due to the divergence of the physical correlation length that slow modes of the Markov chain must decorrelate across. Non-local cluster updates that gen- erate configurations that decorrelate quickly at the critical point are hallmark of statistical physics for simulating short-range (Swendsen
Wang 1987) and long-ran mange (ferromagnetic) spin mod- els (Luijten & Blöte 1995). We here utilize the formulation of Flores-Sola et (2017) to simulate the LRIM. A single cluster update then consists of the following steps:
1. A spin s; chosen at random as the initial seed and put on a stack. Flip the spin orientation
+ -8,
2. If the stack is not empty, pop the top element; otherwise, go to step 1) to construct the next cluster.
3. Generate a number of bonds n, be considered from Poisson distribution Pois(2A:) with
A = Ejzi Ju/keT.
4. Pick the n, bonds categorically according to their probability P; AAA Ae. where Xij
2J/z/ /kgT. This can be done in O(1) using the Hlas-Walker method WWlker 1974).
5. If spin sj # st. put sy on the stack and flip sj.
6. Go to step 2).
We also provide pseudo code implementations of this in Algorithms and 5 Algorithm the pseudo code to construct the Alias Walker method is shown, and Algorithm 4 demonstrates how to sample an event according to its probability from it. Algorithm demonstrates the implementation of the full cluster based simulation.
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Algorithm 3: BuildAlias(I)
Input: Interaction strength J E RLXL
Output: Alias Walker tables (alias, prob) over I,2 offsets; X Σis J
Alias-Walker
A E/ Ej=o J[i,j]:
Flatten offsets (i to indices u E (,,..., I?2 1) and set Pa
Mal T.2:
Initialize two worklists: S {u Pu 1), L {u Pu 1);
while S and not empry do
Pop u E S and E L;
prob[u] < Pu; alias[u] <- u;
Pu < Pv (1 Pu);
pe S 1 then move to S;
else keep v in
return (alias, prob, A);
Algorithm 4: AliasSample(alias, prob)
Draw Unif ..... L 1}. Unif[o, 1];
if r S probfu] then w t-
else < alias/u]:
Map w back offset (Ar, Au) on (O..... I 132:
return Ar. Ay);
Algorithm 5: ClusterMonteCarlo
Input: Interaction strength J. inverse temperature 3, number of desired samples., random seed R Initialize RNG with R; initialize spins s(i,j] randomly to +1;
(alias, prob, A) + BuildAliassD);;
for / samples do
Draw seed location (zo, vo) uniformly from {o,..., I. 132:
Flip spin: a|zo, yo] + - yo):
Initialize stack S - [ wo]];
while + 9 do
Pop (Ir, y) from S;
Draw n Poisson(As]:
for e 1 to do
(Ax, Ay) +- AliasSample(alias,, prob);
v) ((r + Ar) mod L, (y + Ay) mod L);
if s s yo] # s(u, v then
Push (u, v) to S:
s s(u, v) < ssu, u):
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C
DATASET STATISTICS
The diameter
D
max (.EEV(()
d(u, v)
of graph G is defined as the maximum shortest hop distance between any two nodes in the graph. For periodic grid graph of size N, the distances can be calculated using the manhattan distance and the diameter is VN.
The average shortest path of graph the average shortest hop distance between two nodes in the graph. For periodic grid graph of size N = n n it is given as 2(n
SP N(N 1)
E
E
d(u,v)
(EV(G) (G)
Σ
d(u, v0)
topology of all nodes is the same manhatten distance
N(N - 1)
(EV(G)
1
Σ
min Ux + min uy Uy
(N 1)
vEV(G)
2
Σ
min Un, n1 u
symmetry and linearity of coordinates repeated summation of each row
(N - 1)
(EV(G)
1
2n
minf& i, n - i}
(N --1) 1)
T=1
3
1
2n
2 Σ
i
7
(N 1)
2
i=1
1
111
11
2n
1)
(N 1)
2
n
2(n² - 1)
The effective resistance was calculated using the networkx implementation following the Kirchhoff (Elens et al. 2011) and normalized by the number of edges.
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hard to 0.63
easy (p- 1.5)
LRIM-32
-10
NIVESOOO 90905
LRIM-128
A.reeg
A21
Figure 11: Histogram of the ground-truth Energies across the various LRIM datasets (first 1000 samples), binned into 50 bins. From top to bottom are the different system sizes 16,32,64, 128 and 256.
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LRIM-1G-hard
LRIM-32-hard
LRIM-16-easy
Maximum Range 1-WL
LRIM-32-easy
8-
6
Number of Hops
Figure 12: Maximum range of AE values among nodes sharing the same 1-WL equivalence class when restricted to r-hop neighborhoods. Nodes that indistinguishable to message- paassagggpssing neural networks cam initially exhibit vastly different energy targets, requiring expanded receptive fields to resolve these ambiguities. However, they decay much faster than the oracle predictor (Figure P indicates potential overfitting with finite data, though generalization likely requires distances closer the oracle. However, this also means that inherently, 1-WL in distinguishability should pose a major obstacle for MPNNs on our Benchmark.
D EVALUATION
In this section, we discuss how easy it would be for MPNNs to fit the training data. The Weisfeiler- Leman (WL) (Leman Weisfeiler 1968) test of graph isomorphism provides a theoretical frame- work understand the expressive power of MPNNs. Since classical MPNNs equivalent to the 1-WL test, we use the test as a proxy for MPNNs' fitting abilities. By this test, nodes within the same WL equivalence class must be associated with the same outputs, hence they cannot be discerned by one another. We compute 1-WL. labels for all nodes in our datasets up to iteration k, creating equivalence classes of nodes that are indistinguishable to k-layer MPNNs. For each equivalence class, we measure the range of AE values between the nodes, which indicates nodes with very different energies are clustered together. Figure 12 shows the maximum range among all equivalent classes depending on the size of the considered neighborhood. As we can see, the maximum range is not negligible initially, but decreases fast as the neighborhood increases. This results highlights two important insights: First, there are nodes with similar neighborhoods that have very different prediction targets our datasets. This is desirable as it requires information beyond the immediate neighbors to distinguish these cases. As consequence, there is inherent drive towards an increased receptive field, which also necessary to capture long-range dependencies, in order to uniquely shatter the equivalence classes. Second, the curve drops off faster than the analysis of the oracle predictor. That is, because of the finiteness datasets, there exists potential shortcut approximate the true training prediction with fewer than the minimum required number of layers. However, it is crucial to note that this analysis does not provide conclusions about how well models can generalize beyond the training data. In fact, have shown that the number of layers depends on and loosely follows the oracle predictor. We want to make practitioners well-aware of this discrepancy between the number of layers required for fitting the training data and those required to approximate the true target function.
known to be
Lemma D.1. Let fo be a model that predicts AE for a given node on an instance X with periodic grid graph G of size N n x n with diameter n. If Sa only considers the spins of nodes within radius r E D, then there exists configuration X' where So So(X), fo(X"), but IY' - fo(X))l n
Proof. We construct two candidate instances X1, XY, which have the exact same spins as X within radius and are all -1, respectively +1 outside of that. The error of any prediction will then be at
23

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Published as a conference paper at ICLR 2026
least 2 12,
y1,-
222=
Σ #1 L v) (2+g)
uEG
Σ MEG
T2 22 2, d(u, 2) -2t2) (2t0)
def. of AE
Σ uEG,d(u,)>rr
L, (2+0) ()+20)))
E
22 ((u, v) -(2to)
vEG,d(u,))or
E
(o/,u 22 -12t0)
=EG,d(u,o)2r
= =201,y NEG,d(u,))rr
Σ
((t,))-(2t))
1
|Y,
12,1=
Σ
d(u, v) -2+t))
2
uEG,d(u,e)>r
1 (2+0)
2 (N ,2)
n
2
n
2+0
2 n
n.
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E EMPIRICAL EVALUATION
E.
CODE AND TRAINING
All our experiments are implemented using pytorch and the pytorch geometric library (Fey et al. 2025). Our code builds on top of a commmon codebase originated on top of graphgym and further used by TraphGPSSRRampäee
et al. 2022). GRIT et al. 2023 and Benchmarking Positional Encodings (Grötcchla et al. (202)). The experiments were conducted on a variety of GPUs including: RTX 3090 (24GB VRAM), RTX 2080 (12 GB) and Titan RTX (24GB). Furthermore, we used an A100 GPU with 80GB of VRAM for the inference transferability ablation. Unless stated otherwise, the reported model performances are taken over three different model seeds and model selection is done for each combination of model architecture and dataset.
All models are trained using the MSE loss for at most 2000 epochs using the AdamW (Loshchilov & Hutter 2017) optimizer with early stopping of 200 epochs. Further, we limit the compute budget of each model to be at most 48 hours. We use a weight decay of 10-6 and a cosine scheduler with warmup epochs and gradient clipping (12 norm of I).
E.2 MODEL SELECTION
We follow the systematic hyperparameter optimization approach from "Where Did the Gap Go?" by Tönshoff al. (2023) starting with default configurations and conducting linear hyperparameter search. Initial experiments on batch sizes and learning rates showed minimal impact, so we excluded these from further optimization due to computational constraints. On the following hyperparameters we performed linear sweeps to determine the best option. Number of layers, hidden inner dimen- sions, and pre/post-pros prrposssprocssiing layers. Note that for pre-post_mp corresponds to an ML.P with one hidden layer. For each hyperparameter combination in the linear search, we conducted single train- ing run and selected the configuration with the lowest validation LogMSE. The best configuration for each architecture-dattee pair was then trained with three random seeds to obtain the final results reported in Table 2
Table 4: Hyperparameter search space and selected configurations for all baseline models. We perform model selection over key architectural parameters (layers, hidden dimensions, readout) LRIM-16 and LRIM-32 datasets. Bold values indicate the best-performing configuration selected based validation performance for each model architecture. The number of layers was identified as the most impactful parameter and was chosen as a percentage of the system diameter to ensure fair comparison across different graph sizes.
Dusasct
Puram
GIN
GuseGGCN
GaIdGGN- VNS GS
GPS Base
GPS RWSE
GPS LapPe (2. 5 2
52,1.
layesa_post post..p .ap layeza pao ip dim Inner layeza
12, 3)
(2. 5) [0.2 2
2.3)
(2:3)
(2.3)
[0.2]
[.2)
(0.2)
[0,3]
[30, 64, 128)
[52. 64 1283
[30 64. 128)
(92. 64. 1983
[64, 128:
[30, 64, 1283
(1.8, 16, 24.32)
(1. 16 24, 52)
(L. 16, 24, 92)
[1. 16, 24. s0)
(1.8, 16, 24 say
{1. 16, 24, 92)
Inyers-posturp
22.
31
(2. (2.11
ieyersereeтp dinginner layeza
10.
25
[0.2]
10.33
[30, 128)
[52, 128)
(32, 128)
(32, 128:
128)
(52, 128)
(1.8,
(1. 16, 22)
(1.8. 16, 323
(1. 16, 32)
(1.8, 32)
(1, 16, 22)
16.1.5
layeza post.arp layezo.pro.np din-inner layeze
00.2
00.23
00.2
128)
128)
128)
(32,
128)
128)
11. 201
[1.
21.
201
16i 16,0.
layess.post.ap layera,pes din layers
3)
[0.3]
[32, 128)
129)
(32.
[32, 128)
64.
[1. 16.
16.
12,
16.
(1.4,3 20)
(1. 12.
12.16.20
E.3 MODEL ARCHITECTURE
We make use of the GraphGPS implementation for all our GPS and MPNN variants. As depicted in Figure 13 the GPS framework mixes local and global message exchange while using appropri- ate skip-connections, normalizations and feed forward networks similar to the design proposed in Vaswaan et al. 2017). For the RWSE, LapPE or Base variants, the positional encoding part was appropriately modified. The computational cost of computing the LapPE encoding using only k eigenvalues and eigenvectors from Kreuzer et al. (2021) is denoted as O E). although com- puting the full spectra would be more expensive. SSimilarly, the cost of computing the RWSE is
25

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Published as a conference paper at ICLR 2026
O N22as discussed in Zheng et al. 2025). Whereas both the positional encoding and trans- former blocks were skipped for the MPNN implementations where the Local MPNN was either a GIN GatedGCN.
For the GatedGCN- G baseline, we follow the formulation of Southern et al. (002)), which was built the same GatedGCN baseline. Similar to the other paassagepasssing baselines, then capsulate the GatedGCN-VNG formulation (as the Local MPNN) inside the GPS layer formulation without any attention blocks. In the following we restate the formulas from Southern al. (2025) for the general case.
(6) (1) agg (0 10 :j E N..)). 1,10c = upl
(e+1)
((h)
(11)
(e+1) = upya (t)
yu aggva (4) (h j,0cc :j v3))
(t+1)
(12)
h
(e+1)
up (8)
h (1+1) b((+1)) [. >
(13)
And the specific implementation used in combination with GatedGCN.
h
(2+1) Line
=0 120n
(9
(6)
JEN.
h (f+1) ((+1)
-h, + Mean (Q
(e(tt))
(ett))
joon hev):
(e+1)
where (b) h)) o Iw + w!
26

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Output
Output
Node Decoder
Node Decoder
Add & Norm
Add & Norm
Feed
Feed Forward
Forward
Lx
Add Norm
Add & Norm
Add & Norm
Local MPNN
Local MPNN
Global Attention
Positional
Node
Node
Encoding
Encoder
Encoder
Input
Input
Figure 13: Baseline model architectures used in our evaluation. (Left) Graph transformer (GPS) combines local message-passing with global attention, allowing nodes to directly attend to all other nodes in the graph. (Right) Message-passing neural network (MPNN) rely on local neighborhood aggregation through multiple layers expand the receptive field. Both architectures use the identical overarching block structure including normalization, skip-connections and Feed Forward Networks stacked L times. Figure design is inspired by the overview of Vaswani et al. (2017).
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E.4 SCALING ABLATION
26
64
256
26
64
256
AE log MSE 4
-1
256
GatedGCN Layers
GatedGCN Layers
GatedGCN Layers
26
GatedGCN Layers
GatedGCN 16
GatedGCN 32 Layers
64
10a
10
10
10
Number Parameters
Figure 14: Scaling behavior of GatedGCN models with 1,2,4,8, and layers on LRIM-32-hard The plot shows the logMSE (4) as function of the number of parameters across model widths. Each marker annotated with the corresponding hidden dimension and represents single seed run configuration.
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Table Performance measurement of simple CNN baselines on the LRIM datasets. Each CNN con- sists of L stacked 2D convolutions of kernel size K. They are either using circular padding (Circular- All baselines are trained on the 1--M-16 dataset. Addi- tionally, we provide inference results of these models on the LRIM-32-hard and LRIM-64-hard dataset, on which they were not trained, indicated by We report results averaged across three seeds.
CNN) or zero padding
Model
Kernel
Layers
Receptive Field
IRIM- 16-hard
I.RIM-3--
3a-M--2 4
hard 4
Circular-CNe
3
0,2508 1:
0.3284
0.000
0.3506 1:
-0.4437 + c0002
-0.2521
+ azom
-0,1946 + tuxce
-0,9829 1: 50000
0065700
0.00
-0,5336
11
-1.9050 + soce)
-1,0236
-08833 + 0000
-0,4444 1 2000s
+0,2530
0.000
-0.9533 1: Duxcs
-1,4562 + 5-00006
-0,9099
+
-0,7344 +
13
-2,4117 + exe0s
-1,0888
6.00
-0.8184
+ 6004
-28905 3.0%
-1,0552
-0.8199
-0,9504 + 50000
0.6584
4.0m
-0,5537 +: 01005
-2,4080 + 20085
-1.0484
0.947
-0.8185 - 00003
19
.2,3319 +
-1,0552
+
-0.8227 +
31
-2,5400 + 2.129
-1,0501
+ 0.000
-0,8130 + EUROS
Grid-CNN
3
0,4088 +: 20002
0,4015
0.200
0.3854 + DUM01
0.0209 1
-0.0077
4.00
-0.0728 1:
-0,1443
+ 3000
-0,2309
+
-0,3254
+ 0000
-0,2498
30001
003379
0.0s
-0,5019
0,0231 + 5:00
4,0189
-0,0707
+ 01003
-0.2140 + 20002
-0.3306
0.0027
-0.4510
= 00063
-0,2638 4-sre
-0.3977 4.0.00
-0.5254 +
21
-0.2878 + 2xem
-0,4300
+ 9.50
-0,5788 + E00M
7
-0.1407 +
-0,2320
-0.3273
13
-0,2730 + 50007
-0,4095
+ 0.00
-0,5391 +
-0,2909 + 2003s
-0.4312
aoos
-0.5776
+ 00020
-1.0897 +
-0.0464
+ 0.00
-0.1146 :
E.5 COMPUTER VISION ABLATION
Our main intended application for the datasets of the LRIM Graph Benchmark is to test graph learning methods. However, because the topology consists of 2D lattices, in principle other meth- ods can be used as well. To show that it is viable in principle, we include small ablation using proof-of-concept formulations of CNNs and ViTs. We would like to point out that the main aim of benchmark is to provide testing bed, or impactful research tool, to precisely study long-range capabilities in graph learning. The aim is to uncover and specify obstacles that need to be studied more closely. Of course, testing methods specifically for grid-structured data is possible Because data generation known, (by giving more information, or additional grid bias, or re-using the oracle) in order to obtain better scores. What we believe to be more insightful is to develop and test general graph purpose methods and then quantify their capabilities our provided benchmark.
well. possible to test methods that are more and more aligned
The used CNN and VIT baselines as well as their training setup is very minimalist and is to be taken as proof-of-concepl rather than fully optimized baselines. To represent the cyclic grid graph as an image, we duplicate the spins to have 3 channels. All models were trained for 200 epochs using the Adam Optimizer with a learning rate of 0.0003, batch size, and hidden dimension of 64. The CNN architectures consist of a 2D convolution which maps the input to the hidden dimension. Then L 2D convs are applied. Finally, an MLP predicts the energy values for each node. The ViT (Dooviiskkiy et al. 2021) first splits the grid into square patches, flattens each patch using linear encoder, then an Encoder Transformer is used for L layers in combination with sinusoidal positional encoding. The tokens energy values. We report all results of the CNNs in Table and in Table for the VIT. Additionally, there is visual depiction of the CNN results according their size of the receptive field in Figure
then converted back into patches before an MLP is applied predict the
15
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Table Performance of simple Visual Transformer (ViT) on the LRIM datasets. Each VIT slices the circular grid into patches of size p, then L encoder transformer layers applied using 2D sinusoidal positional encoding. report results averaged across three seeds.
Model
Patch Size
Layers
Width of Receptive Field
LRIM-16-hand
VIT
16
-1.6473 1
16
-1.7886 +
16
-1.5524 + 0,9822
16
-206629 t 0.1534
16
-080017
= 0ao0
16
-1.7071
1 00147
-2.1213
0.0811
-2,4185
0,438
+0,8205
0,0333
16
-1,9972
16 -18730
-2.0481 16
-05
Circular-CNN Keme!
AE log MSE
-1.0
Circular-CNh Kemel 5
Circular-CNN Kerne
Grid-CNN Kernel 3
Grid-CNN Kerne
-2.0
-25
11 13
10 21
31
Receptive Field
Figure 15: Visual depiction of the obtained logMSE (L) performance of the CNN baselines reported in Table 5In the LRIM-166 16-a-r dataset plotted function of the width of the receiptive field.
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AE log MSE J
GatedGCN
GPS-RWSE
Base
GPS-LapPE
GatedGCN-VN.
-3
Sim. Oracle
10
Layers
Figure 16: [DIRECTED ABLATION] LogMSE (L) of the layer ablation study on LRIM-16-hard plotting performance a function of number of layers. Note, each combination of model and layer is a separate trained model instance. All models consistently improve with increased depth plateau with a significant gap remaining to oracle predictor. While we would hope MPNNs to ideally closely follow the oracle as the receptive field expands, they plateau before and training becomes increasingly unstable. Note, that no such trend has to apply to the Graph Transformer as they have access to the entire graph even using single layer. However, this comes signficant computational cost of O(LN2) rather than (I.E).
F DIRECTED ABLATION
This section considers directed variant of our proposed datasets, where edges are directed towards down and right direction, but not the other way around. This drastically alters the receptive fields, especially for the message passing baselines. To distinguish between the two settings we add [DIRECTED ABLATION] tag to all captions.
Table [DIRECTED ABLATION] Baseline performance on LRIM-16 16 and LRIM-32 datasets. The number of edges E corresponds to 4N in our datasets. We emphasize the importance of reporting computational complexity alongside performance results, as scalability is crucial aspect in long- range modeling.
Preprocessing
Compatation
16--166 J.
16-eaxy
LRIM.
I.RIM-1-eaay J
GIN GaledGCN
OIL E)
0.268 wae
0.161
1.198 a
0.379 =
OIL E)
-0.871 wms
-1.409 cee
-0.182 a
0.325 1: ozo
GalddCCN-NNG
O(N)
OIL E-
-1.710 wss
-2.266
0.238
GPS Base
O")
22415 Laie
-2.44
-3.098 1
22914 1.
GPS RWS-
O N²)
OUL N2)
-2.938 aaw
-2.953
0.6
-3.130 a
-3.547 + oz8
LapPE
O(3.E)
OIL
-2.758
-3.341
Table 8: [DIRECTED ABLATION] We evaluate how well models trained LRIM- 1--M-11--ard transfer to larger systems without additional training. The results show that performance generally degrades system size increases. GPS variants encounter out- uut-of-memory (OOM) errors LRIM-256 even for inference-only, demonstrating the importance of considering scalability of long-range methods.
Preprocesing
Computation
1RIM-16-hard
LRIM-S2-lanl
LRIM-644harl
J
LRIM-128-tand
LRIM-256-hmml
GIN
OIL.E) OLL E
0.268 + use
0,654 =b uan
0.763 zus
0.788 + um
0.799 can
Gaed
-0.871 4 aro
0.882 e312
0.909 wu
0.914 1
0.916 2 cai
GuNeIGCN-VN,
O(N)
OVL 8 N-
-1.710 4:
1.058 ceo
1.069 mm
1,065 +
1.041
GPS-Base
-2415
-0.50
GPS RWSE
Olk-N") )² OUe 6]
OIL
-2.938 1
-0.66 cmp
0.659 1001
-0.589 J
0OM
GPS- LapPE
O -V2
-2.758 1 ama
-0.289 1 ogs
0.000 10
0.245 1
0OM
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G RANGE OF THE ORACLE AS A TASK
G.1 ANALYTICAL EXPRESSION OF LONG-RANGE MEASURE
In this subsection we provide the derivation Proposition 5.2 and its analytical expression necessary for performing the experiment. We first give an analytical expression of the range measure for "gradient" of the oracle. Recall that in Bamberger et al. (2025), the range measure of node-level differentiable function F for node denoted by pu (..).,, is defined as
(x))),
Pa(F) Σ
do(u, v).
dr,
NEV
Here, Ty the node feature of and dg
- a distance function on a graph = (V, E).
Consider periodic grid graph G of size I, x L to each of whose nodes is assigned an O(1) spin. even and L when L is odd. We consider AE as
The diameter D of G is equal L when F and spins as the node feature of nodes. Hereinafter, we will assume that spins {s:} are embedded into the real line and consider each spin be a variable in Let also (i) be the set of nodes whose shortest path distance from i is shorter than equal to Recall that the oracle for node i with spin s; defined as
AE
Σ
Jij.
(ENg.(1)
Note that we set J.k O when l: = as convention.
Graph shortest path distance: We first give Proposition G.1 the expression of the adapted long- range measures in the case when de V x - IR is the graph shortest path distance of G.
Proposition G.1. Let G be periodic Ising model defined over grid of size L x L, with F = AF. Assume that the binary spins 8; are relaxed to continuous spin values s; E R. Then, when de V x V + R is the graph shortest path distance, the range measure Pi(AE) of a node-level differentiable function AE for a node and its normalized measure (A(AE) are represented
P((AF) Eoster EmENE(O) Jsk
(( E Jik,
P((AE)
OSEST kEN((i)
in which Ne(1) the set of nodes whose shortest path distance from is equal to l.
Proof. The gradient of AE, with respect to spin sy at node ks is
Σ
syJag.
=
OΔE
JENg
(8)
ask
s;Jij.
if = E Nsr O.W.
0,
Then, the measure is
((EE) = Σ
OAE
do(i,))
(9)
Osk
KEV
Ju (i,
(10)
(E
When di k) is the graph shortest path distance, we can group nodes based on the shortest path distance, whose resulting set is Ne(1) for each do(i, k). This completes the proof.
In order to compute ((EE)) and ((EE)) efficiently, we will give an analytical expression of Juk Let H be an Ising model with size (L + 1) x (L + 1), that gives rise to G by iden-
(6)
tifying each node boundary segment of H with their respective other side. Without loss of
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generality, consider that the graph H is injectively embedded into the lattice (c R2) in way that the 1 distance of every pair of neighboring nodes is 1. For S Ve
S D, let also
(m) V V (8 m)-
0 S m S L.
When assuming the coordinate of (0, 0), for each node i in H, there exists at least one (f, m) z² such that [r -ral (m)) and therefore
Jek (f(0(m) (d+a)
Note that the distance rr; r&l which employed for the definition of f (m) and Jak. is the L Euclidean distance through the mapping from H to G.
Theorem G.2 (Even case). Assume L is even and L 2 4. Let also j(0h (m) (f()(m) (att) Then,
1
Σ
(m),
1 15 1S2/
IsmSE
10 (m) - 23 (0).
E= D/2,
Σ Jik
4
I<mSt
10 (m) - 4j 44 (0/)) (0)-8 -8
Σ (m)
(11)
(E)
ISmSf
15mSt-D/2
+4 +4 mm
(0),
t=
Theorem G.3 (Odd case). Assume 1. (2 3) is odd. Let also (m) = (f() -(d+o) Then,
4
Σ
(m),
1 1s85 D/2,
Σ (EN((
Jik
1 Ismse
(12)
4
(m) - -45(D/2 8
Σ (m),,
15mSt
15mct-D/2
Proof. Since the system we are dealing with assumes the periodic boundary condition, the gradient is invariant to the translational action of Z i.c., the value remain unchanged with the -hift action. This implies the gradient constant across the periodic grid. Therefore, suffices to prove the argument for one fixed node.
Even case: Without loss of generality, we consider the coordinate of the center of is (0, 0). We denote the center by i.
For S < D/2, whose visualization shown in Figure 17 (a), node such that do k)
lies on one of segments defined by the intersections of 4 lines x + y =6 with arbitrary sign for the plus or minus, in which (z, y) represents the coordinate of & in the lattice 2². Since f(B (m) is symmetric (on the segments) with respect to reflection across x- and y-axis as well as y O. we get
Σ kEN(()
=4 =
Juk
Σ (m). ISmS!
For - D/2, we begin the proof in the non-periodic grid system HI; see also Figure 17 (b). We first follow the same argument as in the case of 1 S E < D/2, and get the same expression as above. When the periodic boundary condition is reimposed on H, two nodes located on horizontal and vertical boundaries of H respectively are identified with nodes on their respective other sides. Therefore, subtract two contributions, which gives the following contribution for G
Σ (EN
JAk =
Σ (m) (0)
ISmSr
For D/2 < e 1-SD-D1 D- again consider line segments defined by the intersection of c Hl with arbitrary sign for the plus or minus. We again start by considering those lines on H. but also include contributions of spins lying outside of H which correspond to dotted lines and circles in
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Figure 17 (c). We first calculate all the possible contributions the 4 segments, the total of which Among the contributions, total contribution from spins on the segments which
is 4
Σ (100
Ismst
are located outside of is
(2 (0) +8
Σ (m)
ISmEt-D/2
The first term corresponds to the contributions of the 4 intersections of the lines and the second term the of contributions corresponding to spins lying outside and boundary of H. Therefore, an overall contribution of interior spins of H (excluding boundary spins as well) that are on the segments is
4
Σ
- 4j -8
Σ LSmSE-D/2
j (m).
igmse
There remain 8 contributions on the boundary of H. each of which is identified with another bound- ary spin (as in the case of É D/2) in G. Therefore, add Σ ((0 the boundary,
and get the result.
Finally, we have 4 spins whose shortest path distance from i is D, which correspond to the corners of H. All those points are identified as one node in G. Therefore the contribution for = D is dta
V
Odd case: We consider that is embedded in way that in manner similar to the case of the even case, but the center of upper right L subgrid is at (0, 0). We denote this subgrid by G. Notice that for odd L, nodes on the left and the bottom boundaries of H are identified with nodes on the respective other sides whose shortest path distance from is shorter than that of nodes on the left and bottom edges. Therefore, from the non-periodic grid G' accounts for all contributions for P. (AE)..
ignore all nodes on the left and bottom boundaries, and nodes
For < L S D/2, we can proceed in a manner similar to that of S l < D/2 of the even case. Since is non-periodic, the argument also valid for l = D/2. Therefore, the equation
Σ kEM
Jin
Σ (m)
Ismse
holds for S vl < D/2.
The derivation of the expression for D/2 < f D is also straightforward; We apply to G an the even case, and we will get
intermediate argument the proof of D/2 & D- - 1
Σ
J&k 44
Σ (0 ()(0/2)(0)
Σ (m0).
15m5E
Smmet D/2
Finally, we give theoretical guarantee of threshold of a below which the range measure diverges. This result fairly agrees with the figures reported in reported in Figure 2 and Table and it suggests that in terms of the range measure defined in Proposition G.1 the Ising model could potentially retain infinitely long long-range dependency.
Proposition G.4. Given the same assumption Proposition G.1 with L. the range measures pa( (AEE) and (A)) diverge as L + oo when a < 1, and converge when o >> 1.
Proof. Without loss of generality, we set = (0, 0). Then,
(AEE) = Σ Σ
1
(13)
Ii rp dto
1365L kEN((1)
Σ Σ
(14)
I Ix|d+o
1SESL kEN(())
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(8) s
812
(c) D/2 <D<D-
(d) 1= D
Figure 17: Visualization of contributing spins on with size (L + 1) (L + 1) for L = 8(= D), depending on graph shortest path distance do k) Dotted circles are complementary virtual spins located outside of H which drawn by solid lines and circles. Red solid lines represent y =f and red solid circles spins composing (i).).
Because for each é,
(m) S f, 0 < vm < l,
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(a) 1 SES D/2
(b) D/2 <ESD
Figure 18: Visualization of contributing spins H with size (L+1) (L+1) for L. 9 depending on graph shortest path distance da(í, k) = f. Dotted circles are complementary virtual spins located outside of H which drawn by solid lines and circles. Red solid lines represent = y = =e and red solid circles are spins composing Nı(1).
get
P(AE) N
E
Σ
(15)
Fg d+o
1/ED1/2 kEN
Σ
Σ
(16)
pd+a
1S/<D/2 kEN(())
Σ 4.
(17)
Ed+a
1St<D/2
Σ
(18)
D/2
The integral test implies that when a S 1, (AFF) diverges as L - x0. On the other hand, noting that
(
2 2
C
S V m² + (E- (L-- m)2, 0 vm sl.
also get the following upper-bound:
((AE)= Σ
(19)
|r lot d+o
1585L
kEN((()
d+a
(20)
1515h
kENr(s)
d+a
V2
$ Σ
2.
(21)
2 doo
(22)
1
1
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Hence, when > (AEE) converges as L approaches
Finally, the normalization term is evaluated from above as follows:
Σ E JI = Σ Σ
1
(23)
Pa|a+o
d+a
Σ
E
(24)
ISIEL
dta
V2
Σ
48
(25)
ISKSL
-4
Σ
1
(26)
Elto
ISISL
When I, + oo, the upper bound converges for any a > 0, hence so is
Σ Jr,k
ISESLKEN(())
G.2 RANGE MEASURE PLOT VARYING O AND L
We now investigate if there is relationship between the normalized range measure and σ. Figure 19 displays the range measure values using the analytic expression derived in Section G.1] We use the same configuration as those reported in Table 2 with additional configuration for some parameters: The size of the Ising model ranges in [16, 32, 64, 128, 256], which exactly same as those in Table
1
We let takes wider values [0.3, 0.6, 0.9, 1.2, 1.5, 1.8) than the values in Table to evaluate the
impact magnitude of a onto measure. We set r to same as the diameter of the periodic grid across all the configuration.
0=0.3
d=0.9
10a
0=12
0=1.5
log (
0=1.8
50
100
150
200
250
Figure 19: Normalized node range for the graph shortest path distance metric with different sigma. The experiments are run over Ising models with even grid size. for Ising models for solid lines is identical to that reported in Table I The dotted lines correspond to additional a introduced to visualize trend the long-range metric when varying a.
Figure 19 shows clear upward trend in the measure along the increase in the model size L. This result is fairly consistent with a similar trend the measure evaluated for sparse graphs (Bamberger et al. 2025 where graph is considered to show longer-range dependency when the graph becomes sparse. We believe the trend in Figure 19 partially accounts for the performance degradation for smaller and/or larger I. as reported in Figure 2 and Table 2
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H USAGE OF LARGE LANGUAGE MODELS
We have made use of several Large Language Models (LLMs) during the preparation of this work. ChatGPT and Claude were employed assist with spellchecking, improving wording, and short- ening text for clarity and readability. In addition, ChatGPT, Claude, and Cursor were used for analyzing and explaining code, providing code completions, and generating visualizations sup- port our implementation and experiments. These tools were applied as auxiliary aids to polish the writing and streamline the development process, while the core research contributions, experimental design, and interpretation of results remain entirely our own.
I REPRODUCIBILITY
The source code that we used for our experiments as well as the data for all of the LRIM datasets is available on GitHut
For detailed descriptions of the evaluation or experimental setup we refer to Appendix
hhtpp///gittuu..cmiiooo//riii graph henchmark
38


Reviews:
[{"review_id": "0lSypWo5Eg", "reviewer": "Reviewer_6zXs", "summary": "This paper introduces a physics based dataset to measure long range modeling capabilities in graph neural networks, named as Long-Range Ising Model (LRIM) Graph Benchmark. The benchmark utilizes the Ising model with power law interactions, where the target task provably depends on long-range dependencies. The paper provides 10 datasets ranging from 256 to 65k nodes with controllable difficulty through tunable parameter that controls the interaction strength between nodes inversely. Analysis shows that local information is insufficient, theoretical study is given on long rangeness measures, and empirical evaluations demonstrate that both message passing architectures and graph transformers perform lower. The entire dataset is synthetically generated and the graphs are 4-regular and 2D grid like.", "questions": "One observation, GPS shows OOM on LRIM-256 in Table 3. Is there a possibility to include a more approximate alternative for GPS, for instance, specifically to inform the missing scores here?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- For the datasets, the use of the Ising model provides a physics based foundation where long range dependencies are mathematically guaranteed and controllable. This is unlike some prior long range benchmarks in graphs such as superpixels where the long range is not mathematically guaranteed.\n- Compared to previous benchmarks which demonstrate long rangeness of tasks using performance of different model classes, this work has elaborate analysis of the proposed dataset with oracle predictor, theoretical lower bounds and long rangeness metric.\n- The task difficulty can be tuned and is also demonstrated with examples in Figure 3. In addition, there are clear performance gaps between message passing networks and full-neighborhood graph transformers, as in Tables 2,3.\n- The proposed collection of datasets with sizes and difficulty can be used for developing long range graph networks, alongside other recent works/datasets which study this topic.", "weaknesses": "- As acknowledged by the paper, the benchmark is limited to regular lattice structures. This is significant since real world graphs rarely have such regular topology and and message passing GNNs may not be the best architecture;\nthe grid structure may favor certain architectural choices.\n- In addition, methods designed specifically for grid-like data are excluded. however, including them could inform on the necessity of graph-specific networks perform in such settings.\n- A major limitation is the real-world applicability of the datasets which the paper acknowledges.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NnbOgLjRbH", "reviewer": "Reviewer_NHz6", "summary": "The paper introduces LRIM, a physics-based benchmark built on the Ising model that provably depends on long-range interactions, addressing gaps in existing graph-learning benchmarks. Concretely, it proposes a node-regression task to predict per-node energy changes on 2D grid graphs that model different spin configurations of an Ising model. A theoretical analysis shows how the dependence on long-range patterns can be directly controlled in this setting. The provided empirical results report that graphs transformers do significantly outperform local MPNNs in this setting, although at a significantly increased computational cost.", "questions": "1. What is the numerical range of the regression target $\\Delta E_i$? Do these need to be normalized for training?\n2. Given that the graphs are currently regular 2D grids, would it be reasonable to use the same task for benchmarking vision models like CNNs or Vision Transformers?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "I think the suggested task is a valuable addition to the existing set of graph learning benchmarks to study long-range information in a controlled setting. In particular:\n1. Obtaining a provably long-range graph learning task from the Ising model is an original idea and addresses the main problem of prior \"long-range\" benchmarks for which the justification of long-rangedness was purely empirical. \n2. The reported results do show a clear separation between local and global architectures.\n4. The large graph sizes of up to 65k nodes are challenging for standard graph transformers and seem like a good test bed for developing more efficient long-range architectures.\n3. The presentation is clear and key details like hyperparameter budgets are fully provided.", "weaknesses": "The restrictions to only using regular 2D grids is a weakness in the context of graph learning, as the main feature of GNNs is their ability to process arbitrary graph structures. I think this is an acceptable weakness for a benchmark that intends to be complementary to \"real-world\" datasets, but a broader range of graph structures would ultimately be more convincing.\n\nThe set of provided baselines also misses MPNNs with virtual nodes [1] as a standard trick to propagate global information in graphs. It would be very interesting how such architectures perform on this dataset, as VNs allow for global information aggregation but lack the pairwise global interactions of transformers that seem to align well with the suggested task.\n \n[1] Gilmer, Justin, et al. \"Neural message passing for quantum chemistry.\" International conference on machine learning. Pmlr, 2017.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rmuNuNd07d", "reviewer": "Reviewer_PboE", "summary": "The paper constructs a graph regression dataset based on the well-known ising model as a dataset for long-range dependencies.\nThey show that indeed long-range interactions are needed in order to solve the problem and that deeper networks tend to work a lot better than shallow ones. They also provide an oracle value for $k$-hop GNNs which is a lot better than the trained networks, indicating sufficient complexity to be challenging.", "questions": "1. is it really \"interaction\" or rather \"aggregation\" that is happening in the ising model? Especially when it is about energy prediction as in LRIM? \n2. in the monte-carlo simulation that is used to simulate the system, I do not really understand how deterministic this is and how exactly it is used for LRIM.\n3. when going for more complex graphs\n4. How do you rate the possibility of cheating for a model based on e.g. global statistics and thus outperforming the oracle which has limited information (but uses that information optimally). And in that context, how helpful is the provided lower bound?\n5. Do you have an intuition why LapPE did not perform at all? And how it happened to have this odd curve in Fig 4?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The main strength of the suggested dataset is that it addresses one of the key problems in graph machine learning: long-range interactions. Existing datasets are often purely empirical (maybe except for the road networks from Liang et al 2025) instead of being principled.\nThe dataset satisfies a number of desirable properties such as coming in varying complexities and sizes (including large graphs with 65k nodes each) while leaving a significant performance gap between a simple restricted oracle and existing GNNs.", "weaknesses": "Overall, I am not yet convinced that the dataset is really what we are looking for, mostly because the graphs are extremely simple and as far as I understood the task, it is not that much about interactions influencing spin patterns, but rather on aggregating information from far away but in a way that is mostly independent of information that has already been digested.\n\nConcretely, there are a few aspects about the dataset that I consider not too strong:\n- it could have been a lot more clear how exactly the LRIM graphs are generated based on the background that has been described before, especially for graph-learning experts that have not worked with the ising model before. Apart from that, the paper is well-written and easy to follow.\n- The graphs are extremely simple (just lattices, even simpler than the road networks from Liang et al). Thus the task is really an oversimplified edge case for graph learning.\n- There is not that much interaction going on, especially since on a regular lattice all $J_{ij}$ are the same. \n- I did not get convinced that the construction really tests interaction and not just global aggregation (see questions).\n- The provided lower bound states that there exists a solution thats \"very different\", but says nothing about the distribution of such solutions. In particular I believe its possible to \"cheat\" using global statistics.\n\nConcrete (small) things:\n- since LRIM uses lattice graphs, positional embeddings should make looking at the edges irrelevant. (e.g. using a PE that is made for images and is able to encode an x and y position)\n- The task is really about very exact computations which tends to be not the strongest suite of machine learning models. And at some point floating-point precision will start becoming problematic (probably way before -20 where the trivial accuracy boundary is). \n- in the experiments it looked to me that a lot larger MPGNNs would have been possible without exceeding the computational demands of the tested graph transformers. Is there a reason why only \"small\" models have been tested? How does a 50M GatedGCN model perform?\n- 422: Maybe I was misreading the plot, but the numbers of oracle and learned method are not far apart for up to 12 layers (Fig 4). I do not agree with the conclusion made here.\n- 475: As soon as we know the distances, the graph itself becomes highly unimportant. So it is only partially about graph learning, i'd say.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)