Can You Hear Me Now? A Benchmark for Long-Range Graph Propagation
Authors: Luca Miglior
Matteo Tolloso
Alessio Gravina
Davide Bacciu
Venue: ICLR 2026 Poster (2026)
Keywords: graph neural network
long-range propagation
benchmark
dataset
Primary area: datasets and benchmarks
TL;DR: We introduce ECHO, a benchmark to evaluate GNNs on long-range propagation. We show that ECHO genuinely require long-range dependencies, and we establish strong baselines to provide a comprehensive reference point for future research.

Abstract: Effectively capturing long-range interactions remains a fundamental yet unresolved challenge in graph neural network (GNN) research, critical for applications across diverse fields of science. To systematically address this, we introduce ECHO (Evaluating Communication over long HOps), a novel benchmark specifically designed to rigorously assess the capabilities of GNNs in handling very long-range graph
propagation. ECHO includes three synthetic graph tasks, namely single-source shortest paths, node eccentricity, and graph diameter, each constructed over diverse and structurally challenging topologies intentionally designed to introduce significant information bottlenecks. ECHO also includes two real-world datasets, ECHO-Charge and ECHO-Energy, which define chemically grounded benchmarks for predicting atomic partial charges and molecular total energies, respectively, with reference computations obtained at the density functional theory (DFT) level. Both tasks inherently depend on capturing complex long-range molecular interactions. Our extensive benchmarking of popular GNN architectures reveals clear performance gaps, emphasizing the difficulty of true long-range propagation and highlighting design choices capable of overcoming inherent limitations. ECHO thereby sets a new standard for evaluating long-range information propagation, also providing a compelling example for its need in AI for science.

Published as a conference paper at ICLR 2026
CAN YOU HEAR ME Now? A BENCHMARK FOR LONG- RANGE GRAPH PROPAGATION
Luca Miglior Matteo Tolloso'
Alessio Gravina" Davide Bacciu
Department of Computer Science, University of Pisa, Italy
ABSTRACT
Effectively capturing long-range interactions remains fundamental yet unresolved challenge in graph neural network (GNN) research, critical for applications across diverse fields of science. To systematically address this, introduce ECHO (Eval- uating Communication over long HOps), a novel benchmark specifically designed rigorously assess the capabilities of GNNs in handling very long-range graph propagation. ECHO includes three synthetic graph tasks, namely single-source shortest paths, node eccentricity, and graph diameter, each constructed over di- verse and structurally challenging topologies intentionally designed to introduce significant information bottlenecks. ECHO also includes two weal-world datasets, ECHO-Cha rge and CCHO-Energy, which define chemically grounded bench- marks for predicting atomic partial charges and molecular total energies, respec- tively, with reference computations obtained at the density functional theory (DFT) level. Both tasks inherently depend on capturing complex long-range molecular interactions. Our extensive benchmarking of popular GNN architectures reveals clear performance gaps, emphasizing the difficulty of true long-range propagation and highlighting design choices capable of overcoming inherent limitations. ECHO thereby sets new standard for evaluating long-range information propagation, also providing a compelling example for its need in Al for science.
INTRODUCTION
Graphs are fundamental data structures used extensively to represent complex interconnected systems, ranging from social networks and biological pathways, to communication infrastructures and molecu- lar structures. Graph Neural Networks (GNNs) (Sperduti) 1993; Gori et al.] 2005) Scarselli et al.) Michell 2009 Bruna et al., 2014 Defferrard et al. 2016) have emerged as successful method- ology within deep learning, whose research community was initially driven by the development of diverse architectures capable of capturing intricate relational patterns inherent to graph-structured
data, well impactful applications across various domains (Hamilton et al. 2017 Derrow- Pinion
ct al. 202 Gravina et al. 20222] Bacciu et al., 2024] Gravina & Bacciu 2024
Khemani et al. 2024
Miglior
2025 Tolloso et al. (026).
More recently, the research community has shifted its focus towards understanding and overcoming fundamental limitations of the message-passing paradigm underlying GNNs. This shift has been driven by the observation that effectively propagating information over long distances in graphs remains significant challenge. Such challenges have been formally linked to phenomena like over-smoothing CCai & Wang 2020 Oono & Suzuki 2020 Rusch et al. (002)), over-squashing (Alon) & Yahav 2021 Di Giovanni al. 2023), and more generally, vanishing gradients (Arroyo al. 2025), all of which hinder GNN performance in tasks that require capturing long-ram longerange dependencies.
Currently, we are the stage in which such pioneering theoretical studies need consolidation, while looking into methodological advancements that can surpass or mitigate such shortcomings. key enabler of this progress is the establishment of solid and challenging benchmarks that can accurately assess and validate long-range propagation capacities. The availability of controlled synthetic benchmarks, should be complemented by the introduction of compelling application-driven
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datasets which can clearly demonstrate the practical advantages of addressing long-range propagation issues. Long-range propagation capacities, in this sense, have been noted to be central in key areas of science, such in biology (Dwivedi et al. 2022 Hariri & Vandergheynst 2024 biochemistry
(Gromiha & Selvaraj 1999), and climate (Lam et al. 2023).
Existing graph benchmarks have, instead, focused primarily short- medium- tasks (Bo- jehevski & Güinnemann] 2018] Shhhhu et al.] 2018 Wu et al. (2018] Sterling & Irwin 2013 Wale & Karypis]22066 Hu et al. 2020a] Dwivedi et al. 2023), often overlooking the unique challenges associated with distant information propagation. More recently, the growing interest in this challenge has motivated the community to develop a few benchmarks specifically designed to evaluate infor- mation propagation in GNNs. These include the Long-Range Graph Benchmark (LRGB) (Dwivedi) et al. 2022) and the Graph Property Prediction (GPP) dataset (Gravina et al. 2023). While this is a significant step forward compared to earlier benchmarks, it does not fully account for the need to capture the true long-range dependencies present some real-world applications. This is due to limited size of the graphs, the absence of well-defined conditions on the expected propagation range, and the focus of the benchmarks, which is often more aimed at specific issues of over-smoothing and over-squashing, rather than providing broader evaluation of long-range propagation capabilities. Moreover, LRGB and GPP tasks are facing a natural performance saturation, as novel methodologies being developed and optimized on them.
Motivated by this, we introduce ECHO (Evaluating Communication over long HOps), a new bench- mark designed assess the capabilities of GNNs to exploit long-range interactions. ECHO consists of three synthetic tasks and two real-world chemically grounded tasks. The former are designed to provide controlled setting to assess propagation capabilities. They comprise the prediction of shortest-path-basse graph properties (i.e., node eccentricity, single-source shortest paths, and graph diameter) across diverse graph topologies. These have been defined increase the difficulty of effective long-range communication. as they present structural bottlenecks to information flow. The main characteristic of these tasks is that GNNs must heavily rely on global information and effec- tively learn to traverse the entire graph, similarly to classical algorithms like Bellman-Foe Ford (Bellman 1958). The real- world tasks target the prediction of molecular total energy and the long-range charge redistribution in molecules, which are critical and practically relevant challenges in computational chemistry (Dupradeau et al., 2010), as they underlie many fundamental processes such as chemical reactivity, molecular stability, and intermolecular interactions. Accurate modeling of these effects is essential for drug design, materials science, and biology understanding.
Our contributions can be summarized as follows:
We introduce ECHO, a novel benchmark featuring five new tasks specifically designed to evaluate the ability of GNNs to effectively handle long-range communication in both synthetic and real- world settings. ECHO includes three synthetic tasks (collectively referred to as ECHO-Synth) with a total of 10,080 graphs, and two real-world tasks (collectively referred as ECHO ECHO-Chem) comprising between x 100,,000 and <= 196,000 graphs, where the required propagation ranges from 17 to 40 hops.
We propose ECHO- Chem, a novel benchmark consisting of two tasks (i.e., Charge and Energy) designed to capture long-range atomic interactions in molecular graphs. Specifically, ECHO-Charge is a dataset for predicting atomic charge distributions, while ECHC-Energy focuses on predicting the total energy of molecule. Both tasks are built on Density Functional Theory (DFT) (Argaman & Makov (000)) calculations, ensuring quantum- level accuracy. This makes them particularly suitable for evaluating long-range message passing in GNNs, since both charge redistribution and molecular energy depend on subtle, non-local effects. Beyond benchmarking, these datasets also address central challenges in computational chemistry. where modeling long-range interactions remains difficult and computationally expensive, as evidenced by the months of parallel DFT computations required generate our benchmark on the given hardware configuration.
We present a detailed analysis to demonstrate that the tasks in ECHO genuinely capture long-range dependencies, providing rigorous evaluation of GNNs' ability to propagate information over extended graph distances.
We conduct extensive experiments to establish strong baselines for each task in ECHO, providing comprehensive reference point for future research on long-range graph propagation.
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2 ON THE NEED FOR A NEW BENCHMARK
We now elaborate on the need for novel benchmarks specialized on the evaluation of long-range propagation in relation to existing datasets.
The most widely used benchmark for assessing these capabilities is arguably LRGB (Dwivedi et al. 2022). Its introduction in 2022 has certainly marked an important milestone and promoted the development of the field. However, despite initial rapid improvements, performance on LRGB has now plateaued, showing a noticeable deceleration in progress across the last year, as discussed in Appendix B In addition, recent works (Tönshoff et al. 2023] Bamberger et al.. 2025b) question the long-range nature of several LRGB tasks, showing that some are inherently local rather than requiring global information, and that the benchmark itself is highly sensitive hyperparameter tuning. Other henchmarks propose synthetic tasks on generated structures, including the Tree-Neighhorhood (Alon & Yahav 2 Graph Property Prediction (Gravina et al. 2023). graph transfer (Di Giovanni et al. 2023; Gravina et al. 2025). GLoRA (Zhou et al. 2025), and Barbell and Clique graphs (Bamberger
ct al. 2025a). Indeed, most of these tasks are originally designed to address narrow challenges that prevent long-rann range propagation, such as over smoothing (Cai & Wang 2020; Oono & Suzuki 2020
Rusch et al.) 2023) and over-squashing (Alon & Yahav, |021 Di Giovanni et al. 2023). These phenomena, while related, do not necessarily capture the full spectrum of challenges associated with long-range communication. Moreover, despite being designed to test the ability of GNNs to overcome these limitations, these datasets typically involve small graphs with limited-size diameters. This inherently restricts the propagation radius, creating a significant gap between the benchmark tasks and real-world problems that require much deeper propagation across significantly larger structures.
The limitations above suggest the need for a new benchmark that reflects the challenges and oppor- tunities in long-range GNN research. An effective benchmark should provide tasks that explicitly test model's ability to traverse extensive graph structures, effectively aggregate global information, and adapt to diverse topological constraints. Moreover, as the field has matured and wide range of models have been established, ranging from graph transformers (Shi et al. [021 Rampašek et al. 2022 to multi-hop GNNs (Abu-El-Haija al., 2019 Gutteridge et al.) (023) and others (Shi et al. 2023). it seems timely to introduce new benchmark that can accurately assess the long-range propagation skills of these families of models, now that they are well understood and consolidated.
ECHO addresses this scenario by a suite of synthetic and real-world tasks with clearly defined long- range propagation needs, providing a clear target for the evaluation of this property. ECHO tasks require computing shortest paths between all nodes, long-range charge redistribution,
molecular total energies, with clearly defined propagation ranges between 17 and 40 hops, depending on the specific graph structure. This explicit range ensures that models failing to capture dependencies within this span are underreaching and have poor long-range capabilities.
The
and ECHO Energy molecular tasks have strong value per se, proposing a novel, practical, and high-impact challenge for learning models in computational chemistry (Dupradeau et al.] 2010). Previous popular benchmarks in this domain (Sterling & Irwin 2015 Wale & Karypis| 2006 Hu et al., 2020a) Wu et al. 2018 Dwivedi et al. 2022) focused on the prediction of molecular-level properties, such as solubility or HIV inhibition, which are predominantly short-range tasks. This is evident when they can be reduced to the problem of counting small- maull-dimensionall local substructures (i.e., with length smaller than 7) (Bouritsas al., 2023). Differently, ECHO ECHO-Charge Charge and
are the first graph benchmarks that target long-range interactions at the atomic level, i.e., the microscopic scale. Both benchmarks are not only inherently long-range but also partic- ularly challenging, as they require accurate modeling of charge distributions, energy stabilization, and the complex interplay of atomic interactions. This makes them computationally expensive to solve with current computational chemistry tools. We provide further details on the computational complexity of the underlying quantum simulations in Appendix C
Therefore, ECHO-Charge and ECHO-Ene set a new standard for evaluating long- rangrange graph information propagation, as well as they provide a compelling application of AI for science and chemistry, enabling faster predictions with potential impact on drug/material design understanding biological functions.
Contemporaneously with our work, Liang et al. (225) proposed a synthetic benchmark, which we view as a complementary effort to our ECHO in addressing the long-range propagation problem. While Liang al. (2025) focuses on a single synthetic task on large graphs (up to 569k nodes), our
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ECHO benchmark proposes five tasks (as discussed before) that provide controlled setting to assess propagation capabilities, are inherently long-range, and extend beyond current standards. Our goal is to provide practical, accessible benchmark that balances long-range complexity and usability for the broader community, avoiding digital divide concerns while still reflecting real-world scientific challenges.
3 THE ECHO BENCHMARK
In this section, we introduce a suite of datasets designed to rigorously evaluate the long-range information propagation capabilities of GNNs. Our benchmark consists of two complementary components: set of algorithmically constructed tasks (collectively called ECHO-Synth) and set chemically grounded real-world datasets (collectively called ECHO-Chem). Detailed dataset statistics are reported in Appendix E
The synthetic component includes classical graph-theoretic problems (i.e., single-source shortest path, node eccentricity, and graph diameter) posed across diverse graph topologies designed to induce structural bottlenecks and challenge multi-hop message passing. These tasks isolate long-range dependencies and enable controlled analysis of model behavior under varying topological conditions.
The proposed real-world benchmarks target practically relevant and physically grounded tasks in computational chemistry: SCHo-CCarre focuses on predicting long-range charge redistribution at the atomic level, while ECHO-Energy addresses the prediction of molecular total energies. Both problems are rooted electronic structure modeling, reflecting realistic quantum phenomena such as charge transfer and energy stabilization, and build upon prior work in quantum-accurrae deep learning models for molecular systems (Ko et al. 2021 Zhang et al. 2022).
3.1 THE ECHO- SYNTH DATASET
The algorithmic dataset is designed to benchmark GNNs tasks that require long-range information propagation across diverse set of graph topologies. It focuses on three graph property prediction tasks: Single Souree Shortest Path (ssap), Node Eccentricity (ecc), and Graph Diameter (diam). Among these, sssp and ecc are node-level tasks requiring the prediction of a scalar value per node, while di am is a graph-level task requiring single prediction for the entire graph. We refer to this dataset as ECHO- -Shthh
These tasks draw inspiration from Corso al. (2020); Gravina et al. 2023 and were intentionally selected due to their heavy reliance on global information. For example, solving sssp from given source node requires identifying shortest paths to all other nodes (Dijkstra 2022), since the information spans the entire graph. Eccentricity builds on this by requiring the longest shortest path from each node, demanding complete graph awareness. Diameter even more global, involving the longest shortest path between any two nodes (Cormen et al. 2009). Classical algorithms like Dijkstra's (Dijkstra) 2022) and Bellman- traversal, illustrate the challenge these tasks pose for GNNs, which rely localized message passing. To prevent models from relying on input features rather than learning structural patterns, each node is assigned uniformly distributed random scalar feature ~ 2((0, 1) Additionally, for the ssap task, a binary indicator is included to mark the source node. This ensures that the model can distinguish the source while maintaining uniform input statistics across tasks.
(Bellman) 1958). which perform complete graph
'While SCHO-Synth draws inspiration from Corso et al, 22020; : Gravina et al, 2023)), it departs from them Both
in a key aspect. Both Corso et (2020) Gravina et al. (2023) use small graphs that are mostly sampled from distributions that yield highly connected structures with small diameters and limited long-range dependencies. In contrast, ECHO- BCHO-Synth is explicitly designed to rigorously stress-test long-range capabilities, leveraging larger graphs and employing topologies deliberately designed to introduce bottlenecks, i.e., requiring substantially more propagation steps for accurate predictions. These design choices significantly increase the long oangrange rigorous benchmark for evaluating long-ran long-range propagation. We opted for random uniform node features rather than zero vectors to introduce stochasticity (which makes the synthetic tasks less trivial), and to provide a unique identifier for the nodes in the tasks and prevent the trivial scenario in which all nodes share identical initial representations, hindering the expressiveness of certain architectures (Sato et al. 2021
difficulty of the tasks, making ECHO-Synth
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0600000 www.oo....oo.ooo 000.00000000000000
(a) line
(b) ladder
(c) grid
(d) tree
caterpillar
(f) lobster
Figure 1: Visualization of the proposed topologies in the synthetic dataset. All graphs consist of 30 nodes to ease visualization, and different node colors correspond to different random initial features.
Dataset Construction. This dataset includes six distinct families of graph topologies i.e., line, ladder, grid, tree, caterpillar, and lobster (see Figure [], each selected highlight different structural and propagation characteristies. The 1 line graph (Figure Tab serves as simple but non-trivial baseline. To introduce non-local interactions, we modify it with stochastic residual connections: each node has a 20% chance of forming an edge to another node 2-6 hops away. Building on this, the ladder topology (Figure Tbb consists of two parallel line graphs connected by one-to-one cross-links enabling richer routing possibilities and redundancy in message pathways. The grid d topology (Figure Ich represents 2D lattice structure where edges are independently removed with a 20% probability. This results in irregular neighborhoods and broken spatial symmetries.
To model hierarchical structures, we include cree-structured graphs (Figure [cl generated through preferential attachment. A new node connects an existing one with probability proportional to ki, where k; represents the degree of the i-th node (with a = 3), leading to the formation of high- degree hubs and reflecting connectivity patterns often in natural networks. The caterpil lar topology (Figure Te) augments a central linear backbone with peripheral nodes attached randomly along the spine, combining features of chain-like and tree-like graphs to create moderate branching and directional flow. Extending this idea, the lobster graph (Figure [f] adds third hierarchical layer: nodes in the outermost layer connect only intermediate nodes, resulting in deeper branching while preserving an overall elongated structure. This configuration is especially useful for testing the limits of multi-hop message passing under structured constraints. Beyond their long-range dependencies, the complexity of the synthetic tasks is further increased by the presence of topological bottlenecks, which pose significant challenges GNN based on message passing (Gilmer al.) 2017). Bottlenecks emerge in graphs where information flow between distant nodes is constrained to pass through small subset of intermediary nodes, thereby restricting the bandwidth of information flow. This structural constraint can increse the risk of over-squashing, phenomenon in which exponentially growing information is aggregated into the low-dimensional node representations (Alon) & Yahav 2021 1). As result, critical signals may be compressed lost during propagation, severely limiting the model's 's capacity to distinguish and preserve meaningful long-range interactions (Topping
al. 2022 Di Giovanni al. 2023

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Graph families in synthetic dataset are explicitly designed to expose models to such bottlenecks. For example, in the 1ine topology information between distant nodes must propagate sequentially through single path. making each node along the path critical bottleneck. Similarly, tree- structured graphs inherently introduce bottlenecks at branch points and hierarchical layers, where entire subtrees depend on narrow pathways for communication with the rest of the graph. The caterpil 1 alar and lobster graphs further reinforce this pattern by adding additional peripheral layers while maintaining centralized backbones, exacerbating the bottleneck effect in their hierar- chical layouts. Even in the more uniform grid d topology, bottlenecks are implicitly introduced through random edge deletions, which can disrupt regular pathways and force information traverse suboptimal and congested routes.
Dataset Split. To support robust evaluation, we generate graphs with target diameters in the range d E [17, 40), capturing diverse long-range interaction scenarios. For each of the six graph topologies and each diameter value, we produce 70 unique graphs, yielding total of x x 6 10,080 graphs. To ensure consistent and unbiased evaluation, we partition these graphs into training, validation, and test splits in a stratified manner. Specifically, for each topology and diameter combination, we assign 40 graphs to the training set, 15 to the validation set, and 15 to the test set. This strategy guarantees that all splits share the same distribution over both graph topologies and diameter values, which are uniformly sampled. Consequently, models are evaluated on data that is statistically aligned with the training set, avoiding distributional shifts and ensuring fair comparison across methods.
3.2 THE ECHO-CПEя DATASETS
Molecular property prediction is cornerstone application of GNNs, with common benchmarks involving graph-level prediction tasks such as molecular fingerprint (Duvenaud et al. 2015). solubility, toxicity and various chemical properties (Coley al. 2017 Hu et al.) (020b)). One fundamental task in this domain is the prediction of atomic partial charges, which are continuous, atom-level properties that reflect the electron distribution within a molecule. Accurate charge prediction is essential for modeling molecular interactions, reactivity, and electrostatic behavior. Figure 2 illus- trates this task the 3D molecular graph of caffeine, where each atom is colored according to its predicted partial charge. Complementary to this, another central quantum property of molec- ular systems is the total energy, which governs stability, chemical reactivity, and conformational preferences. Thus, predicting molecular energies is equally important for chemistry applications. Traditionally, both atomic charges and molecular energies are computed using quantum mechanical methods, especially Den- sity Functional Theory (DFT) (Argaman & Makov 2000) or related quantum chemical simulations. While these methods provide high accuracy, their computational cost, arising from solving complex equations, limits their scalability to large molecular datasets or accuracy simulations require several minutes to process single molecule. We report a quantitative description of DFT simula- tion efficiency in Appendix C
tasks. Specifically, high-
significant challenge for Machine Learning (ML) methods addressing these prediction tasks is effectively capturing long- range dependencies across molecular graphs. Specifically, here we will refer to "long-range" in the graph space (e.g., nodes sep- arated by many hops), rather than purely spatial distance. The ftrre-dimensionall configuration of molecules greatly intensi- fies this task complexity, as distant atoms in the graph topology can still exert significant influence on electronic properties and total energy [Jensen] 2017 Ko et al. 2021 Shaidu et al.) (2024). Specifically, the total molecular energy is computed consider- ing several ceaantu--mechane atomic charges are influenced by non-local electronic effects (Ko et al.) 2021 Shaidu et al.) 2024).
Figure 2: The 3D molecular graph of caffeine annotated with atomic partial charges. Blue indicates re- gions of negative partial charge, while red corresponds to positive charge accumulation.
long-range interactions (Jensen 2017), and, similarly, the partial
Such nri-trivial, long-range interdependencies become increasingly challenging to model accu- rately as molecular graph diameter grows. To systematically address this challenge, we introduce ECHO-Charge and ECHO-E CCHO-Energy, with the specific aim stress long-range dependencies in

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real-world scenarios. ECHO-Charge is formulated as a node-level regression problem, where the model must predict the partial charge of each atom in a molecular graph. while ECHO-Energy is formulated graph-level regression problem, requiring the prediction of the total molecular energy. We note that the total energy cannot be computed as the sum of per-atom energies, since it consists of several quantum-mechanical long-range interactions, e.g., Seectron-nuclerr electron-electron, and nuclear-nuclear contributions at the chosen level of theory (Jensen (017). We collectively refer to Cchree and ECHO -nnergy as ECHO- -hhmm
Beyond serving as rigorous benchmarks for GNN architectures, these datasets have strong potential for practical impact ML applications for science and chemistry. Capturing these sophisticated long-range interactions can significantly improve efficiency of predicting atomic partial charges and molecular energies, while also serving as accurate and computationally inexpensive initialization for subsequent quantum mechanical simulations. Such improvements could substantially accelerate computational chemistry workflows, facilitating rapid exploration of the large molecular space.
Dataset Construction. Comprising x = 170,000 (BCHO CCargee and Re 196,000 (ECHO Energy) molecular graphs selected from the ChEMBL database (Zdrazil et al. 2024), our benchmarks include molecules with graph diameters between 17 and 40, where the interplay between the molecule size and the task ensures the need to work with sigmificant long-range dependencies that thoroughly stress model capabilities. In both ECHO-Charge and ECHO-Energy. each graph represents single molecule (see Figure 2). and each node (i.e., atom) is labeled with the atomic number, essential for chemical identity, and spatial distance from the center of mass of the molecule, to provide geometrical context. Edges correspond to chemical bonds, and are labeled with bond type (single, double, triple, or aromatic) and bond length. Notably, this encoding of spatial information is invariant under the action of the E(3) group, meaning that relative geometric features such as distances remain invariant under global 3D rotations, reflections, and translations of the molecular structure. This ensures that the spatial representation respects the underlying symmetries of molecular physics, essential for learning physically consistent models.
To generate the datasets, we employed a two-step approach. Firstly, the generation process began with molecular 3D structure generation starting from ChEMBL SMILES [Weininger] (988) strings for all molecules satisfying the given diameter constraint. In order to generate molecular conformations we opted for coordinate optimization using the Generalized Amber Force Field (GAFF) (Grimme) et al. 2010), a well-established force field specifically designed for optimizing wide variety of organic and medically relevant compounds. These optimized structures served as initialization for the subsequent quantum chemical calculations to determine accurate structures, partial charges, and molecular energies. Specifically, we employed Density Functional Theory (DFT) to match the required chemical accuracy required for reliable molecular property annotation. All computations were run with the ORCA package for quantum chemistry (Ncese) [0222] Neese et al. 20200 Neese 2023). detailed description of the quantum simulations provided Appendix C along with
information about the computing platform in Appendix D
Dataset Split. To evaluate model performance under consistent and reproducible conditions, we employed random uniform sampling strategy to split the original datasets. This approach ensures a balanced distribution of molecular structures, charge ranges, and energy levels across the training, validation, and test sets, therefore minimizing potential sampling bias. For ECHO-Chargo rge, we adopt an 80/10/10 split for training, validation, and testing, while for ECHO- Eetto- we use a 90/5/5 split.
4 EXPERIMENTS
Baselines. We consider diverse set of GNNs baselines that capture core directions in the devel- opment of graph neural architectures, spanning from classical GNNs more recent approaches that demonstrate strong empirical performance in capturing long-range dependencies. As classical baseline models, we include GCN (Kipf& & Welling 2017), GIN (Xu et al. 2019), GINE (uu et al. 2020b) and GCNII (Chen et al. 2020). which represent standard message- paassaggg frameworks with strong theoretical grounding. We also consider multi-hop GNN, i.e., DRew (Gutteridge et al. (203) which adaptively rewire the graph to facilitate more effective propagation across distant nodes. We
We added GINE as haseline to RCHO-Cha rge and OCHO-Knerggy henchmarks to overcome the limitations of GIN to process edge attributes.
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evaluate GPS (Rampásek et al. 2022) and GRIT (Ma et al. 2023b). two effective graph transformer that enables long-range propagation via attention mechanism between any pairs of nodes. Finally, explore the performance of a family of GNNs that draw on principles from dynamical systems theory. namely differential-equation inspired GNNs (DE-GNNss) This includes GraphCON (Rusch) et al. 2022), which is designed to address the over-smoothing issue, as well models explicitly designed to perform long-range propagation, whose architectures are based on non-dissipative or nort-Hamittoniaan dynamics, such A-DGN (Gravina al. 2023), SWAN (Gravina al. 2025). and PH-DGN (Heilig et al. 2025).
Model Architecture and hy hyperparameter selection. All models share a unified backbone design to enable a fair comparison. In particular, each model is composed of a linear embedding layer, stack of GNN layers, and task-specific readout module. For node-level tasks, the readout is two-layer MLP applied directly to the node representations. For graph-level tasks, node representations are first aggregated using the mean, max, and sum operations, concatenated, and then processed by a two-layer MLP. This standardization ensures that differences in performance are attributable to the core propagation mechanisms rather than auxiliary architectural choices.
Training follows consistent protocol across all models. We minimize the base- 10 logarithm of the Mean Squared Error loss (MSE), 1o (MSE()uue - Nurget)), since the predicted values can be very small in magnitude and this scale- sensitive loss emphasizes small differences. We use the Adam (Kingma& & Ba 2015) optimizer and adopt Early Stopping based on validation loss with a patience of 50 epochs. The maximum number of training epochs is set to 1000. This procedure ensures convergence while preventing overfitting, and serves as reference setup to facilitate reproducibility of our results. In order to ensure fair and robust comparison across all methods and datasets, we employ an extensive hyperparameter optimization protocol. Specifically, for each -dttss- pair, we perform a Bayesian Optimization based on Gaussian Process prior (Snoek et al. 2012) in the chosen hyperparameter space, spanning 100 trials to explore the respective search space efficiently. We report the complete set of explored hyperparameters for each model, as well as with the selected Finally, the best configuration found is validated through four independent training runs, each initialized with a different random seed.
hyperparameters, in Appendix [F]
Results on ECHO-Synth dataset.
We report results on the syn- thetic benchmarks in Table [ All the values are reported using the Mean Absolute Error (MAE). Additional metrics are reported in the Appendix H Table 18 We start observing that mod- els employing global attention mechanisms significantly outper- form traditional message--pssinng frameworks. Specifically, GRIT demonstrates a superior perfor- mance on the sssp task, achiev- ing a remarkably low MAE of 0.121. In line with literature find- ings (Dwivedi et al. 2022), this result suggests that incorporating transformer- like global attention substantially mitigates inherent limitations in localized message- passing, which are pronounced in classic architectures such GCN and GIN. This is further supported by the analysis in Appendix[] [ which shows that the highest attention scores are often assigned to node pairs that are not directly connected and often far apart in the graph. Interestingly. differential- equation-inspired architectures, particularly those employing non-dissipative or port-Hamiltoniar formulations (SWAN, A-DGN, and PH-DGN), consistently perform well across tasks, with similar performance metrics. GRIT achieves the lowest MAE on the d am task (1.014), closely followed by SWAN and A-DGN. This highlights the benefit of incorporating attention or non-dissipative dynamics to improve long-range information propagation. Moreover, DRew, reveals its effectiveness in the ecc task, attaining the lowest MAE (4.651). This success emphasizes the advantage of multi-hop information propagation, thus effectively addressing topological bottlenecks critical for accurately
Table I:
Test MAE (mean with standard deviation subscript) for each model across the three synthetic tasks: di am, ecc, and sssp. Lower is better. Values are color-coded by performance, with darker green indicating lower error.
Model
diam 4
ecc 4
sssp +
A-DGN
1.151 + 0.008
4.981
0.087
1.176 +: 0.10
DRew
1.243 0..04
4.651
0.020
1.279 0.0011
GraphCON
2.969 0.199
5.474
0.001
5.734
0.0111
GCN
3.832 + 0.202
5.233
D.034
2.102 + 0.09
GCNII
2.005 + 0.093
5.241
0.030
2.128 + 0.49
GIN
1.630 1: 0.161
4.869
0.092
2.234
1: 0.271
GPS
2.160 +: 0.098
4.758
0,021
0.472
GRIT
1.014 0.046
5.091
0.158
0.121
0.013
PH-DGN
1.627 + 0.398
5.068
D. 126
1.323 + 0.485
SWAN
1.121 + 0.070
4.840
0.045
0.896 + 0.232
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capturing node eccentricities. Differently, GraphCON does not inherently outperform traditional methods, and show notably weaker performance relative to other models of the same architectural family (e.g., A-DGN and SWAN). Thus, mere message-passinne dynamics that only mitigate the over-smoothing issue does not ensure superior performance in long-range tasks.
Finally, traditional message-passeane passing models like GCN demonstrate consistent limitations across all benchmarks, indicative of fundamental constraints in purely localized message-passine architectures required in our ECHO ECHO-Synthh benchmark suite.
when facing extensive long-range dependencies This limitation is most evident in the di am task, where GCN records the highest MAE (3.832), underscoring inadequate capacity for global information aggregation.
Results on ECHO-Chem tasks.
We detail the performance of all eval- uated models on the atomic partial charge and energy prediction task in Table 2 Additional metrics are re- ported in the Appendix H Tables [9 20) As anticipated, architectures capable of handling long-range depen- dencies demonstrate clear advantage on both benchmarks, given the na- ture of the task which requires precise modeling of subtle interatomic inter- actions spread across the molecular graph.
Table 2: Test MAE (mean with standard deviation as sub- script) performance across models on the ECHO-Energy and ECHO-Charge rge tasks. Lower values are better. Cells are color-coded by performance, with darker green indicating lower error (independently normalized per column).
ECHO-Charge (x10
Model
ECHO-Energy
A-DGN
12.486 :: 1.621
6.543 1: 0.146
DRew
11.325 :: 2.394
9.086 0.443
GCN
28.112 1.239
8.421 0.512
GCNII
13.235 = 2.630
8.829 =
GIN
47,851 + 10.154
10.784 0.059
GINE
23.558 +: 7.308
7.176 :: 0.371
Notably, GPS achieves the best perfor- mance on ECHO-Ener Energy and it is competitive on ECHO Chaare, con- firming the utility of global attention mechanisms in capturing distant influ- ences that modulate quantum chemi- cal properties, albeit the cost of increased computational complexity (as shown in Appendix I
GPS
5.257 0.842
6.182 0.219
GRIT
25.508
2.507
7.134
GraphCON
14.295 = 0.807
19.629 0.105
PHI-DGN
16.080 = 1.123
7.915 1: 0.266
SWAN
12.629 1.157
6.109 +: 0.103
Models like A-DGN and SWAN also produce competitive performance, appearing consistently among the top performers, with SWAN emerging as the best model ECHO-Charge, Their success suggests that imposing non-dissipative priors not only improves the propagation dynamics but also guides the model toward chemically plausible solutions. Interestingly, while DRew outperforms classical GNNs, especially in the ECHO-Energy taks, performs comparatively worse than DE- GNN and transformer baseedde--b-adddddd based models.
Traditional message- paagggg--aingggggg networks, particularly GCN and GIN, again lag behind. These results again confirm the hypothesis that localized aggregation, without mechanisms to improve propagation effectiveness integrate distant node information, is inadequate for atomic-level charge modeling. The ECHO-Chem benchmarks thus clearly illustrate the necessity for architectures that either in- corporate global attention or embed non-dissipative dynamics effectively tackle the intricate and non-local dependencies inherent in these molecular tasks.
We provide visual depiction of charge prediction accuracy on non-trivial molecule from the test set in Figure 3 the figure compares prediction errors between GPS and GCN. GPS shows consistently lower errors, especially at peripheral atoms, highlighting its ability capture long-range interactions, while GCN accumulates errors at structurally distant or chemically sensitive sites. This comparison illustrates the benefit of long-range information propagation on complex graphs and chemical structures.
Lastly, we note that although the energies and partial charges errors are small in absolute magnitude across baselines, even subtle deviations (as stated in Dupradeau et al. (2010)), the order of 10-4 to 10-6 e, can lead to significant downstream effects in molecular modeling and reproducibility of results. Therefore, predictive models must target this level of granularity to produce chemically meaningful output.

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Published as a conference paper at ICLR 2026
Additional Experiments and Analysis.
22
We provide further results and a detailed evaluation of baseline performance in Ap- pendix G Specifically, we investigate how the radius of the explored neighborhood af- fects each method (Appendix[G.]) G.I) and how model performance varies across graphs with different diameters (Appendix G.2 Our results consistently indicate that deeper networks outperform shallower ones, con- firming the long-range nature of these benchmarks. Moreover, in Appendix G.3 we examine the impact of the readout depth and show that the final performance ap- pears to be independent of this design choice. In Appendix [.4] we further an- alyze model performance across different graph topologies in ECHO- Synth. The results indicate that, although absolute per- formance varies slightly with the underlying graph structure, the relative ranking of the models remains consistent, reinforcing the robustness of our findings. In Appendix[] we also visualize GPS's attention patterns, highlighting the importance of connecting distant nodes to facilitate information flow and improve performance. Together, these analyses reinforce the importance of long-range information propagation in the ECHO benchmark. Finally, Appendix rreporsss runtime measurements, illustrating key trade-off between accuracy and efficiency: transformer-basee models like GPS achieve strong performance but are computationally demanding, whereas models such as A-DGN provide more balanced alternative.
(b)
Figure 3: Visualization of prediction errors for the ECHO-Charge task using two different GNN archi- tectures: (a) GPS and (b) GCN. The coloring repre- sents the logarithm of the absolute prediction error, [yg([ytrue - Upred)- Lower values (in green) indicate better prediction accuracy, while higher values (in or- ange) correspond to larger errors.
5 CONCLUSION
In this paper we propose ECHO, new benchmark for evaluating long-range information propagation in GNNs. Our benchmark includes two main components, ECHO-Synt Synth and set of chemically grounded real-world datasets (collectively referred to ECHC-Chem), that target long-range com- munication in both synthetic and wolld settings. The synthetic tasks are designed to predict algorithmic and long-range- ree--annnb--ddsiieee by-design graph properties, while the real-world tasks focus predicting atomic charge distributions and molecular total energies, both of which critically depend on long- range quantum interactions. We provided detailed analysis to demonstrate that the tasks in ECHO genuinely capture long-range dependencies, and we established strong baselines for each task to provide a comprehensive reference point for future research. Our results highlight the limitations of current GNN architectures when faced with long-range propagation challenges, and we believe that ECHO will serve as a critical step toward building more robust, scalable, and generalizable GNNs capable of handling the full spectrum of graph-based learning tasks, posing a challenge to the community to push the boundaries of GNN design and evaluation. Not only ECHO provides solid benchmark, but it also leaves ample room for future architectures to improve and advance GNN architectures capable of more effective information propagation.
ETHICS STATEMENT
The research conducted in this paper conforms in every aspect with the ICLR Code of Ethics. Our study does not involve human subjects, sensitive personal data, or applications with foreseeable harmful consequences. No ethical concerns are anticipated regarding data usage, methodology, or findings.
REPRODUCIBILITY STATEMENT
We provide all necessary details to reproduce our ECHO benchmark in Section 3 and Appendix C and describe the setup of each experiment in Section 4 and Appendix F thus ensuring sufficient
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information to replicate our results. We openly release data and code replicate our results and perform new evaluations at /t/ttps://githbbb http://hh.......ccaaoao/------aa
ACKNOWLEDGMENTS
This work has been partially supported by EU-EIC EMERGE (Grant No. 101070918) and NextGener- ation EU programme PNNR Partenariato Esteso PE00000013 "FAIR" Future Artificial Intelligence Research" Spoke 1 "Human-centered Al".
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A LLMS USAGE
Large Language Models (LLMs) were used as general-purposs assistive tools to improve the writing quality of this paper. Specifically, used LLMs to help with grammar correction, rephrasing for clarity, and suggesting some improvements to the overall structure of the text. All LLM -ggnerrtt text was carefully reviewed and edited by the authors ensure that it accurately reflects the authors' intentions and scientific content. No LLMs were used to generate scientific content, including but not limited research direction, hypothesis formulation, experimental design, data analysis, or interpretation of results.
B DISCUSSION ON LRGB
One of the most widely used benchmarks for assessing the long-range propagation capabilities of GNNs is the Long Range Graph Benchmark (LRGB) (Dwivedi et al. 2022). The benchmark proposes five tasks: two molecular property prediction tasks (Peptides-func and Peptides-struct) one molecular bond prediction task (PCQM-Contach) and two computer vision tasks (PascalVOC-S)) and COCO-SP). However, despite initial rapid improvements, performance on LRGB has plateaued. Since its introduction 2022, there has been a noticeable deceleration in progress. Considering set of 30 models on the Peptides-func task, we observe a performance improvement of 6.5% in the first year, but only by 1.3% in the second, and no significant gain in the third year (Dwivedi et al.
2022. Rampašek et al.
2022
Tönshoff et al.] 20033 Shirzad et al. 2023 Cai et al., [023] [023]Glio Glickman &
Yahav 2023: He et al.
2023 Gutteridge et al. [023] Giusti et al. 2023; Ma et al. [2023a] | Ngo et al.
2023; Michel et al. 2023 J Ma et aal.322230 Ding al. 2024 Choi et al. [0244] Geisler et al. 2224
Behrouz Hashemi 2024, Wang et al. 2024 Errica et al. 2024]] Gravina et al.
[2025] Heilig et al.
2025 Eliasof al.) [2025] Hariri et al. 2025 Trenta et al. 2025). A similar trend exists for the other
benchmark tasks as well.
Furthermore, recent analysis LRGB (Bamberger et al. 2025b). well as the benchmark's 's
sensitivity to hyperparameter tuning (Tönshoff et al.] (023). raises additional concerns about the long-range nature of its tasks. The analysis reveals that only subset of tasks genuinely require longer interactions, while the peptides tasks are effectively local. This highlights the need for more focused benchmarks that explicitly and systematically test the long-range propagation capabilities of GNNs.
C CHEMICAL SIMULATION TECHNICAL INFORMATION
This appendix provides detailed information on the computational pipeline used to derive partial atomic charges in the CCao-e dataset and total energies in Eeeto-Energy... The pipeline comprises three primary stages: (i) 3D structure generation from SMILES, (ii) quantum chemical computation of partial charges, and (iii) geometry optimization.
3D Structure Generation from SMILES. Since subsequent charge optimization steps require pre-optimized 3D coordinates, all structures were geometry -pptmmize prior to simulation using Open Babel OoBoye et al. 2011) and its Python interface, Pybel (O'Boyle et al.) 2008) Initial molecular geometries were generated from SMILES strings using the General AMBER Force Field (GAFF) (Grimme al. 2010). GAFF was chosen over alternatives such as MMFF94 (Halgren 1996) due to its favorable trade-off between accuracy and computational cost, and its strong performance in predicting both energies and geometries. The optimization procedure involved 100 steps of coarse minimization followed by 500 steps of local refinement for each molecule. The SMILES strings were converted into 3D conformers, which were then minimized yield low-energy structures. These structures were exported in SDF format for subsequent compatibility. The average time required for 3D structure generation per molecule (considering only those satisfying the RCHO-Cha rge and SCHO-Energy dataset diameter criteria) was 562 + 124 ms.
Quantum Chemical Computations with ORCA. To compute partial atomic charges and total energies, we employed the ORCA quantum chemistry software suite (version 6.0.1) (Neese 2022 2023 Neese et al. 2020). All calculations were performed using the B3LYP, a hybrid density func- tional (DFT) method that mixes Hartree- Fock exchange with Becke's 's exchange and Lee-Yang- Parr correlation functionals to balance accuracy and efficiency in quantum chemical calculations (Argaman)
& Makov 2000).
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Table 3: Mean time required for computation of partial charges of single molecule with differ- ent configurations of the ORCA tool. Variance is computed over 30 random molecules from the ECHO-Charrg/ECH--EEeng dataset. "Denotes the chosen basis set for DFT computation.
Method
Setting
Times (s)
HF-3c
LooseSCF
10.441.3
HF-3c
TightSCF
28.1 14.3
B3LYP def2-TZVP* (DFT)
LocseSCF 146.5
B3LYP def2-TZVP" (DFT)
TightsCF 634.5+21.3
We provide a summary of required times for computation with both full DFT computations and HF methods in Table 3 Under our configuration, the average runtime for single quantum chemical calculation was 634.5 + 21.3 seconds per molecule, requiring = 2 months of computational time on hardware configuration. Simulations
exploiting full thread parallelism provided by
ORCA.
Celf-Consistent Field (SCF) Convergence Settings. To further improve the accuracy of the simula- tions, we employed the Ti git.tCFFF setting in ORCA, which enforces tighter convergence thresholds in the self-consistent field (SCF) procedure, thereby reducing numerical errors in the electronic structure calculation.
Charge Extraction. Atomic partial charges were extracted using the Löwdin population analysis method (Szabo & Ostlund 1989). These charges were used as supervision signals in our dataset generation pipeline.
D HARDWARE RESOURCES
All quantum chemistry simulations were conducted on a dual-socket Intel Xeon 6780E machine with total of 288 physical cores (144 cores per socket, thread per core). Each socket is equipped with 108 MiB I.3 cache, for combined 216 MiB of shared 1.3 cache, along with 288 MiB of 1.2 and 27 MiB of L1 (data instruction) cache across the system. The CPUs support AVX2 and FMA instruction sets, enabling efficient linear algebra operations, which are critical for electronic structure methods.
The machine is configured with two NUMA nodes, each associated with one the sockets. Each NUMA node has over 500 GiB of local RAM for total of approximately I TiB of RAM. The high memory capacity and bandwidth are critical for quantum chemistry workloads. particularly those using density functional theory (DFT) correlated wavefunction methods, which require extensive memory for large basis sets and integral evaluations.
The large number of physical cores allowed us to parallelize over both molecular batches and internal basis function evaluations, providing efficient scaling for density functional theory (DFT) and semi-empirical caleulations.
For model training and inference, we used a separate compute node equipped with NVIDIA H100 GPUs
E ADDITIONAL DATASET INFORMATION
We report in Table 4 the detailed statistics of the proposed datasets. In Table we provide summary of the input and target features used in the ECHO S-HH--yythhh and ECHO- ECHO-Chem Chem (i.e., ECHO- ECHO-Charge and ECHO Energy) datasets. Figures of the graphs in the datasets, including distributions of the number of nodes, number of edges, average node degree, graph diameter, and node eccentricity. Additionally, Figure jillustrates the correlation between the number of nodes and the graph diameter, highlighting structural differences between real and synthetic data. These insights support the design choices for model evaluation across diverse graph regimes.
and report detailed statistics on the structural properties
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Table 4: Statistics of the proposed dataset.
Dataset
# Graphs
Avg Nodes
Avg Deg.
Avg Edges
Avg Diam
Node Feat
Edge Feat Tasks
BCNO Bynth
10.080
83.69
2.33
211.63
28.50
None
1ine
1,680
75.60+27.32
2.37+00.10
90. 10.10.33..00 89
28.50-a.02
None
ladder
1,680
36.52:15.a2
2.92
82,54
28. n8.50
Nonc
grid
1,680
193.10 10
2.95
288.32 29
n8.50
tree
1,680
60.42
1,96.0.01
59.42.117.1
28. 28.50
None
cacerpillar
1,680
34.71+7.96
1.94=0.02
33.71+7.96
28.30-6.92
None
labster
1,680
21.79:22..4
1.97.0001
80.79.225.4
28.50
Nome
ECHO Chazge
170,367
12.49
2.09 =0.00
151.32++25.u 16
23.54-2.64
BCHO-Energy
196,528
13.73
153.84 58 23.61g2.59
Table 5: Summary of dataset properties.
Dataset
Node Features
Edge Features
Target
SCH0- Synth 3CH0-Chem
Random scalar, source indicator for asap Atomic number, distance from center of mass
None
diamm sssp, ecc
Bond type. bond length
Partial charges, Total energy
2500
2500
2000
2000
Frequency
Frequency
1500
1500
1000
500
500
0
100
200
250
500
750
1000
1250
Number of Nodes
Number
Edges
1000
3000
800
Frequency
Frequency
2000
600
1500
400
1000
200
0
2,00
2.25
2.50
2.75
3.00
3.25
25
40
Average Degree
Diameter
40000
Frequency
30000
20000
10000
0
10
15
20
25
30
35
40
Node Eccentricity
Figure 4: Statistics the ECHO-Synth dataset, reporting the distributions of the total number of nodes, total number of edges, average degree, diameter, and node eccentricity.
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30000
25000
25000
Frequency
20000
Frequency
15000
10000
5000
5000
40
60
80
100
120
140
100
150
200
250
300
Number of Nodes
Number of Edges
50000
30000
Frequency
25000
Frequency
40000
20000
30000
15000
20000
10000
10000
5000
1.9
2.2
20
Average Degree
Diameter
Figure 5: Statistics of the ECHO-Charge dataset, reporting the distributions of the total number of nodes, total number of edges, average degree, and diameter.
30000
30000
25000
25000
Frequency
20000
Frequency
20000
15000
15000
10000
10000
5000
5000
60
100
120
140
100
150
250
300
Number of Nodes
Number of Edges
60000
50000
30000
Frequency
Frequency
40000
20000
30000
20000
10000
10000
0
1.9
2.2
20
35
Average Degree
Diameter
Figure 6: Statistics of the ECHO-Energy Energy dataset, reporting the distributions of the total number of nodes, total number of edges, average degree, and diameter.
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40
35
Diameter
30
30
25
25
20
100
200
300
400
60
80
100
120
140
Number of Nodes
Number of Nodes
(a) Correlation in OCEO-Syntie
(b) Correlation in ECHO-Charge.
Diameter
30
25
20
100
Number of Nodes
(c) Correlation in ECHO-Energy.
Figure 7: Correlation between the number of nodes and graph diameter in ECHO-Synth, ECHO-Charge and ECHO-Energy.
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F HYPERPARAMETER SELECTION
Tables to 16 report the hyperparameter search space and the best values selected for each task (di ddiam, ecc, ssap, SCHO-Charges and SCCHO-Energy: across the different GNN architectures we considered. For details on specific hyperparameters, we refer the reader to the original papers. Each table includes the name of the hyperparameter, its search range or categorical options, and the optimal value obtained for each task, as identified through hyperparameter tuning on the validation set.
Another strong evidence supporting the long-range nature of the ECHO benchmark, implicitly comes from our hyperparameter optimization process. Specifically, Bayesian Optimization consistently selected configurations with large number of GNN layers. This suggests that, even without explicit guidance, the hyperparameter optimization procedure identifies deeper models as necessary to minimize validation enror, further reinforcing the notion that the task demands long-range information propagation.
Table 6: Hyperparameters and their best values across tasks for A-DGN.
Hy Hyprraaraeeter
Search interval
diam
SSS5
BCHO-Charge
ECHO-Bnerqy
Number of layers
40
27
28
40
34
16
Hidden dimension
16 256]
68
65
45
130
216
Learning rale
10-5 10-21
0.00101
0.00229
0.00473
0.00072
0.0085223
Weight decay
10 10 $1
0.00098
0,00000
0.00003
0.00001
0.00044851
0.001 0.5]
0.19254
0.32934
0.10560
0,25667
0.4215
0.001 0.5
0.41827
0,46803
0.21252
0. 0.19499
0. 0,15304
Graph convolution
NaiveAggr: GCN
NaiveAggr
NaiveAggr
NaiveAggr
GCN
NaiveAggr
Table 7: Hyperparameters and their best values across tasks for DRew.
Hyperparameter
Search interval
diam
ss8p
ecc
ECHO-Cha Charge
ECHO Energy ECHO-Energy
Number of layers"
4]
4
4
k-hop
10]
10
10
10
10
10
Hidden dimension
[16 256]
249
78
119
232
241
Learning rate
110
10
0.00037
0.00797
0.00126
0.00036
0,00023597
Weight decay
[10
10
0.00068
0.0001
0.00003
0.00001099
Employ delay
True, False]
False
False
False
True
True
Table 8: Hyperparameters and their best values across tasks for GCNII.
Hyperparameter
Search interval
am
sssp
ece
BCHO-Charge
ECHO-Energy
Number of layers
[10 40]
32
39
30
37
21
Hidden dimension
116 256]
81
40
64
33
103
Learning
10-
10
0.00260
0.00032
0.00005
0.00345
0.0047014
Weight decay
[10 8 10
0.00000
0.00009
0.00009
0.00002
0.000035227
0.0 0.9
0.70544
0.07902
0.04742
0.17158
0.10116
'While the layer search goes up 4 layers, DRew performs multi- hop aggregation (up 10 hops per layer), yielding effective receptive field of 4 10 40 hops, comparable to the ranges explored by the other architectures.
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Table 9: Hyperparameters and their best values across tasks for GCN.
Hyperparameter
Search interval
diam
ssap
BC:0-Charge
ECHO-Energy
Number of layers
- 40]
26
40
26
Hidden dimension
[16 256]
48
42
40
109
160
Learning rate
10 10
0.00007
0.00004
0.00023
0.00079
0.00052181
Weight decay
100 5 10
0.00007
0.00009
0.00002
0.00002
0.000043613
Table 10: Hyperparameters and their best values across tasks for GIN.
Hyperaarameter
Search interval
diam
ssap
BC:0-Charge
ECHO-Energy
Number of layers
[10 40]
29
34
25
11
19
Hidden dimension
[16 256]
58
170
78
197
90
Learning rate
]1 10
0.00002
0.00003
0.00006
0.00002
0.000046134
Weight decay
110 10
0.00003
0.00036
0.00091
0.00069
0.00046213
Table 11: Hyperparameters and their best values across tasks for GINE.
Hyperparameter
Search interval
diam
sssp
ecc
ECHO-Charge
ECHO-Energy
Number of layers
40]
22
31
Hidden dimension
16 256
85
Learning rate
10- 10
0.00014
0.0005028
Weight decay
10 8 10
0.00004
0.00033338
Table 12: Hyperparameters and their best values across tasks for GPS.
Hyperparameter
Search interval
diamm
sssp
ECHO-Charge
3CHO-Energy
Number of layers
17
17
Hidden dimension
[66 256]
40
162
216
192
Learning
[10
10 2
0,00004
0.00031
0.00034
0,00005
0,000024067
Weight decay
[10
0.00015
0.00029
0.00007
0.00005
0.00038179
GNN Backbone
[GCN]
GCN
GCN
GCN
GCN
GCN
Number of Backbone Layers
1
1
1
Number of attention hends
2
2
2
2
2
Table 13: Hyperparameters and their best values across tasks for GraphCON.
Hyperparameter
Search interval
diam
ssap
eCC
EC:0-Charge
ECHO-Energy
Number of layers
40] 16 256]
37
25
35
32
Hidden dimension
63
72
151
144
96
Learning rate
[10 5 10 2]
0.00088
0.00013
0.00007
0.00292
0.00032
Weight decay
[10 8 10
0.00038
0.00001
0.00001
0.00026
0.00059
00.001 1.0
0.57880
0.95470
0.98433
0.78163
0.82108
Table 14: Hyperparameters and their best values across tasks for GRIT.
Hyperparameter
Search interval
di am
s5Sp
RCHO-Charge
ECRO-Rrergy
Number layers
[1 40]
32
32
Hidden dimension
16 256
256
128
128
128
64
Learning rate
[10
10
0.00082
0.00048
0.00003
0.00034
0.00076
Weight decay
[10 10
0.00032
0.00047
0.00001
0.00034
0.00094
Atlention dropout
00 0.5)
0.433
0.008
0.014
0.351
0.178
Number attention heads
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Table 15: Hyperparameters and their best values across tasks PH-DGN.
Hyperparameter
Search interval
ssap
ECHO-Caarge
ECHO -aeEoyEnergy
Number of layers
17
Hidden dimension
16 256]
28
66
120
103
166
Learning rate
10
10
0,00150
0.00178
0.00037
0.00033
0.0038211
Weight decay
[10
10
0.00054
0.00082
0,00081
0.00063
0.00044422
00.001 1.0
0.34977
0.16491
0,36140
0.68993
0.40992
0..11 1.0
0.47190
0,90892
0,63323
0.87607
0,15544
9
0.01 1.0)
0.70474
0.92918
0.99675
0.91251
0.1 0.11011
P conv mode
NaiveAggr, GCN
GCN
GCN
NaiveAggr
mode
NaiveAggr, GCN
GCN
GCN
GCN
NaiveAggr
NaiveAggr
Drubled dimension
True, False]
False
False
True
True
false
Final state
p.g.2g]
pq
P
P?
P4
param.
Dampening mode
param+,
MLP4ReLU.
param+
DGNRcLU
param
param+
param+
DGNReLU.
External mode
MLP4Sin, DGNtanh]
MLP4Sin
MLP4Sin
MLP4Sin
MLP4Sin
MLP4Sin
Table 16: Hyperparameters and their best values across tasks for SWAN.
Hyperparameter
Search interval
di
assp
ECHO-Chazge
ECHO-Enezgy
Numher layers
40
32
38
Hidden dimension
16 256]
167
97
195
163
217
Leurning rate
10-3 10
0.00040
0.00107
0.00086
0.00063
0.00012157
Weight decay
[10 10
0.00057
0.00011
0.00010
0.00016
0.00028686
00.001 1.0)
0,54847
0.45462
0.07451
0.38229
0.67265
0.001 0.5
0.41480
0.28342
0.45928
0.07794
0.3156
[-10- 1.0
0.34233
-0.20976
0.37682
-0.36245
-0.67256
(AndSSynNaiveAgge (ASNA),
Graph convolution
BounsedGCNConn (BGC).
ASNA
BNA
BNA
ASNA
BoundedNNaiveAggrr (BNA)) [True, False]
Attention
True
False
Fulse
False
True
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G ADDITIONAL EXPERIMENTAL ANALYSIS
In this section, we investigate how the radius of the explored neighborhood influences the performance of each method, as well as how the models perform across graphs with varying diameters. We also analyze how the performance of the models changes across different graph topologies and the influence of the readout depth in ACHO-Synten
G.1 LAYER-WISE PERFORMANCE ANALYSIS
In Figure X we evaluate the impact of the radius of the explored neighborhood (i.e., the number of GNN layers) on test MAE across all tasks. We divide the results into three regimes: shallow (< 10 layers), medium (10-17 layers), and deep (> 17 layers). Therefore, in the shallow regime, GNNs perform short-range propagation; in the medium regime, they capture medium-range dependencies; and in the deep regime, they are able to model long-range interactions. A consistent pattern emerges across most tasks: deeper networks, especially those tailored for long-range propagation, tend to perform better, thus confirming the long-range nature of the proposed benchmarks. Specifically:
On the di am task (Figure 8a performance trends are model-dependent. Long-range models such as DRew and A-DGN remain stable slightly improve, and PH-DGN exhibits large performance improvement moving from a shallow to medium depth regime. This task, being graph-level and heavily reliant on global information by design, clearly benefits from increased depth and non-dissipative architectures which are able to perform many message- meessgg-passiigg steps across multiple hops.
For the ecc task (Figure (b), we observe consistent performance gain with increasing depth across nearly all models. Again, long-range architectures like A-DGN and SWAN, or the multi-hop GNN, DRew, show strong improvements in the deep regime, outperforming the others. This aligns with the intuition that eccentricity, being node-level but globally- informed property, benefits from many pessaae-ppssingg layers to capture distant context, highlighting the strength of long-range architectures.
In the case of sssp (Figure Sc again observe strong depth-related improvements, with the exception of GraphCON. Notably, SWAN, GPS, and DRew achieve large gains in the deep regime. Traditional models such as GCN and GIN or GraphCOO exhibit plateau or degradation, revealing limited depth scalability.
On the ECHO-Charge task (Figure &d). the behavior differs. This task involves precise regression of atomic partial charges, where small errors matter. Most models show stable MAE across depths, except for GCN and GIN, which degrade significantly in the deep regime. Importantly, models with explicit long-range message- message-passing capabilities (A-DGN, SWAN, PH-DGN, and GPS) retain high accuracy even at >> 17 layers. This suggests their robustness in fine-grained, long-range molecular prediction tasks.
Finally, on the BCHO-Enezge task (Figure Se), performance trends more model- dependent and exhibit higher variability, while still underlying the benefit of deeper regimes. Generally, non-dissipative architectures such as PH-DGN and A-DGN benefit from increased depth. The best performance is achieved by the transformer- model GPS, supporting the intuition that this graph-level task requires globally informed representations.
Overall, the observed patterns reveal clear correlation between the number of message- saasggggaaiinggggge layers and performance: models require many layers to perform well, confirming that these benchmarks require information to propagate over long distances. Remarkably, architectures explicitly designed to support many message- pessaag-passing passing steps consistently outperform others, further confirming the long- range nature of our proposed benchmarks.
G.2 PERFORMANCE ACROSS GRAPH DIAMETERS
Figure reports, for the best configuration of each model (as selected during model selection), the test MAE across varying graph diameters for all tasks. This analysis highlights how different models handle increasingly long-range dependencies.
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PH-DGN SWANN
A-GN DRew
GON
GINE
GraphCON
GRIT
GIN
GPS
GCNI
5.4110
0'10
52x10
(1
1x10
51x10
$x10
4.8
4.8x10
4.7x10
10"
Shalow (410)
Medium [1G-17]
Deep (+17)
Shallow (410)
[10 171
Deep (+17)
(a) di am task
(h) ecc. task
Bost MANE duos Buskd
10"
10*
Stallow (+10)
Mediu [10-17)
(-17)
Shallow (c10)
Medum [10-17]
Deep (+17)
Number of Layers Depit)
Number of Laye's (Death)
(c) sssp task
(d) ECHO-Charge task
10
Tout Log
10!
Shallow (=10)
Medium (10-17
Deep (-17)
Numberol Layer (Dapt))
(c) ECHO- -neegyy task
Figure 8: Test MAE different numbers of GNN layers across tasks.
For the diam task (Figure 9a). most models show robust performance for small moderate diameters. with a slight increase in MAE for very large diameters. Notably, GCN, GraphCON and GCNII architectures exhibit substantial degradation as diameter increases, suggesting poor scalability in capturing global structure many pessaag--asssng steps. Again, non-dissipatiee architectures (i.e., A-DGN, PH-DGN, and SWAN), DRew, GPS, and GRIT remain consistently accurate across all graph diameters, demonstrating their capacity to generalize across different graph scales. Finally, we observe that models with lower generalization capabilities tend to bias their predictions towards the statistical mean of the dataset. Consequently, these models achieve low error rates in the central range, where the ground truth aligns with the mean, but show high errors at the tails, as they fail to distinguish graphs with extreme diameters from the average case.
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PH-DGN SWAN
A-GN DRew
GON GIN
GINE
GraphCON
GRIT
GPS
GCNI
1.5
1.0
(17-20)
(20-21)
(2f--30
(30-35)
(76-40)
(17-20
(20-25)
[25-30]]
(20-06)
(25-40)
Rance
diam task
(b) ECC task
0.000
0.018
0.016
0.014
0,010
0.008
0.006
0.004
(17-20)
(2-25)
225-33)
(90-20)
(17-23)
(20-25)
(25-30)
(20-35)
(35-00)
(c) task
(d) ECHO-Charge task
[17-20]
(20-20
(25-30)
(20-30)
(35-40)
Demmeker Range
(e) ECHO-Energoe task
Figure 9: Test MAE for the best configuration of each model (as selected during model selection) at different graph diameters across synthetic and molecular tasks.
The ecc task (Figure 9b) reveals a characteristic U-shaped curve. Performance improves as diameter increases from small to moderate values, and deteriorates again for very large graphs. Here, all models follow similar performance pattern that is highly correlated with the node eccentricity distribution depicted in Figure 4 creating an inverse relationship between sample frequency and error. In particular, in the center of the plot, is possible to see the GNNs that learn the dominant pattern; in addition, simply predicting values close to the statistical mean minimizes the expected loss this dense region, and results a lower error. Conversely, graphs exhibiting lower eccentricities are statistically less frequent, leading to noticeable degradation of the performance across extreme values.
In the sssp task (Figure 9c increasing graph diameter consistently correlates with rising MAE, Model performance divides into three groups, with GraphCON exhibiting the worst performance both in terms of overall MAE and degradation with increasing diameter. GCN, GCNII, and GIN show similar values across the task and similar degradation trends. Finally, non-dissipative models, Graph Transformers, and DRew once again demonstrate remarkable and consistent performance
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difficult graphs. This trend reinforces the long-range nature of the task, where deeper more expressive models are required to maintain strong performance.
On the molecular EcHo-Charge task (Figure 9d), test MAE consistent across all ranges, but subtle trends emerge. Models like DRew and GPS show stability and even slight improvements for larger molecular graphs. Interestingly, GINE performs substantially better than its counterpart GIN, suggesting that edge-level attributes play a crucial role in the ECHO-Charge regression task.
On the molecular task ECHO-Energy. (Figure 9е clear trend emerges: test MAE increases with graph diameter across all models. However, architectures such as SWAN and DRew exhibit substantially lower sensitivity diameter, maintaining nearly constant MAE across the full range. Additionally, we note that all models exhibit general performance drop when processing molecular graphs with a diameter greater than 35, We attribute this behavior the original ChEMBL dataset's distribution, which includes fewer graphs with diameters in the 35 40 range, as illustrated in Figure 6
Overall, this complementary diameter wise analysis underlines the necessity for architectures capable of handling variable and large receptive fields. It also highlights that while shallow models may perform competitively on small graphs, their limitations become apparent regimes requiring long-range reasoning.
G.3 IMPACT OF READOUT LAYER DEPTII
To examine the impact of the readout depth, we conducted an experiment varying the depth of the readout (from to 3 layers) the ECHO- -Synh tasks. Table 17 presents the performance of GCN with varying readout layers. The results indicate that increasing the readout depth to three layers does not improve GCN performance on the synthetic tasks. Therefore, the final performance appears be independent of the readout depth. Since the additional layer does not yield measurable benefits, we consider the increased model complexity unjustified. For this reason, we retain two-layer readout to prioritize computational efficiency and performance.
Table 17: Ablation study Readout Depth for GCN ECHO-Synth tasks. Metrics are reported as Mean Absolute Error (MAE) + Standard Deviation.
Model
diam
ecc +
sssp $
GCN (1 layer readout)
6.219
5.493
2.367=0.083
GCN (2 layers readout)
3.832.1.2622
5,233
2.102
GCN (3 layers readout)
5.743
5.172
2.075 20.545
G.4 PERFORMANCE ACROSS DIFFERENT GRAPH TOPOLOGIES
In this section, we analyze the models' performance on different graph topologies in ECHO-Synth Figures I0 II and 12 show respectively di am, ecc and sssp. The results show that although absolute performance varies slightly with the underlying graph topology (e.g., GPS performs better on lobster graphs than on tree-like topologies in SSSP), the relative ranking of models remains consistent, reinforcing the robustness of our findings. Interestingly, we observe that the line topology consistently the most challenging one. This is expected: in line graph, every message must pass through a sequence of intermediate nodes, making each node a critical bottleneck for information propagation and amplifying the need for long-range communication.
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Matel
A-DGN
DRew
GCN
GCNII
GIN
3
GINE
MAE
GPS
GRIT
2
GraphCON
PH-DGN
SWAN
Caterpillar
Line
Grid Graph Topology
Ladder
Lobster
Tree
Figure 10: Results by different topologies on ciam task.
6.00
A-DGN
DRew
GCN
GCNII
5.50
GIN
5.25
GINE
MAE
GPS
5.00
GRIT
GraphCON
4.75
PH-DGN
SWAN
4.50
4.25
Caterpillar
Line
Grid Graph Topology
Ladder
Lobster
Tree
Figure 11: Results by different topologies on ecc task.
Model
A-DGN
DRew
GCN
GCNII
GIN
MAE
GPS
GRIT
GraphCON
PH-DGN
SWAN
Caterpillar
Line
Grid
Ladder
Lobster
Tree
Graph Topology
Figure 12: Results by different topologies on sssp sp task.
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H EXTENDED RESULTS
In Table 18 we report additional results on ECHO-Synth benchmark. In particular, we report MAE, MSE and loss (defined as log (MSE) obtained on the Similarly, report the same metries for ECHO- -Charge and ECHO BCHO-Ennegg Energy in Table 19 and Table 20 respectively.
Table 18:
Test performance (mean + std) of different models across ECHO-Synth tasks. Lower is better. In bold the best model.
Model
Test Loss 4
Test MSE 4
Test MAE
diam
A-DGN
-2.531=0.010
4.81820.108
1.151 10 038
DRew
-2.635 020
3. 3.75610.11
1.243 047
GCN
-1.848-0.055
22.872
3.832 262
GCNII
-2.227=0.026
9.696=2.668
2.005=0.098
GIN
-2.356_0.066
1.258
L6.30=0.161
GPS
-2.192-0.025
10.45420.610
2.160
GraphCON
-1.995
16.427 41.419
2.969
GRIT
-2.630-o.0as
20.295
1.0144:0.04 0.06
PH- DGN
-2.416
6.699 72.778
1.627 +0.398
SWAN
-2517_00022 023
4.950 265
1.121
CCC
A-DGN
-1.619-a.006
35.967=0.499 #0.492
1.981 00.077
DRew
-1.69640.00 002
32. 247 10.148
4.651 00.00
GCN
-1.606_0.005
39.706=0.44m
5. 5,233
GCNII
-1,603 000
39.911:0.08
5.241 030
GIN
0.6668
34.454 1. 201
4.869 1. 092
GPS
-1.682-0.00
33.346 226
4.758 021
GraphCON
-1566600.001
43.505=20.01
5.47420.001
GRIT
1.618
38.667
5,091
PII-DGN
1.630
37.510 1.44 1.416
5.068 126
SWAN
1.671 000007
$4.208$u0.o74
4.840
s3ap
A-DGN
-2.566_u.ve
4.425
1.176 140
DRew
2.386__0.001
6.58922..11
1.279 011
GCN
2..37
9.713
2.102 094
GCNII
-2.213_0.17
10.369 575
2.128 429
GIN
-2.138
11.868
2.23 1.23140.271
GPS
-3.115
1,255
0.472
GraphCON
-1.488=0.00m
52.104
5.73440.011
GRIT
4.11910.264
0.147 10.083
0.121 1.0.000000
PH-DGN
-2.616
4.656+3.013
1.323 + 485
SWAN
-2.782 0.205
2.905
0.89644
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Table 19: Test performance (mean + std) of different models across the ECHO-Charge task. Lower better. In bold the best model. Test loss (MSE)) is computed on the normalized dataset, while test MSE and test MAE are reported the original (non-scaled) data.
Model
Test Loss 4
Test MSE x +
Test MAE x10
A-DGN
-3.84010.00
1.456
6.543
DRew
-3.44440.06
3.669 10.49
9.086.1.447
GCN
-3.508 086
3.126
8.421 10.512
GCNII
03.46220.m1
3.490 147
40.02
GIN
-3.245
5.750
10. 10.78440.058
GINE
-3.648
2,2844 +2844+004002
7.176
GPS
-3.821 #0.018
1.620 4-0005
6.182.4..21
GraphCON
-2.879
13.25 13.25++..20 +0.200
19.62944..18
GRIT
-0.02220.017
1.765
7.134
PHIDGN
-3.59520.00 024
2.562
7.915.0.260
SWAN
3..007
1.251
6.10920.1
Table 20: Test performance (mean + std) of different models across the ECHO-Energy task. Lower better. In bold the best model. Test loss (MSE)) is computed on the normalized dataset, while test MSE and test MAE are reported on the original (non-scaled) data.
Model
Test Loss
Test MSE x10
Test MAE x10 1
A-DGN
4.85740.085
1.415
0.125
DRew
5..007
1.281 733
0.113 024
GCN
-1.210+0.010
4.561 17.176
0.281 +0.012
GCNII
-4.884+0.1900
1.560
0.132-+0.026
GIN
-3.800+o.160
12,215
ad.
GINE
-4.418 10.65
5.225 536
0.236
GPS
5.786 07786.0..10 0.1188
0.18010 1..0000.000 .0.0445
0.053 0.000
GRIT
-4.348 10.084
5.122 492
0.255 02.025
GraphCON
-4.817+0.080
0.975 242
0.143+20.00s
PH-DGN
40.44
1.359 40.
0.161
SWAN
-4.825+0.107
2.652
0.126
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I RUNTIMES
To assess the computational efficiency and predictive performance of all models, we report both training and inference runtimes measured NVIDIA HI00 GPU, well the mean absolute error (MAE) across tasks in the ECHO benchmark (see T1b Training time is measured as the average per-epoch duration over 10 epochs, while inference time computed as the average forward pass duration over 10 independent runs on the test set, using a batch size of 512. The three metrics correspond to the best hyperparameter configuration selected for each model. This comprehensive evaluation allows a direct comparison of models not only in terms of accuracy but also with respect to their scalability and practical deployability. We note that DRew's reported runtime does not include the preprocessing step, which involves computing the Flrss-Warshall algorithm (Cormen al. 2009). a procedure with cubic time complexity in the number nodes.
Table 21 highlights that while transformer-basee models like GPS achieve strong performance long-range tasks, particularly on the real-world CCHO-Charre/ECCOOEBEE -nnrrgy dataset, they do so at the cost of significantly higher computational overhead. In contrast, architectures such as SWAN and A-DGN strike a more favorable balance between efficiency and accuracy, suggesting the potential of non-dissipative DE-GNNs in overcoming the limitations of standard message passing.
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Table 21: Training and inference runtime (in seconds, mean + standard deviation) on the ECHO Bench- mark. Results for ECHO-Synth were measured on an NVIDIA H100 GPU, while ECHO-Chen tasks were measured on an NVIDIA L40S GPU. Training time refers to the average time per epoch computed over 10 epochs. Inference time refers to the forward pass on the test set, computed over 10 independent runs. both cases the batch size is set to 256. For each task, the reported values corre- spond to the best configuration of each model as selected during model selection. To ease we also report the performance of each model alongside its runtime. Note that comparison, MAE values should be multiplied by x10 for correct interpretation. DRew's ' reported runtime does not include the pre- propescsss step, which involves computing the Floyd- Warshall algorithm, a procedure with cubic time complexity in the number of nodes.
Metric
Model
ECHD-SYnch
BCH0-Chem
dien.
sesp
ecc
SCHO-Chazze
ECHO Eaeo--
Training (5)
1.430++..100
1.460
1.710
12.847=0.648
36.476+a.uu
Inference (s)
A-DGN
0.028 20.002
0.01840.000
0.027 2..000000 o01
0.191
2.121=0.397
MAE
1.151 go.03s
1.176
4.981 1 037
6.543
12.48641.00
Training (8)
1.920 20.000
1880020.000
1.760.00.10
17.618 20.325
17.318:1.061
Inference (s)
0.100 2o
0.043....0 001
0.057 1.0 002
0.606
0.847 20.4720.060
DRew
Pro-processing (6)
48.108 00, .43
48,108=0.948
48. 10,108 943
315.379 1.002
104.337=1.600
MAE
1.243+00.07
1.279:+0.011
4.651+0.008
9.086+0.47
11.325+22.04
Training
L.700
1.830 020
1.620=0.110
19.013-0.286
23. 23.03940.89
Inference (s)
GCNII
0.U71
0.U62 20.021
0.059.0
0.829
1.466
2. 2.005g0.008
2.128 429
5.241 g0.000 630
8.829 021
13.235 630
Training (8)
1. 1.480
1.790 o60
1.450 100
9.418 1:0 388
8,976
Inference (5)
GCN
0.048 20.01
0.065 2.00 003
0.046 002
0.376 472
0.385_6.38
MAR
3.832 20.202
2. 1.102 1.102200.04
5.233
8.421 40.612
18.112+1,,880
Training (6)
1.410 20
1.370
1.340
9.066
10.259
Inference (4)
GIN
=0.001
0.011900.001
1.01640..o0 001
0.065+0.00 048
1.305=00,209
MAE
1.600:0.100
2.2344:02771
4.869 092
10.784=00008
47.851
Training
N/A
NIA
A/A
18.978=0.778
13.615+00.220
Inference (s)
GINE
N/A
N/A
N/A
0.138 003
1.311_1.833
N/A
7.176 371
23.558
Training (s)
9.720 270
14.580 1..8802.c00 210
11.960 20 050
788. 788.9855.4 go 166
383.794211
Inference (5)
GPS
4.536=00.00
7.026 001
6.235 076
34,264-2..4
89.699
MAE
2.160 +0.008
0.472+0.050
4.758+0.021
6.182+0.220
5.257 5..57+0.842
Training (6)
0.990
0.920 040
0.940
7.471 4:0 429
9.037
Inference (8)
GraphCON
0.006+00001
0.00440..001
0.006+0.cor
0.001
0.146++0.21
1.080-0.534
MAE
2.969.20.180
5.734-20.011
5.474 001
19.629-20.95
14.295 807
Training (8)
2.136-+0.008
2.8988+0 +6.
2.150+0.655
55.961-0.962
35. 35.246400655
Inference (5)
GRIT
0.579 20.004
0.703-20.177
0.632:2..322
1.474
1.940=0.43b
1.01 1.14:00.00 0.040
0.121 0.003
5.091 +0.158
7.134 000
25.508 207
Training (s)
2. 2.840 _2 060
4.480 000
3.010 o60
39.405_1.P 1.5
43. 43.65311.199
Inference (4)
PH-DGN
0. 00.001
0.375 002
0.299 000
1.218 685
2.31 -00.00
MAE
1.627 +0.308
1.323 3.23+t0.44
5.068
7.915
16. 16.080
Training (5)
2.330 20.10
2. 2.130 050
2.090
54.528
20.774
Inference (8)
SWAN
0.203 00.002
0.099.+..oo 000001
0.168 0668:...00 0.01
0.595 820
1.873-0.101
MAE
1.121 20.070
0.896.0.228
4.810 ..05
6.109 103
12.629....0 807
J VISUALIZATION OF GPS ATTENTION PATTERNS
In this section, we analyze the attention patterns of the GPS model within the sssp task. These patterns are illustrated in Figure 13 We observed that, starting from the first layer, the highest attention scores are often assigned to pairs of nodes that are not directly connected and that are usually far apart in the underlying graph. Notably, the model appears to identify one or few nodes as central hubs that aggregate and redistribute information from these distant nodes. This mechanism effectively reduces the maximal traversal distance to only few hops, allowing distant nodes to communicate more easily. Therefore, this mechanism effectively reveals how the model routes long- range communication through structural shortcuts, thus confirming the long-range nature of the proposed tasks.
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x10-1
x10-2
2.0
Attention Score
-1.5
-1.0
(a) Ladder.
(b) Lobster.
Figure 13: Visualization of the first-layer GPS attention scores in sasp for the Ladder and Lobster topologies, averaged across all heads. The top 40 attention weeghtt node pairs are highlighted in color, while the original graph topology shown black.
35


Reviews:
[{"review_id": "KwiITYuIj7", "reviewer": "Reviewer_UETf", "summary": "This paper introduces a benchmark for evaluating long range information propagation in graph learning networks. The benchmark consists of 5 tasks- 3 synthetic graph-theoretic problems (single-source shortest paths, node eccentricity, graph diameter), and 2 real-world datasets focused on molecular graphs (charge and energy) for predicting atomic partial charges and molecular energies. The task design intends to stress on long range propagation based, for example, on the hops these datasets cover. Experiments are done using multiple GNNs, Graph Transformers and differential equations inspired models and insights are presented for models performing good and bad.", "questions": "na", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- There is a gap of datasets on long rage testing in GNNs and although there exists the benchmarks that this paper discusses they have limitations. So the proposed datasets address this.\n- The datasets include both synthetic and real world graphs with the real ones standing a good contribution due to the curation as well as the performance trends.\n- Empirical results show that models arguably good for long range propagation such as GPS, A-DGN, SWAN, DRew, etc outperform MPNNs, with particularly strong evidence that depth matters.", "weaknesses": "- The primary contribution of this work is the collection of molecular dataset as the similar synthetic dataset is known in the prior literature.\n- The strong claim in lines 146-147 about the molecular dataset on long range could be disputed since prior molecular dataset also involve long range affects although they could be synthetic targets.\n- The paper lacks close discussion/comparison/adaptation of known insights with 2 works which would make it stronger, particularly on quantifying ECHO datasets' long rangeness. First with [1] which has a strinkingly similar multi-task synthetic dataset. Second, with [2] which quantifies long rangeness of a graph dataset.\n\n[1] Corso, Gabriele, Luca Cavalleri, Dominique Beaini, Pietro Liò, and Petar Veličković. \"Principal neighbourhood aggregation for graph nets.\" Advances in neural information processing systems 33 (2020): 13260-13271.\n[2] Liang, Huidong, Haitz Sáez de Ocáriz Borde, Baskaran Sripathmanathan, Michael Bronstein, and Xiaowen Dong. \"Towards Quantifying Long-Range Interactions in Graph Machine Learning: a Large Graph Dataset and a Measurement.\" arXiv preprint arXiv:2503.09008 (2025).", "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": "aQ7oIocmss", "reviewer": "Reviewer_5Ph3", "summary": "- The paper argues the need for newer and better-motivated long-range benchmarks than those currently in use by the community\n- The paper introduces a new benchmark, ECHO, for long-range interactions in GNNs. The benchmark includes:\n\t- Echo-Synth: three synthetic tasks (two node level and one graph level), requiring the computation of shortest-path-based properties\n\t\t- Long-range dependency is assured as these properties require transversing the entire graph, and all synthetic graphs have diameters of at least 17 hops\n\t- Two real-world tasks based on atomic partial charges; Echo-Charge, a node-level regression of the partial charge of each atom in a molecular graph, and Echo-Energy, graph-level regression of total molecular energy\n\t\t- Long-range dependency is again ensured by graphs with minimum diameters of 17 hops, and argued based on the nature of the underlying task\n- The paper includes a comprehensive suite of experiments on a fairly comprehensive range of architectures, including classical MPNNs, a multi-hop MPNN, differential-equation-inspired GNNs and a graph Transformer.", "questions": "- [Q1] Line 284: What is 'total molecular energy'? Is it simply the sum of individual node energies?\n- [Q2] Line 1109: Your hyperparameter search for Drew only goes up to 4 layers, whereas it goes up to 40 for other models. As I understand it, Drew is an MPNN, with a receptive field determined by the number of layers. Does this not affect its LR capability? Why such a small search window?\n- [Q3] On the task of predicting atomic partial charges — \n\t- Is this a problem for which GNNs have already shown promise? Do you have a source for this? It was not obvious to me from the text\n\t- Can you contextualise the experimental results a little? I.e. do any of the models do a good enough job to reasonably replace/approximate expensive DFT computation? \n\t- To put it another way, it isn't clear to me whether GNNs are actually reasonable for solving this problem, or whether this is just an instructive task for GNN benchmarking\n\nMisc:\n- Line 208: Is this an appropriate use of 'skip connection'? I would tend to associate it with residual connections\n- Line 454: I would suggest adding a sentence here about the contents of the experiments in Appendix G, as I think they are quite interesting\n- Typos:\n\t- Line 104: 'on the need ~~of~~ **for** a new benchmark'\n\t- Line 380: 'differently, graphcon do not'", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- The need for new long-range benchmarks and the shortcomings of existing ones are well-argued.\n- The tasks, both synthetic and real-world, are at least as well motivated as existing benchmarks; I can see ECHO becoming a new and sorely-needed standard benchmark for the community.\n- The paper is very well written, and Figures 1,2 and 3 are used to great effect to illustrate the tasks introduced; I think this will be a great boon to adoption.\n- The authors evidently put a lot of time and effort into this paper; in addition to the experiments in the main text, the comprehensive additional experiments in the Appendix, and apparent 2 months of compute time required to produce the -Charge and -Energy datasets are evidence of this.\n- I did not run the code, but it appears thorough, looks easy to run and is well-documented.\n- I did not closely read the Appendix, but the inclusion of runtime measurements is welcome, as is the illustration of effective hub nodes in GPS, and the large depth and large diameter experiments in Appendix G", "weaknesses": "- [W1] The authors argue that the -Charge and -Energy tasks  are 'inherently long-range' due to (i) their large size/diameter, 17-40 hops, and (ii) the underlying task; they say: \n\t- \"The three-dimensional configuration of molecules greatly intensifies this task complexity, as distant atoms in the graph topology can still exert significant influence on electronic properties and total energy.\"\n\t- [W1.1] This makes sense to me, but do you have a source for it?\n\t- To put it another way; large diameter/graph size and intuition about the underlying task are strong indicators that a task is long-range, but do not prove it necessarily. Your experiments, in which deeper models/Transformers dominate, would also indicate that long-range interactions are present but this is somewhat circular (i.e., if the dataset is an evaluator of model LR capability, its validity as an LR dataset  cannot be based on model performance).\n\t- [W1.2] With this in mind, I would like to see a more theoretically motivated justification for the long-rangedness of the real-world datasets. (Echo-Energy is a graph-level task, but this is not necessarily inherently long-range; it could well be simply a sum over local information applied at readout.) For example, you cite Bamberger et al. (2025), who propose a range measure, as evidence for the questionable long-rangedness of LRGB. Why not apply this measure to your benchmark?\n- [W2] I think experiments would be improved by the inclusion of more than one graph Transformer, perhaps something more recent than GPS — e.g. GRIT [1], Graph-Mixer [2]\n\n---\n\n[1] Ma, Liheng, et al. \"Graph inductive biases in transformers without message passing.\" International Conference on Machine Learning. PMLR, 2023.\n\n[2] He, Xiaoxin, et al. \"A generalization of vit/mlp-mixer to graphs.\" International conference on machine learning. PMLR, 2023.", "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": "iVNPBRo3pK", "reviewer": "Reviewer_5FWW", "summary": "The paper introduces a suite of new inductive benchmark datasets specifically designed to evaluate long-range propagation capabilities in graph neural networks (GNs). The benchmark collection includes five tasks, of which three are synthetic and two are real world. The synthetic tasks consist of predicting property predictions on different topologies, two of which are node-level and one is graph-level. The real-world tasks consist of chemically grounded tasks, one being a node-level regression task where the goal is to predict atom charges, and the other is graph-level where the goal is to predict total energy. The paper argues that these datasets are indeed long-range. The benchmark is intended to fill the gap left by prior datasets (e.g. LRGB) as they received criticism recently for poor hyperparameter tuning and their long-range property having been questioned.", "questions": "1. GPS is an architecture that combines several components: positional encoding, MPNN layers, and fully connected attention. Could you please add more information on what components were used and tuned (especially what MPNN architecture and number layers)?\n2. In the synthetic datasets it says that the node features are random uniform. Wy not set the node features to zero?\n3. Could you do an ablation on the post-pooling MLP depth for the graph-level real world task?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is very well written and very clear.\n2. The paper addresses an open problem consisting of finding a good long range benchmark for GNNs, which is particularly relevant given the recent criticisms of LRGB (hyperparameter tuning, and long-range nature of the task).\n3. The hyperparameter tuning is more rigorous than previous work, directly addressing a criticism of previous benchmarks.\n4. The ECHO-Charge and ECHO-Synth is original and may foster more work applying GNNs to chemistry applications.\n5. The paper verifies the long-range nature of the ECHO-synth benchmarks by showing that the performance improves with number of layer.", "weaknesses": "1. **Long range evidence for real world tasks is lacking** The evidence for the long-range nature of the real world benchmark (ECHO-Charge and ECHO-Energy) is limited. Indeed there seems to be no correlation between depth and performance (Fig 7d). The only argument is that long-range architectures (such as GPS and SWAN) outperform standard supposedly not long range architectures. This argument is very similar to the LRGB arguments which were shown to be limited. On the other hand, the hyperparameter tuning in this work is more thorough.\n2. **Hyperparameter tuning can be improved** The main criticism of LRGB in Tönshoff et al is that increasing the MLP depth post-readout improves performance on the proposed tasks. However, in this work the post-readout depth is set to 2 without tuning. In light of this, more thorough look at this hyperparameter would be interesting to the community and would further improve the hyperparameter tuning.\n3. **[Minor] better evaluation of different topologies** In the synthetic datasets, the motivation for all different topologies are for example to include bottlenecks, however performance per graph topology was not evaluated.", "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": "bFmVT5mUIx", "reviewer": "Reviewer_DTw9", "summary": "This paper introduces a new benchmark for long-range graph propagation in graph learning, motivated by the fact that existing benchmarks (as the LRGB) have recently been questioned to actually capture long-range ability and somewhat plateaued. The authors introduce ECHO, consisting of a synthetic and real-world part. The synethetic tasks consists of single-source shortest path, node eccentricity, and graph diameter, and are to be predicted on graphs sampled from 6 different (sparse and high-diameter) graph topologies. The real-world tasks on molecules are predicting atomic charges (ECHO-Charge) and total energy (ECHO-Energy). The new benchmark is evaluated on a wide range of common graph learning methods, including standard MPNNs and graph transformers. Results indicate that models with global information exchange are performing better on ECHO.", "questions": "- Could you please refer to the points from “Weaknesses”?\n- Do you have any intuition for the U-shaped error in fig. 8(b), and, to some extent, (a)?\n- Superficially, ECHO-Synth resembles the graph property prediction (GPP) dataset from [2] (same tasks but different graph topologies).  Since [2] is cited, it would help to clarify the relationship, i.e., if/how ECHO-Synth builds on or departs from GPP, more explicitly (perhaps in sec. 3.1).\n\n[2] Alessio Gravina, Davide Bacciu, and Claudio Gallicchio. Anti-Symmetric DGN: a stable architecture for Deep Graph Networks. ICLR 2023.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- I personally think this is an important and timely contribution. I think that the graph ML community urgently needs new and well-designed long-range propagation benchmarks, as performance on the LRGB has plataued and been questioned to really probe long range propagation. Yet, most current methodological works still use the LRGB, especially the peptides tasks, often as the sole empirical basis for any claims regarding long-range ability.\n- The benchmark design is quite good (synthetic and real-world), and it is clear that significant thought and computational resources went into the creation and evaluation on the baselines. For example, the benchmark is large-scale (200k graphs for the ECHO-Charge/Energy), and in the synthetic case, the graph topologies are properly stratified in train/val/test.\n- The experimental rigor is high. Hyperparameters were extensively tuned on the benchmark (app. F), which is crucial given how untuned baselines have led to incorrect conclusions in the past.", "weaknesses": "- The graphs in all tasks are quite small (order of roughly 50-200 nodes). The paper could be more upfront about the benchmark not probing scalability, even though many current methodological works target methods that both capture long-range interactions and scale well. It might also help to explicitly position the work against newer long-range benchmarks that explicitly test scalability. [1]\n- Not really a weakness but a suggestion: For the synthetic tasks, readers might initially think that the targets are simple monotonic functions of graph size within each topology, reducing the problem to a topology classification. I checked the data and this does not seem to be the case, but additionally grouping plots like Fig. 6(a) by topology could better convey task difficulty and make the reported MAE values more interpretable.\n\n[1] Huidong Liang, Haitz Sáez de Ocáriz Borde, Baskaran Sripathmanathan, Michael Bronstein, Xiaowen Dong. *Towards Quantifying Long-Range Interactions in Graph Machine Learning: a Large Graph Dataset and a Measurement.* https://arxiv.org/pdf/2503.09008", "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)