SONAR: Long-Range Graph Propagation Through Information Waves
Authors: Alessandro Trenta
Alessio Gravina
Davide Bacciu
Venue: NeurIPS 2025 poster ()
Keywords: graph neural networks
long-range propagation
Primary area: deep_learning

Abstract: Capturing effective long-range information propagation remains a fundamental yet challenging problem in graph representation learning. Motivated by this, we introduce SONAR, a novel GNN architecture inspired by the dynamics of wave propagation in continuous media. SONAR models information flow on graphs as oscillations governed by the wave equation, allowing it to maintain effective propagation dynamics over long distances. By integrating adaptive edge resistances and state-dependent external forces, our method balances conservative and non-conservative behaviors, improving the ability to learn more complex dynamics. We provide a rigorous theoretical analysis of SONAR's energy conservation and information propagation properties, demonstrating its capacity to address the long-range propagation problem. Extensive experiments on synthetic and real-world benchmarks confirm that SONAR achieves state-of-the-art performance, particularly on tasks requiring long-range information exchange.

SONAR: Long-Range Graph Propagation Through Information Waves
Alessandro Trenta"
Alessio Gravina"
Department of Computer Science
Department of Computer Science
University of Pisa Pisa, Italy
University of Pisa Pisa, Italy
alessandro. ttrettapph..umpp......
alessio. seresio.graiiaadde unipi.
Davide Bacciu
Department of Computer Science
University Pisa
Pisa, Italy
davide bacciueunipi.it
Abstract
Capturing effective long-range information propagation remains fundamental yet challenging problem in graph representation learning. Motivated by this, we introduce SONAR, novel GNN architecture inspired by the dynamics of wave propagation in continuous media. SONAR models information flow on graphs oscillations governed by the wave equation, allowing it to maintain effective propagation dynamics over long distances. By integrating adaptive edge resistances and state-dependent external forces, method balances conservative and non- conservative behaviors, improving the ability to learn more complex dynamics. We provide rigorous theoretical analysis of SONAR's energy conservation and information propagation properties, demonstrating its capacity to address the long-range propagation problem. Extensive experiments on synthetic and real- world benchmarks confirm that SONAR achieves state-of-the-art performance, particularly tasks requiring long-range information exchange.
Introduction
Graph neural networks (GNNs) 67 76,
10. 18 have become a powerful framework for process- ing graph-structured data, enabling applications across various domains such social networks
159
molecular biology 36, 89 43 5l and more [8 58
Most GNNs are built upon the
Message-Passing Neural Networks (MPNNs) framework (36), where information is exchanged
between neighboring nodes, enabling effective learning from local graph structure. Despite their widespread use and success, persistent challenge in GNNs the effective modeling of long-range dependencies within graphs, as information propagation tends to degrade over extended distances. This is caused by phenomena such as over-smoothing
19, and, more generally, vanishing gradients [3] Several recent approaches have been developed to better cap- ture long-range dependencies in graphs, including graph rewiring [35
7 over-squashing
77 multi-hop GNNs 52 and graph transformers
[ 20], patrrenttallquuttonnn based GNNs (DE-GNNs) [13
39
40
72 88). Graph transformers, in particular, have gained popularity due to their use of attention mechanisms, which allow any pair of nodes to exchange information directly. However,
78
23 90
Equal contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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their quadratic computational cost poses scalability challenges. Moreover, recent studies have shown that, on some long-range tasks, they may actually perform worse than standard MPNNS 82]
To address the long-range propagation problem, we draw inspiration from wave propagation in continuous media, where signals travel vast distances with minimal energy loss (29 We propose SONAR (Structured Oscillatory Graph Neural Network with Adaptive Resistance), a novel DE-GNN with MPNN-like architecture that models information propagation through the lens of wave-based dynamics. Unlike diffusion-basse baaee propagation schemes such as [59] 85 89], which are inherently dissipative and struggle to capture long-range interactions 3] wave-based dynamics preserve signal energy and enable effective communication across distant nodes. This mechanism is analogous to acoustic wave propagation in the ocean, where, under specific temperature and pressure gradients, sound waves can become trapped in the SOFAR channel [30]. This natural waveguide allows acoustic signals to travel across thousands of kilometers with minimal attenuation. SONAR adopts a similar principle in the graph domain: by treating information as wave propagating through the graph topology, maintains signal fidelity over many hops and facilitates robust long-range interactions. Specifically, it adopts a mathematically grounded formulation to model wave dynamics [34] resulting in inherently linear propagation dynamics with strong theoretical guarantees. Moreover, SONAR introduces adaptive resistances that enable each edge to control the flow of information, along with external forces that balance conservative and non-oonservative dynamics, allowing the model to capture more complex behaviors and improve performance on downstream tasks.
The key contributions of this work are the following: (i) We propose SONAR, novel MPNN-like architecture based on the wave equation for graphs, which enables the balance and integration of non-dissipative long-range propagation and non-conservative behaviors. Additionally, it incorporates adaptive resistances that allow each edge to modulate the information flow. (ii) We theoretically prove that, in its conservative form, SONAR does not dissipate the energy of features in the graph, i.c., information is preserved. In its continuous form, the sensitivity matrix between any two nodes never vanishes but oscillates following a cosine function. (iii) We employ additional dissipative and forcing terms, allowing SONAR to mediate between pure conservative and non-conservative behaviors. This gives SONAR the flexibility to learn to filter out irrelevant information. (iv) We conduct extensive experiments to demonstrate the benefits of our method. SONAR consistently achieves on par or better performance than existing methods across diverse tasks and benchmarks. Notably, SONAR outperforms existing ttate-of-the-a methods on synthetic long-range benchmarks, where accurate modeling of distant node interactions is crucial, highlighting its strength in long-range propagation.
2 Related Work
Long-Range Propagation in GNNs. Effectively modeling long-range dependencies remains central challenge in deep learning for graphs 777 3. GNNs usually rely on local neighborhood
[2] [9] due to challenges such as over-smoothing [11 68 73 and over-squashing 2 83 19], which are linked
aggregation, which limits their capacity to capture interactions between distant nodes
to the problem of vanishing gradients [3]. Therefore, GNNs based on message passing exhibit a
performance degradation in tasks requiring more global context, such as molecular property prediction
24 To address these challenges, a variety of strategies have been proposed. Graph rewiring methods, including SDRF 83 DIGL [35]. FoSR [57] and DRew [6), modify the graph topology to
facilitate long-range communication. Other approaches include regularizing the model's weight space
39
41 40), exploiting port- Hami-tanilnnn dynamics (52), filtering messages in the information flow
28
using graph adaptive method based on learnable ARMA framework (25), or using multi-
hop information in a single update (21). Graph Transformers 66 72 78 80 90) offer an alternative paradigm by enabling direct message passing between any pair of nodes via attention mechanisms. While effective, most of these methods often introduce additional computational complexity due to denser graph shift operators or all-pairs interactions.
GNNs based on Differential Equations. Recent advancements in the field of representation learning have introduced new architectures that establish a connection between neural networks and dynamical systems. Building on foundational work in recurrent neural networks [6 47), this perspective has been extended to GNNs (0). Indeed, works like GDE (11, GRAND 13, PDE-GCN (26). DGC [87)
GRAND++ 81 propose to interpret GNNs as discretisations of ODEs and PDEs. Methods in this
class impose a bias on node representation trajectories to follow the heat diffusion process [3 87
exploit non-dissipative dynamics [39 40), or hamiltonian dynamics [56] 91 52 GraphCON
74
2

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further explores oscillatory dynamics to preserve the Dirichlet energy encoded in the node features and mitigate oversmoothing. In GraphCON, each node acts as an oscillator that exchanges infor- mation with its neighhors through simple GCN 59 or GAT 85
In contrast, SONAR adopts mathematically grounded formulation of graph calculus 3 to model wave propagation on graphs 34, resulting in inherently linear dynamics that offer stronger theoretical guarantees for onggrangge information propagation. Moreover, SONAR introduces adaptive resistances (which allow each edge to modulate how information is transmitted) well as external forces that enable the mod- eling more complex dynamics. The spatio-temporal evolution of graphs has been studied in
27
2 55
SONAR
In this section, introduce SONAR (Structured Oscillatory Graph Neural Network with Adaptive Resistance), a novel GNN architecture whose information flow designed as the wave equation on graphs, enabling long-range information propagation between nodes. Figure provides visual representation of the overall architecture and wave propagation dynamics.
(12)),
Blocks
Block 2
MLP
Linear Recuprence
Block
SONAR Eq.
SONAR Eq.
X
(1-),0
(a) Deep SONAR architecture
(b) SONAR propagation dynamics
Figure 1: Illustration of (a) deep SONAR architecture and (b) the SONAR propagation dynamics of
one block.
3.1 Notations and Preliminaries
We consider graph G (V.E) the system of interacting entities called nodes, where is of n nodes and & C V x V is set of m edges. The structural information expressed by & can be encoded in the adjacency matrix A {0, 1) nx i.e., a binary matrix where A,j if (i, & and zero otherwise. We define the neighborhood
node as the set of nodes u directly linked with o, i.e., (()) = {u] Aue 1). and the degree of node v as the number of its neighbors, deg(v) = (V)). Lastly, we consider the Laplacian matrix L and two normalizations, i.c., symmetric and random-walk normalized Laplacian matrices, which respectively defined as
L A.
= I -D-I-DD**ABB (D*)*A(D***
-
(1)
where I is the identity matrix, D is the diagonal degree matrix with D; deg( (1:) and generic node. + is the pseudo-inverse of D. Following (34), the definition of graph Laplacian can be generalized its weighted version and to any node function (u) V R as
(af)(=) (L*)() (D* (o) (A*)(v)
Σ ane (s(v f(u) (EN(e)
where D and A a the weighted degree and adjacency matrices, and awo R_ is weight on the directed edge (u, v) E E.
Rdd, which provides the evo- lution of the node representation at time in the system. The term X(t) [xo(t), xx, xt)] Raxd is the matrix of all node states at time t. We note that in the domain of anfferentialequaatinn the continuous dynamics expressed by the neural network is discretized into classical GNN layers [39]. Therefore, given the initial condition (node input features) x, (0) Xx, each GNN layer 11..., computes node states x5, which approximate x, hl), with h being the integration step size.
Each node in the graph is associated with a hidden state vector x,, (t) x,,(t)
3

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The wave equation in physics. Waves in physics describe the propagation of oscillations in continuous medium. Common examples are electromagnetic waves [62] and sound waves [48]
8 c² 8' where ft,, is scalar function describing the wave and is its propagation speed, which is an intrinsic characteristic of the conditions and medium. Directly descending from their equation, waves have interesting properties: (i) they can travel indefinitely without losing amplitude and energy at constant speed [29], and (ii) they can be linearly composed and decomposed, allowing for superposition of waves propagating from different directions [62]. More specifically, the total energy E(()) of a wave on R is defined as
These are described by the wave equation 8r2
dz2
E(t) = /
1010 0t (()
(8f) 0z
2
-
dx,
and can be shown to be constant 29 in the whole space and even on bounded domains. Our objective to exploit these properties to propagate energy and information indefinitely graph.
Waves following a non-conservative dynamic can be modeled with the addition of a dissipative term D(t, x) = k f (t, z)), which dampens the wave and dissipates energy, or with an external forcing term F (t., ar), which represents other environmental factors acting on the wave.
3.2 The SONAR wave propagation
To model information propagation on graphs as oscillations in continuous medium, we formulate the dynamics of SONAR through a differential equation that follows the graph wave equation. Specifically, we start by defining the evolution of node states as:
(t) LX(t),,
(4)
where (() is the second-order derivative of X(),, W E Rdxd is a learnable weight matrix, and X contains the initial node states. To more closely mimic wave propagation in physical media, we adopt the weighted graph Laplacian in Equation 2 allowing different different edges to modulate how signals travel through the graph. In other words, each pairwise communication between nodes can be viewed as residing in distinct medium, where the wave properties, e.g., speed and amplitude, vary depending on the properties of the medium. This is analogous propagating signals at different depths in the ocean, where specific water temperature and pressure alter the wave propagation capabilities. Therefore, in our setting, the Laplacian weight Oue can be interpreted as an inverse resistance term that governs the ease with which information flows between nodes u and v. Although Que can be implemented as static (fixed) weights, in our experiments in Section f we employ neural networks to learn such weights in order to adaptively model the dynamics of the propagation on the specific task
To further cast the system dynamics in the more general setting, then introduce the possibility of trading between the conservative behaviour typical of undampened wave equations (discussed in detail in Section 3.3) with con-conservative dynamics. We introduce dissipative and external forcing terms to alter the conservative evolution of node states. Therefore, Equation 4 can be rewritten to include the new terms as
(t) = (t - -D((()) o X(t) + F(X(t))
(5)
where (()(t)) E x and F(X(t)) E gaxd denote the state-dependent dissipative and external forcing terms, respectively. @ is the Kronecker product. We note that without D(X and (t)( the purely conservative inductive bias of the wave equations constrains node states to evolve along conservative trajectories, which may limit the model's 's effectiveness and hinder its ability to capture the more complex dynamics required for downstream tasks. Therefore, the dissipative term allows irrelevant information to be dissipated during propagation, while the external force introduces additional flexibility to control the dynamics based on input-dependent signals. In our experiments in Section we employ neural networks to learn both the dissipative and external forcing terms.
Discretization and implementation details. As for standard DE-GNNs, a numerical discretization method is needed to solve Equation (5). We solve the equation with finite difference scheme 31
Note that disconnected nodes always have zero adaptive resistance. Thus, the resistance aua is calculated only between connected nodes.

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To this aim, we first introduce the auxiliary velocity variable to be the first-order derivative of the node states, i... V(t) (t), to rewrite Equation
as the first-order system:
X (t) V(t) (it) - o +()+F(XX())
(6)
This system can be now discretized by iteratively solving
- hVe+1, + & V V -((X)))
(7)
where is the step size of the discretization method, and X = X and v = Wv,,, with Wv Edxd learnable matrix for the initial velocity. Note that edge features can be seamlessly incorporated into Equation 7 as detailed in Appendix A.1 As discussed above, we can discretize time into GNN layers, such that the embedding x from Equation 7 encodes information that is hops away from v.
We note that, when dissipation and external forcing are not employed, information is linearly aggregated. To increase effectiveness, we can build a deep architecture for our SONAR, following principles established in modern architectures 66 45 12 Therefore, define SONAR block by applying an MLP to the output of Equation 77 after L propagation steps, introducing nonlinear dynamics into the model. Multiple SONAR blocks can then be stacked form deep architecture. where the output of block i becomes the initial condition of the subsequent block (i + 1):
X (O)L SONAR (X (0)0)
(8)
x (++1),0 MLP(X
This concept is visually summarized in Figure la
In our experiments in Section 4 the Laplacian weights dae are implemented as the output of an MLP that takes the states x,, and x input, followed by ReLU activation to ensure positive output. The dissipative term modeled using an MLP applied to the current node state, also followed by ReLU to enforce non-negative outputs. The external forcing term is computed via an MLP without activation constraints, allowing it to output both positive and negative values. As a result, each propagation step (i.e., iteration of the discretization step) L in each block is associated with its own set of learned Laplacian weights, as well as dissipative and external forces. Each SONAR block is equipped with its own set of parameters for learning the Laplacian weights, the dissipative and external forcing components, as well as its own learnable matrices for initial velocity w and message passing aggregation weights (ii) (which are shared within the block). We consider both the number of blocks and propagation steps as hyperparameters in our experiments.
3.3 Theoretical Properties of SONAR
We now provide theoretical statements about energy and information conservation properties of SONAR, showing that our model effectively performs long-range propagation between nodes. Ap- pendix B provides the proofs for the statements.
SONAR allows for long-rane range propagation. To prove that SONAR has a conservative behavior, we start by defining the energy of the system, adapting Equation B the graph domain,
E(t) = Σ 1 VX, NEV
(x(()
(9)
0t
polential energy
kinetic energy
The energy contains two components: the potential energy, which measures how much the signal varies across the graph (i.c., spatial variation f and the kinetic energy, which captures how node states change over time (i.e., layer-wise variation). In other words, the kinetic energy tells how fast the wave vibrating, while the potential energy reflects the tension in the system. By maintaining a constant energy, we therefore ensure that node information is preserved during propagation.
Thhe potential energy is equivalent with the Dirichlet energy in 37 employed to measure over-smoothing.
5

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To better highlight how node coupling and SONAR convolution affect energy conservation in the graph, we now focus on single feature X(t E R" The generalization to multiple features is left to Appendix B The following theorem shows that without dissipation and external forcing (i.e., with propagation similar to Equation (4), the wave propagation through the graph conserves the energy.
Theorem 3.1. Let X(t) R" be the node states at time t, obtained as the solution the graph wave equation in Equation 5 with initial condition X(0) X. null dissipative and external forcing terms. Then, the energy (()) in Equation
(9) is conserved, that is, E(()) E(() VI 0.
This result implies that information encoded in the node states neither amplified nor dissipated over time (i.e., while traversing the graph), but redistributed in lossless manner across the graph. This property enables SONAR to perform deeper propagation, making it particularly suitable for tasks requiring long-range interactions.
To provide clearer picture of the long-range capabilities of our SONAR, we follow the recent literature [83 19 and evaluate the long-range propagation ability by measuring the sensitivity of the node states. Specifically, we first measure how sensitive a node state at an arbitrary time ( with respect to the initial state of another node, i.e., 0x, (0),))))) Then, we compute the same sensitivity for the discretization version of our SONAR, i.e., 0xt//xx. and compute its norm similarly to [9]. We consider SONAR with null dissipative and external forcing terms, and initial conditions (() = X R,, V(0) = 0. The explicit solution of our system (following (34) is
(t) = cos (tvL) X,
(10)
where cos [ =
of VL 2n We drop the Laplacian superscript " to ease notation
and explicitly write the sensitivity matrix in the following theorem.
Theorem 3.2 (Sensitivity matrix, continuous case). Let (t) be the state for node v at time t. Then,
the sensitivity (x,(t) between nodes u and u is
(0)
(x. ix,, (0)
= COS (tVL)
(1I)
The proof is consequence of the solution in Equation (10). Theorem 3.2 shows that the influence of node u on node u oscillates following cosine function, but never vanishes definitively.
We now consider a full discrete message passing with state vectors X E Raadd and measure the sensitivity after e steps (i.e., layers).
Theorem 3.3 (One-step sensitivity matrix, discrete case). The sensitivity matrix 8x. Equation with null dissipative and external fxternal terms is given by
E Raxd of
ax!
0x
0x 1-1 21
L W.
(12)
The step size h balances two components: the residual information from the node itself, related to the term 2I... and the signal coming from neighboring nodes, related to L... W. This last term is also proportional to the difference between node features (see the definition of the graph Laplacian in Equation] 2) encouraging the exchange of information when neighboring nodes are different.
Similarly [19,, we now assess sensitivity of our SONAR in Equation (7) for the full non- conservative case, i.c., with dissipation and external forcing.
on node states X(t) E paxd Let the dissipation coefficient be such that (D(X))] < k and the external forcing be F = X'Wrr. Let the initial velocity be calculated as V(0) = XW Finally, let
Theorem 3.4 (Sensitivity bound, discrete case). Consider the SONAR in Equation
max W Wrl. W 1...1). Then, the sensitivity matrix has the following upper bound
OxL
s (wd) ((11+ 12+ + 1)) +) 1))1+H2A))
(13)
Ox!
L
vm
where max,gv N, max.ev (()' is the maximum degree the graph, d is the number of node features, and h is the step size of the discretization.

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This result demonstrates that the sensitivity of SONAR remains well-controlled across layers. We note that classical MPNNs usually include a factor c in the bound, with c, the Lipschitz constant of the nonlinearity a. In practice, this term often decay extremely fast, limiting the ability of standard MPNNs to propagate information over long distances. By contrast, the linear propagation dynamics of our SONAR allows for stable long-range information flow. Therefore, together with the previous theoretical results, it holds the capability of SONAR to perform long-range propagation effectively.
We remark that Theorem 3.1 and Theorem 3.2 do not depend on the choice of the discretization of the solution, while the results in Theorem 3.3 and Theorem 3.4, which explicitly depend on the step size, can be easily extended to any other integration procedure.
Complexity Analysis. SONAR consists of a stack of blocks, each with complexity of MPNN (e.g., 159 (- Specifically, each iteration of Equation (7) is linear in the number of nodes (n) and edges (m), therefore it has a complexity of O( + m)) Assuming that I. iterations are performed, SONAR block has a complexity of [L.n + m) + p), where is the complexity of the MLP at the end of the block, as defined in Equation (8)
Experiments
In this section, we empirically validate the practical benefits of our method on popular graph bench- marks for long-range propagation as well as heterophilic node classification tasks. In Sections [ and 4.2) assess SONAR on synthetic benchmarks that require the exchange of messages between far-away nodes, thus performing long-range propagation. Specifically, we consider the graph transfer tasks from [4)] and the task of predicting three graph properties from (3) With the same purpose,
we verify our method on the real-world long-range graph benchmark 24 in Section 4.3 Moreover,
we assess the performance of SONAR on heterophilic tasks from 770 in Section 4.4 In Section
we empirically assess the long-range capabilites of SONAR in terms of the sensitivity metric (dis- we report additional ablation studies to provide a more
cussed in Section 3.3). In Appendix C.2 comprehensive understanding of SONAR, discussing runtimes and the role of adaptive resistance, dissipation, external forces, and step size. The performance of SONAR is compared with sfte-of-the- art methods, such as MPNNs, DE- GNNs, higher-order GNNs, and graph transformers, detailing the employed baselines in Appendix A.2 We report the hyperparameter space used in our experiments in Appendix A.4 All the experiments are performed on server with NVIDIA H100 GPUs. We openly release the code at hhttts://// /ttp::://gitubb.... com/ hwtg//aravinnaaaa.
4.1 Graph Transfer Task
Setup. We consider the graph transfer task proposed by 19 under the experimental setting of [4]. The objective of this experiment is to transfer a label from a source target node placed at increasing distance L, and measure how much information is propagated through the graph. We initialize nodes with a random valued feature, and we assign values "I" and "0" source and target nodes, respectively. We consider three graph distributions, i.c., line, ring, crossed-ring, and four different distances = (3. 5, 10, 50). Thus, we consider short to extreme long-range scenarios. As & increases, the task becomes progressively more challenging, demanding more effective long-range information propagation. Due to oversquashing, the performance is expected to deteriorate with larger l. Consequently, addressing this problem requires methods with increasingly robust mechanisms for maintaining information flow across distant nodes. For this experiment, we use the same data, hyperparameter space, and experimental setting of 44))
Results. The results of SONAR and baseline models on the graph transfer task are shown in Figure 2 where we report the test mean squared eтror with standard error bars as a function of the distance between the source and target nodes. This task specifically designed evaluate a model's ability to propagate information across varying distances, making it critical benchmark for assessing long-range capabilities. SONAR clearly outperforms all baselines at large distances (most notably at 50 hops) empirically validating its ability to support long-ran rangrange information propagation. Moreover, SONAR achieves state-of-the-ee performance across all tested distances, including shorter settings such as 3, 5, and 10 hops. Notably, while other models exhibit significant degradation as the distance increases, SONAR maintains a nearly constant error, demonstrating that can propagate information effectively over both short and long ranges with consistent accuracy.
7

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GIN
GCN
GAT
+
GPS
A-
SWAN
PH-DGN
SONAR
10-1
10-
Mean Squared Enor
10
10-2
10
10
10
10-4
10
10
10
10-1
10
35
35 10
10
Source-larget distance (#hops)
Source-Target distance (#hops)
Source-larget distance (#hops)
(a) Line
(b) Ring
(c) Crossed-Ring
Figure 2: Information transfer performance on (a) Line, (b) Ring, and (c) Crossed- -ingg graphs. Overall, SONAR transfers the information more accurately distance increases, achieving lower error than non-dissipative methods (i.e., A-DGN, SWAN, PH-DGN) and transformers (i.c., GPS).
4.2 Graph Property Prediction
Table 1: Mean test set (MSE)() and std averaged on random weight initializations on Graph Property Prediction tasks. The lower, the better. First, second, and third best results for each task are color-coded.
Setup. We consider the three graph property prediction tasks proposed in [39] and investigate the perfor- mance of our SONAR in predicting graph diameter, single source short- est path (SSSP), and node eccen- tricity on synthetic graphs sampled from multiple distributions. These tasks inherently demand the ability to capture and transmit information across the graph, they are based on shortest-path computations. In- deed, similarly to standard algorith- mic approaches (e.g.. Bellman-Forde Dijkstra's algorithm), accurate solu- tions depend on the exchange of mul- tiple messages between nodes. Con- sequently, models that struggle with long-range propagation result poor performance. For this experiment, we use the same data, hyperparam- eters space, and experimental setting presented in [39].
Model
Diameter
SSSP
Eccentricity
MPNNS
0.7424 0.066
0.9499 0001
20,025
GAT
0.8221 0.822100.0762
0.6951+40
0.7909
GrapSSAGE
0.8645_00.001
0.2863-2-0.188
0.7863-20.0200
GIN
0.6131 +0.0000
-0.5408 40.4199
0.9504+0.000
GCNII
0,5287 20.0570
-1.1329 20.0133
0,7640
-0.5891 0700200
3.9579920.0760
0.0515 1818
DE-GNNs
DGC
0.6028 00.000
-0,1483=0.022
0.8261 +0.0082
GRAND
0.6715-20.0900
-0.0942
0.6602++..150
GraphCON
0.0964-00.0220
-1.3836
-13883620.00092
0.683320.0077
A-DON
-0,5188
-3.2417 10.117100770
0.4296
.0.5981 10.416 1140
-3.5425 0.0080
0.07719.0.2100
PH-DGN
-0.5473+0.1074
.4,2993
-0.9348
Graph Transformers GPS
-0.1121
3,5990
0.6077
Multi-hop GNNs DRew-GCN delay
-23692
-1.5905
2.10044 00.04
1.4018-to.109
-1.6023 20.007
-2.0291 22.91.0000000
Our SONAR
33.2906
.6.5517 10.000
.3,1187 10.0020
Results. We report results using the (Mgi) metric in Table This benchmark focuses on predicting shortest-path -attssst-aah-hhee patt--aseeee based properties that inherently require the model to capture the global graph structure, thus long-range information. SONAR sets new performance standard by outperforming all existing methods by at least one order magnitude across all tasks. While standard MPNNs struggle to model such properties due to their limited ability to propagate information over long distances, SONAR consistently surpasses DE-GNNs (its direct competitors), multi-hop GNNs, and transformer- basedd models, which are typically more computationally demanding. Notably, SONAR improves over the best-performing baseline by 0.89 points on Diam., 1.01 on Ecc., and remarkable 2.46 the SSSP task, further demonstrating its effectiveness in capturing long-range dependencies.
4.3 Long-Range Graph Benchmark
Setup. To assess the performance on real-world long-range benchmarks, we consider the "Peptides- func", "Peptides-struett, "PascalVOC-SP" tasks [24). The first two tasks involve predicting the functional or structural properties of graphs derived from peptides, which are large molecules based amino acid chains. Similarly to previous tasks, accurate predictions require effectively capturing long-range interactions, as these properties are determined by the interplay between distant regions of the graphs, i.c. peptides. The "PascalVOC-SP" is node-classification task superpixel

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graphs, which is considered to be more difficult than the peptides ones in the long- Longerange context 66. We use the same data and experimental setting in [24]. including the 500k parameter budget.
Results. We present the performance of SONAR alongside leading base- lines in Table 2 with a more ex- tended comparison in Appendix C.1 SONAR achieves significantly bet- ter results than standard MPNNs and generally outperforms most DE- GNNs (i.e., SONAR's model class) and multi-hop GNNs across all tasks. On func, SONAR ranks second overall, only one point be- hind DRew, which relies on expen- sive graph rewiring. On struct, SONAR achieves a performance with standard deviation overlapping with top models. On PascalVOC-SP SONAR improves the Fl score of transformer models by a remarkable 8.3%, ranking first with or without positional encoding. Overall, com- pared to transformer-based models with quadratic complexity, SONAR achieves comparable or better perfor- mance while maintaining linear com- plexity. We conclude that the long- range capabilities of SONAR are also evident in real-world tasks.
Table 2: Results for Peptides-func, Peptides-struct and PascalVOC-SP averaged over 3 training seeds. Baseline re-
sults are taken from [24 46 51 6.
40 (2.. Note that all
MPNN-based methods include structural and positional en-
coding. The first, second, and third best scores are colored.
Peptides-fune AP
Peptides-struct MAE $
Pascal VOC-SP F1
Model
MPNNs
GaadGCN
58 28.64
0.342020.000 0013
0.2873 0219
GCN
59.30
0.349610 0013
0.1 11128
00c0
GCNII
55.43
0,3471 4000000
0.1698 00.0000
GINB
54.98.0077
13344720.00015
0.1265 p076
Multi-hop GNNs
DIGLL+MPNN+..n DRew-GCN
68.30
0.261610.000
0.2921 1:0 0038
69 0.77770
0.2781 40.0022
0.1848 +0.00107
DRow-CCN+LapPE
71.50 2..44
0.25361 0015
0.1851
2:0 0092
MixHlop GCN
65.92 10.32
0.2921 10
0,2506 1:0
MixHlop GCN+LapPE
08.45
0.201420.0023
0.2218 0.00174
Transformers
GraphiGSSLLaPEE
63.35 2..41
0.2500 000s
0.3748
20.0109
Graph VIT
69.42.0.75
0.2449 00-0016
GRIT
69.8820.88
0.2460 ::0 0012
SAN-LapPE Transfomer-LapPE
63.84=1.21
0.2683+0..0m
0.3230
21.26:11.28
0.2529
0.2529
00.0018
0.2694
20094-00.0006
DE-GNNs
GRAND
57.89 10.62
0.3418 0015
0.1918 0007
GraphCON
60.22
0.2778
0.2
A-DGN
59.75=0.44
0.2024
0.2349 +22349+0.000 +0.0054
SWAN
67.51 20.59
0.2485 0008
0.3192 0250
PH-DGN
70.12
0.2465 0.2465-40.0022 40000000
Ours
SONAR
68.42-0.11
0.2525 +-000
0,4058 +0.0000
SONAR+LapPE
70.47
0.2456.2o.om
0.4082 0037
4.4 Heterophilic Tasks
Setup. To further evaluate the performance of our SONAR, we assess its the effectiveness capturing complex relational information in heterophilic settings, where nodes belonging to same class are often connected through longer and sparser paths, we consider the five node classification tasks introduced in (70]. Specifically, we consider the "Roman-empire", Amaazon-atinngs" "Minesweeper", "Tolokers", and "Questions" datasets. We adhere to the same data and experimental setting presented in (70).
Results. We report the results in Table 3 (extended comparison in Appendix C.I.p SONAR achieves the best performance on both the Roman empire and Minesweeper tasks, surpassing the second-best models by 0.8 and 2.8 points, respectively. It also performs competitively on Amazon- ratiing and Tolokers, with less than one point difference from the top models and overlapping stds. Remarkably, SONAR outperforms all deterophily-desiinatee GNNs by up to 20 points, except for the Questions dataset. This shows the flexibility of our approach on different tasks.
4.5 Empirical Sensitivity Analysis
We empirically assess the long-range capabilities of SONAR compared to a standard GCN, which can be considered as the most comparable baseline to our proposed SONAR. Both rely on the Laplacian operator and message- message-passing paradigm, but GCN lacks energy preservation guarantees for long-range information propagation, adaptive resistance, and external forces. For this analysis, we consider the Line task in Section 4.1 with L = 50 and measure the norm of the sensitivity matrix in Equation (13) between any node v and the source node u. Specifically, we compute the sensitivity considering increasing distance between u and (i.e., 10. 20, 30, 40, 50) with respect to increasing values of model recurrences (i.e., number of explored hops). The results, reported in Table confirm our theoretical findings in Section 3.3 the information propagation of SONAR leads to sensitivity norms that never vanish, even at higher distances. On the contrary, the dissipative dynamics of the GCN cause exponential decay in the influence from the source node to distant nodes.

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Table 3: Mean test set score and std averaged over 4 random weight initializations on heterophilic datasets. The higher, the better. First, second, and third best results for each task color-coded.
Roman-empire
Amazon-ratings Acc
Minesweeper AUC
Tolokers AUC
Questions AUC
Model
Acc
MPNNs
GAT
80.87
49.09=0.00
92.01 +0.08
83.70
21.43
Galed-GCN
74,46
_0.00
87.54
77.31 14
76.61
GCN
73.69
48.70_6.00
89.75 40.00
83.64
76.09
SAGE
85.7440.07
53.63-0.00
93.51 40.67
82.43+0.44
76.4440.02
Graph Transformers
Exphormer
89.03-1..37
53.51 0..46
90.74.20.52 53
83.77
73.94 1.06
NAGphonner
74.34.0.77
51.26
84.19
78.32
68.17
GOAT
11.59
44.61 =0.50
81.09+1.02
83.11+1.04
75.76+1.66
GPS
82.00+0.011
53.10-0.44
90.63 0.077
83.7140.48
1..73
Ser (RWSE)
87.04
49.92_0.6m
91.08 10.58.
84.3810.81
77.14 17.14
20.71
51.17 oo
76
83.23 64
77.95 77.95 20.06
GT-sep
87.32=0.33
52.18=0.50
92.29=0.44
82.52+00.2 02
78.05
Heterophily- Desigmated GNNs
eestrophily-Desiigaaeee
FAGCN
65.22
44.12
88.17 88.17
77.75+
17.24
FSGNN
79.92.10.50 56
52.74_0.63
90.08 70
82.761. 61
78.86
GBK-GNN
74.57=0.44
45,98
90.85 20.5
81.01
74.47
GPR-GNI
64.85 20.27
44,88 20.34
86.24
72.94
55.48 20.99
JacobiCuny
71.14
43.55
89.66
68.6610.00
73.88 1.110
SONAR
89.82
52,22
96,29
83.57
74.96
Table 4: Sensitivity across different distances and recurrences in the Line-50 graph from Section 4.
SONAR
GCN
N. Recurrences-
50
75
100
N. Recurrences
50
75
100
Distance
Distance 1
x10
x100 6
x100 5
x x10
10
0,0115
0.0413
0.573
1.955
10
0,7078
0.0
1.1470
0.3562
5.084
30
0,0273
0.0616
1.269
4.341
30
0.3725
0.9779
0,5259
0.0745
0.0131
0.0253
0.584
1.061
0.0
0.0
50
0,0002
0.0324
0.143
1.004
50
0.0
0.0
0.0
0.0
5
Conclusions
In this work, we introduced SONAR a novel DE-GNN architecture that models information prop- agation using wave dynamics governed by the graph wave equation. SONAR offers principled approach for long-range information propagation by balancing conservative and non-conservative behaviors by integrating adaptive edge resistances and state-dependent external forces. Our theoretical analysis demonstrates that SONAR's energy preservation ensures stable and effective propagation of information over long distances. Moreover, the sensitivity analysis confirms that SONAR main- tains nan-vanishing influence between distant nodes, both in its continuous and discretized forms. Empirically, SONAR achieves sfate-of-the-an performance on a range of challenging benchmarks, i.e., synthetic and real-world long- long-range tasks, as well as heterophilic tasks. For such a reason, we believe SONAR represents step forward in the design of GNNs capable of effective long-ramg ong-range propagation. Future work can focus exploring alternative discretization methods, e.g., adaptive multistep scheme 4 and extend SONAR to temporal graphs (38].
Impact Statement. This work aims to advance the field of machine learning on graphs, with a focus on enhancing long-ran long-range information propagation. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Acknowledgments and Disclosure of Funding
The work has been partially supported by EU-EIC EMERGE (Grant No. 101070918).
10

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A Experimental Details
A.1 Edge Feature Aggregation
Together with the usual node features X2 E RR, some graphs provide edge features E E RRxd, made of dd-dimensional vector eur E Rd. for every edge (u, v) E. While classical MPNNs add this information during the convolution, we adopt a different approach to preserve our wave-like propagation and theoretical properties. particular, we follow [26), transforming the edge features and injecting them into the node space. To do so, we define the average Avg and gradient Grad
operators, which shift node features into edge features 26 33 as
(AvgX) nr Guo + Xu);
(GradX) = Gur x.);
(14)
where Ruxn is a matrix where Gwv = Tuy and The is the geometric mean of the degrees of nodes u and v. Their adjoint operators, which coincide with their transpose, transform edge features into node features [33 26). Therefore, before starting the propagation of information, the initial condition a concatenation of the initial node features and the transformation of the edge ones:
Xnew
X + Avg
+ Grad " E)
(15)
A.2 Employed Baselines
In our experiments, the performance of our method is compared with various state-of-the-art GNN baselines from the literature. Specifically, we consider:
classical MPNN-based methods, i.e., GCN
[59].
GraphSAGE
49
GAT
85
Gat-
edGCN 19], GIN (89), GINE [54], GCNII [55
heterophily-specific models, i.e., H2GCNN 93
CPGNN (92], FAGCN 8 GPR-GNN 17.
FSGNN (66). GloGNN 63 GBK-GNN
22
and JacobiConv [6];
DE-GNNs, i.e., DGC 87 GRAND
13
GraphCON 75 A-DGN
39
SWAN 40
PH-DGN 52]:
Graphh Transformers, i.e., Transformer 84
23
GT 79].
SAN 61
GPS 772.. GOAT 60
Exphormer 880, NAGphormer [4.. GRIT 65], and Graph VIT 51:
Higher-Order GNNs, i.e., DIGL
35 MixHop
[I. DRew [46), and GRED 20
A.3 Employed Datasets
In our experiments, we evaluate SONAR's performance on widely used graph benchmarks. Specifi- cally, we consider long-range propagation tasks, including the graph transfer tasks from 44). as well as the three graph property prediction tasks introduced in [39 ("Diameter", "SSSP" and "Eccentric- ity"). Additionally, we assess SONAR on the "Peptide- func". "Peptide-struct" and "PascalVOC-SPP tasks from the real-world tiveness, we include five heterophilic tasks: "Roman-emin eemiree". "Amazon- rattings", "Minesweeper". "Tolokers", and "Questions" [70]. We highlight that the three graph property prediction tasks are problems fundamentally linked with information propagation, as node signals must travel from each node across the entire graph. Therefore, they require nodes to iteratively exchange information with increasingly distant nodes. This process shares similarities with classical algorithms like Bellman- Ford Dijkstra's algorithm. Indeed, Dijkstra's 's algorithm works by progressively expanding frontier of visited nodes, where each node updates the shortest known distance to its neighbors.
Graph Benchmark (LRGB) [24). To further evaluate its effec-
In Table 5 we report the statistics of the employed datasets.
A.4 Hyperparameter Space
The hyperparameter space employed by SONAR in experiments is reported in Table 6
24

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Table Dataset statisties.
Dataset
#Nodes
#Edges
#Graphs
Task
Graph Transfer
100
396
1,200
Node Regression
Diameter
25-35
22 553
7,040
Graph Regression
SSSP
-5
22 553
7,040
Node Regression
Eccentricity
-5
553
7,040
Node Regression
Peptide-func
444
10- 928
15,535
Graph Classification
Peptide-struct
444
10 928
15,535
Graph Regression
PascalVOC-SP
-
1,044 2,942
11,355
Node Classification
Roman-empire
22,662
32,927
Node Classification
Amazon ratings
24,492
93,050
1
Node Classification
Minesweeper
10,000
39,402
1
Node Classification
Tolokers 11,758
519,000
1
Node Classification
Questions 48,921
153,540
1
Node Classification
Table 6: The grid of hyperparameters employed during model selection for the graph transfer tasks (Transfer). graph property prediction tasks (GPP), Long Range Graph Benchmark (LRGB), and heterophilic benchmarks (Hetero).
Values
Hyperparameters
Transfer
GPP
LRGB
Hetero
Optimizer Leurning rate
Adam
AdamW
Adam 0.001, 0.00U5
0.001
0.003
0.001
Weight decay Dropout
10
0,
0. 0.5 122,
N. recurrences
distance/2, distance
5, 10, 20
1,2,4.6.,,12116
Embheddnn dim
64
117, 136
128, 256, 512
N. Blocks
1,
1,2,3,4.5
from to
0.05, 1. 0.5.
0.001, 0.1, 0.5,
0.01, 0.02, 0.025, 0.05, 0.1
0.0001, 0.0005, 0.001, 0,005, 0.01, 0.1
Dissiparion
True, False Troc, Falsc
Use Exteraa Force
Adaptive Resistance
True, False
D D AD
I D
B Proofs of the Theoretical Results
In this section, we prove the theoretical results in Section 3.3
B.1 Proof of Theorem 3.1
To prove Theorem 3.1 first need this preliminary lemma:
Lemma B.1. Let X R be a feature vector for the nodes in graph G. Then, holds that
Σ |vx,10) -I V
Σ (x. - 2
1
Σ
Apu - xx,) = (16)
2 U.VEV
VEV BEN.
The lemma involves only one feature per node, but it can easily generalized multiple features by considering norms and scalar products.
25

---

Proof of Lemmal B.I The first equivalence comes from the gradient definition in a graph (see [44] while the second follows from expanding the sum of neighbors. We now show the last equivalence:
1 Σ (X,, x,) == ΣE
uu (x
+ x 2x.X,)
2 N,VEV
+ Σ x Σ A Aon - Σ
A nu Xp
X deg(( (J) + Σ xydeg(u) -2 Σ
AueX
2
2
2 D - Σ
AuxxxXo
2
2
Σ Xy NEV
X Dror
Σ
A AnxXa Xu
(17)
D mm
(ouoXu
A Anuku)
NEY
Sunuur Avu)xu
NEV
X, (D anddu Auv)Xu
Σ Lx, = X
MP
where dw is 1 if u and 0 otherwise.
Proof of Theorem 3.7 We show the theorem for d (one feature), since involves the norms of the
feature vector (t). To improve clarity, here we consider the initial considtion to be X(0) = X = X.
The wave (4 has an explicit solution (see (34 given by
x(1) = cos (VVL) X.
(18)
where
22
1 L
0C
cos (IVL)
=
+
1)"
(19)
4!
n!!
n=0
The energy in 9 can be calculated as
(x()) 2
1
E(1)
Σ
VX,
Ø1
(20)
H(cos [tVL) X)
1 2
X(I) LX)) +
ØL
where we used the Lemma fo the first term and the explicit solution for the second. Using the explicit solution, the first term is equal to
X(1) (()TLa LX(() cos (IVL.) x) L(cos (IVL. X
(X)T cos (IVL) Loos (IVL) X
(21)
= VI (tVI) x!²
/ooos (IVI. VLX
26

---

where the last equality follows from the fact that cos (tvL) and vL commute, which follows from The second term can be calculated as
the Taylor series expansion in 19p.
a
Σ
20
(-1)
2m
VL
x
Σ
20
(2n-1 (2m - 1)!
VL X.
(22)
Ot
2m!
which, using the Taylor series expansion of sin (lvL) L is equal to
(x
VL sin (tVL) X = sin (IVL) VLX
(23)
a1
Finally, the sum of the two terms is equal
E(t) 2
1
|ooss
(tVL)
VLX
+ sin (tvI) v vIX
1
(x)T VL cos (IVT.) COS (IVL. VLX + (X) T sin (IVD) sin (IVL) VIX)
2
(X)
VI c (tVL) VLX + (X) ( V (tVL) vLX)
2
1
(xx) VL (COB² (WVL) sin? (IVL) VLX)
2
1 (X)
(LVVLLLLX)
(()) LX) E((),
2
(24)
Where the transpositions for VL, sin, and cos vanish as they are all symmetric matrices (since L symmetric and positive definite, we can choose symmetric square root).
We now prove general version of Theorem 3.1 with features X(I) Raxd and W I.
Theorem B.1. Let X(t) E Baxd be the node states time I, obtained as the solution to the graph wave equation in Equation (5) with initial condition X(0) X. null dissipative and external forcing terms, and W I. Then, the energy E(t), defined as
E(t) = Σ
1 2
(t)l
1 +
oX(t)
(25)
2
Ot
VEV
conserved, that is, E(t) = ((0) Vt > O.
Proof. The proof is almost identical to the one from Theorem Theorem 3.1 Since features are not coupled, the solution to the equation is the same as Equation (IO). and the same steps apply. Since work with multiple features, scalar norms become vector norms, while the latter become matrix norms.
B.2 Proof of Theorem 3.2
While the proof of Theorem 3.2 is a direct consequence of the explicit solution, we now prove more general version of it for node states with d features and W I.
Theorem B.2 (Sensitivity matrix, continuous case). Let (t) RR be the state for node at time t.
(xx(t) Then, the sensitivity matrix 0x. (0) between nodes u and v is
(x,(1) COs (IVI. I.
(26)
Ox,, (0)
Proof. Since W = I, the analytic solution the SONAR wave equation Equation 4 is the vector
version of Equation 10)
X(1) cos (IVL) x cos (WLL) X(0)
(27)
27

---

Now, by differentiating with respect to x, (0), we obtain
xx, (l)
cos
(tvL I.
(28)
xx (0)
While Theorem [BT and Theorem B.2 are not the most general statements with any possible feature coupling, we are the first to show proots for the wave equation for graphs involving multidimensional features. Most of the works in the mathematical literature on the graph-based Laplacian and graph wave equation involve only the scalar case with single-valued functions (3 53
B.3 Proof of Theorem
Proof. We start by recalling the Sonar update with null dissipative and external forcing term
x X + hV
(29)
y = h (LX'W),
Substituting the second equation in the first one, we obtain
X = X hV L
=x' + X' X
L''''W
(30)
2X xt-1 L
now write the update for single node w
#+1 2x
x 2-1 - 112
L x W
(31)
NEV
Therefore, if we consider any node u E V, the one-step sensitivity matrix can be directly calculated by differentiating Equation 31 by xu. Since the dynamical system is causal, we have that and, finally,
Ux
8x
0x 21 L W,
(32)
0x
with Lnv denoting the (u, v)-th entry of the identity matrix.
B.4 Proof of Theorem 3.4
Proof of Theorem 3.4
Let us consider one update of the wave equation solution from equation
7
x x + +hve+1
(33)
vi+1= = - - O -
We will now show that
0x
S (wd)
++)+)+++AA))
(34)
Tx
1
and
Ov!
< (wd)
((L+MN+&+
(35)
xx
1
by induction, following [19]
28

---

Base case L f=1 1
Using the update equations, it is clear that
I dow + dech
(36)
0x1 0.9
0xa
and
Ovl.a
@vyga
0x
8em
hk
+ (/ ))
h Lowl
|
(37)
0x 0,3
0xx U.3
Oxu 1
Ox 0.3
Since
0v 0x
W
w.
have that
0v..
hk ht)
Boulwhh S deu (1 + h(N + +k+ 1))
A agt wh
(38)
0xu 0,5
which, by summing over 3 and maximizing on
ov!
S (d) (1+ I))I+ AA)
(39)
xx! Aa
and, by applying the same reasoning equation 36
0x!
$ (wd) ((1 + h + + A²A)
(40)
0x4
Inductive step, l > + 1.
We start by noticing that
0x
i P.o
60m
+ h
(41)
Ox 0,3
ixu
0
Ox.." 0.8
which can be further decomposed as
0x(+1,a θxLaa
Ov
Oxti
gv @vll
h
+ h
(42)
0x 0.3
0x.1
0.3
Ox.0"
p,7
Ox
0.8
Ov
Oxll 0.3
It is easy to see that
L-e-1a 1,a
< h Level w + hawdru I &omn (N + 1) + whAww 100
0x
(43)
σv! f-La
< 8o (1 + hkk))
Ov
We now calculate 0vt+1,a
&yaa
axt.a or
axtaa w 0x
< 6wn
(1 + hkk))
+ -wh
+ hac Lywul
0x0.0
0x
0x0.3 0,3
(44)
Ov w Ox0
Ox w
Ox
< 6wn
(1 + hk)
+ wh(NN + 1)
+ AewAAss
Ox!
Ix
We now expand this last expression by inductive hypothesis and discard the terms with O(h)) as they become (1h3) in the next step. Summinn over and maximizing over a we get
(wd)++1 (((1 + h(N + k + 1)) +A)
(45)
0x
0.B
which proves the inductive step for the sensitivity matrix V. We are now ready to prove the final
part. Starting from equation 42 plugging in equation 43 we get
@xs+1.a
0x
Ox
@vt+L.a
Ov Txm 1.)
0x113
0.3
Ox
Ox
0x!
@v
1.1
Ōx
Ox
@v Ox!!
-
+ hw N + 1) + whhww)
+ +h(11 + hk)6
axx!!
Ox
1
<(wd)" ((1+ k 1)) 12))+
va
+ h(0erhhWNN + + + wh Awd) ( + h 12(N + + A²A)")
+
wu
+ Sow wh(1 + hkk)())) ((1 k +
(46)
29

---

Summing over the last two terms and discarding the terms O we have
I (wd) ((1+h++h*) 1))++ +
TM
(47)
+ (red))hh(1 - h(N + k + 1))6kw + hm
((1+++1 4 + ()
Now we put oow in front over the first term (and sum over w) to join the two terms together obtaining
S (wd)) (((1 + h (N + 1)) A²A) +
Oxl.
+ (wd)) wh((1 + h(N + k + 1))6 + hwww) (11+h+ + k + +)))))
-w e1 ((
+ h 2 + k + 1))6 + h Aww) ((1+h))))) +k+ 1))1 +))))
= ((1++) + 1)) +)))) (
(48)
Summing over 3 gives the additional factor d, while maximizing over a gives us the thesis.
0
C Additional Results And Comparisons
C.1 Extended Comparison
To further evaluate the performance of SONAR, we report more complete comparison for the LRGB tasks in Table for the heterophilic tasks in Table Specifically, in the LRGB setting, we include more multi-hop DGNs and ablate on the scores obtained with the original setting from 24 and the one proposed in [82). The latter incorporates added residual connections and 3-layers MLP decoder. In the heterophilic setting, we include more MPNN-based models, graph transformers, and seterophiilyddesinnaaae
GNNs. In both tables, color the top three methods. Different from the main body of the paper, here we also include sub-variants of methods in the highlighted results, providing additional perspective the findings. Notably, our SONAR achieves state-of- sate-of-theeatt -trrh--rrt performance across all considered tasks.
C.2 Ablations
To better understand the contribution of each component in SONAR, we conduct a series of ablation studies.
Adaptive Resistance, Dissipation, and External Forces. We analyze the role of the dissipation mechanism, the external force term, and the adaptive edge resistance shaping the model's 's dynamics and performance. Each of these components plays a crucial role in controlling information propaga- tion over the graph. By systematically removing these elements, assess their individual impact on long-range information flow and overall predictive performance.
Table the mean and standard deviations on the test set for some tasks we analyzed in Section i.e., graph transfer line-50, SSSP, ppptides-struct, and KomaneEmpire. We report the results for each possible combination of adaptive resistance (i.e., adaptively re-computing the edge resistance at each step), dissipative, and external forcing components. We note that the importance of each component in SONAR varies depending the task and the nature of the information it involves. In the graph transfer task, all components of SONAR are essential, particularly dissipation and external forcing intermediate nodes contain random features that must be effectively filtered out. In the SSSP task, re-computing the resistances at each step leads to a performance drop, indicating that a constant propagation pattern is preferable. For peptides -eptides--stuut, the external forcing term proves to be the most impactful, while dissipation offers little benefit. The best results on the Roman Roman-Empi Empire re dataset achieved using the purely conservative form of SONAR, highlighting the value of non-dissipative signal propagation in that setting.
Lastly, to understand the importance of the energy preservation behavior of our SONAR, we compare its performance with that of standard GCN, which can be considered as the most comparable base- lines to our proposed SONAR. Both rely the Laplacian operator and cessage-passing paradigm,
30

---

Table 7: Results for Peptides-func, Peptides-strun struct and Pascal VOC-SP averaged over 3 training seeds. Baseline results are taken from [24
46
51
82
65
21
40 52 21]. Re-evaluated methods employ
the 3-layer MLP readout proposed in 82 and positional encoding. The first, second, and third best scores colored. means 3-layer MLP readout and residual connections are employed.
Note that all MPNN- methods include structural
Peptides-fune AP1
Peptides-struct MAE
Pascal VOC-SP
Model
FI
MPNNs
GatedGCN
58.66 o0.17
0.3420
0.2873
GCN
59.30
0.3496
0.1268+0.0mm
GCNII
55.43
0.3471
0.1698
GINE
24.98
0.3547
0..005
Multi-hop GNNs
DIGL+MPNN
64.69
0,3173 +0.0007
0.2824+40.0039
DIGL+MPNN+LapPE
68.30=0.20
0.2616::0.
0.2921 10.0038
DRew-GCN
69.9620.78
0.2781
0.1848
DRew-GCN+LapPE
71.50 00444
0.2536 10.0015
0.1851100000
DRew-GIN
69 29.40
400009
0.2719
DRew-GIN+LapPP
71.26
0.2606+0.000
0,2692
DRew-GatedGCN
67. 60.33
0.2699
0.3214
DRew-GGatcdGGCC++aappp
69.77
0.2539 1:0 0007
0.33 1..002
GRED
70.85
0.2503 10.0019
GRED+LapPE
71.33 0..11
0.2455 1000013
MixHop-GCN
65.9220.36
0.2921 00.0033
0.250620.0133
MiXHHop-GCN+LLpPE
68.43
0.2614 0.00023
0.2218+0.0174
Transformers
GraphGPS+LapPE
65.35
0.2500 +0.0005
0.3748
Graph VIT
69.42.0075
0.2449 :0.0011
GRIT
69. (9,888.m 20.62
0.2460::0.0012
SAN+LapPE
=..84=1.21
0.2683 0043
0.3230.200009
Transformer+LapPE
63.2641.228
0.2529
0.2694+0.o00s
Modified and Re-evaluated
GCN
68 68.60=u.50
0.2460+ -000007
+.2078++o.0031
GINE
66.21 20.67
0..003340.0017
0,2718 0.2718+0.0004
GatedGCN
67.65 0..47
0.2477 ::0 .009
0.3880.10.0040
DRew-GCN+LapPE.
69.45
0.2517 0.0011
GraphGPS+I.apPF
65.34_10.11
0.2509 10.0014
0.4440.:0.066
DE-GNNs
GRAND
57.89 00.62
0.3418 0..001
10.0097
GraphCON
60.22 00.88
0.2778 0.000118
0.2108 1.0.0991
A-DGN
59.75+0.44
0.2874
0.2349
SWAN
67.51 +0.39
0.2485 0.0000
0.319 0.3192
PH-DGN
70. 40.45
0.2465 0.00000
Ours
SONAR
68.4220.11
0.252540.om 0038
0.4058 0.4058=0.0053
+0.0039
SONAR+LapPA
70.47
0.2486
0.4082
31

---

Table Mean test set score and std averaged over random weight initializations on heterophilic datasets. The higher, the better. First, second, and third best results for each task are color-coded.
Roman-empire
Amazon-ratings
Minesweeper
Tolokers AUCT
Questions
Model
Ace
Acc
AUC
AUC
64
MLP-2
66.04
49.55
50.92
74.58 70.75
69.97
SGC-1
44.60.10.5
40.69.0.42
82. 02.04-10.77
73.80
71.06
MI.P-1
64.12:1..1
38.60
50.59 10.88
11.89
70.33
Graph-agnostic
ResNet
20.88
45, 45.90=0.62
50.89
72.95 0.0
00.34
ResNet+SGC
73.90
50.66-0.48
70,88
80.70
75.81
ResNettadj
52.25 40.40
01.83
50.42
78.78 1:111
75.77
MPNNs
GAT
80.87
49.09
92.01 10.68
83.70
17.43
GAT-sep
88.75.10.11
52.70
93.91 10.35
20.43
76.79
GAT (LapPE)
84.80 14.80+4.44
44,90 00.70
03.50+00.54
84,99 40.54
76.55 +0.84
GAT (RWSE)
86.62++0.68
48.58 48.58
92.53 40.63
85.02
77.83
GAT (DEG)
85. #0.66
51.65=20.00
93.04
84.22+00.4
77.10
Gated-GCN
74.46.10.5
43.00.20.32
87 1..54
71.31
76.61
GCN
73.69.0.74
28.70
39.75.0..2
83.64
76.09 1.22
GCN (LapPE)
$3.37.1..55
44.35
94.26.10.40
84.95
77.79
GCN (RWSE)
84.8440.66
46.40=2.65
93.84++0.48
85.11
77.81
GCN (DEG)
84.21 40.47
50.01 20.69
44.14
82.51
76.96
SAGE
85.74.0.07
53.63 20.39
93.51 +0.57
82.43
76.44
Graph Transformers
Exphormer
89.03....3 1.037
53.51 0.446
90.74:1..553
83.77 1:078
13.94:11.06
NAGphormer
74.34
51 51.2620.72
84.1940.06
78.32
68.17.1.53
GOAT
71.59
44.61 20.50
81.09
83.11
75.76+1.66
GPS
82,00
53.10_2.42
90.63
83.7 83.71 20.48
71.73+1.47
GPSuce-Perfomee (LapPE)
83.96
48.20=a.007
93.85 4-0.41
84.72+t.77
77.85
GPSucN+Paformer
(RWSE)
84.72++0.68
48.08 =8.05
92.88 +00.00
84.81 +0.86
76.45
GPSucN+Pariomee (DEG)
83.38
48. 48.93
93.60
80.49
74.24
GPS GATHeremrereeee (LapPE)
85. 15,9931..5
48.86=0.38
92.62.10.79
84.62.0.54
76.71
(RWSE)
87.04.0.55
49.9220.68
91.08 1.0.8
84.38
17.14
(DEG)
85.54 85.54
51.03
91.52.10.46
82.45 10.89
76.51 11.
GPS GCnn-Taffeereee
(LapPE)
OOM
OOM
91.82++0.41
83.51
OOM
GPS SGCN
(RWSE)
OOM
OOM
91.17+0.51
83.53+1.06
OOM
GPS Transformer (DEG)
OOM
0OM
91.76+00.11
80.82+0.95
OOM
GPSGAT+ Transfmmee
(LapPE)
OOM
OOM
92.29 4:0.61
84.70
OOM
GPSGAI+ Transfommr
(RWSE)
OOM
OOM
90.82..0.56
84.01 1:0.96
OOM
GPS APPSGAATTaasoonn
(DEG)
OOM
OOM
91.58
81.89 81.89+0.85 4..85
OOM
GT
86.51
51.17
91.85
83.23
77.95
GT-sep
87.32.0.39
52.1820.60
92.29
82.52+0.92
78.05
Heterophily-Designated GNNs
CPGNN
63. 83.96
39.79 u.777
$2.03+a..40
73.36
45.96+11.93
FAGCN
65.22+4.c0 60.66
04.12
88. 47.77.
77.75
77.24
ESGNN
79.92
52.74
90.08
82.76
78.86
GBK-GNN
74.57 10.47
45.98
90.85
81.01
74.47
GloGNN
59.63 10.69
36.89
$1.08
73.39
65.74
GPR-GNN
64.85 +0.27
44.88
86.24+0.61
72.94
55.48
H2GCN
60. 10.11400 +0.62
36.47
89.71
73.35
63.59+1.44
JacobiConv
71.14
43.55
89.66+0.40
68.66+0.65
73.88
Our
SONAR
89.82
52.22
96.29
83.57+1.44
74.96
32

---

Table 9: Mean test set results with standard deviations for different combinations of SONAR
components. The last line presents the results from standard GCN for the comparison with non-conservative method. First, second, and third best results for each task are color-coded.
Adaptive Resistace Dissipation
External Forcing
Line-50 (oR3o(MSF) 4
SSSP 1og10 MSR 4
Peptides-truct MAR
Roman-Empp Emmpie Acc
-6.40699 4004
4053
-3.223 20.
0.2611 4:0 0075
88, $8.244.0.20
-5.7542++.0026
-3.8182++.2225
0.2592+00 0168
88. 6.1++1111
-6.25445 +3.4
-2,7857
2..857++0.1731
0.25066::.000
89.27 20.77
-5.3827 4078
-2.8108
20,200
0.2560 0098
89.81
-6,3598
-6.7515
(.7515-00.0880
0.2631
88,95+00.7
-5,9058 5875
5876
-6.5939
2.0, 29125
0.2508 2.0 0036
89. 89.12-10.82
-4,9749 277777 2717
-6.5241
50.5738
0.2486 1.0 0006
89.22 20.41
-5.4487 70.7870
6.3226 0.2266
0.2503 0006
89.42=00077
GCN resalts for comparison
-3,5032+0.0001
0.9499.
0.3496 0018
73.69+0.74
Table 10: Mean train and test errors of identical SONAR models with different step sizes on the
Ring-10 10 task from Section 4.1
step size h
mean frain loss
mean test loss
0.05
0.0020273
0.0019732
0.10
0.0019947
0,001 9512
0.50
0.0039039
0.0039458
1.00
0.0085834
0,0086960
but GCN lacks energy preservation guarantees for long-range information propagation, adaptive resistance, and external forces. We note that GCN fails in solving these tasks since it performs dissipative signal propagation. As a result, information about distant nodes becomes increasingly indistinguishable and ultimately lost. In contrast, our SONAR successfully solves these tasks because it performs a non-dissipative propagation. This is enabled by explicitly preserving the energy of the system (Equation (9) in its design, with the result of preserving signals during propagation (as theoretically discussed in Section 3
Overall, these results highlight SONAR's ability to balance conservative and non-conservative behaviors, enhancing its ability to learn more complex and task-specific dynamics. We believe that the modular design of SONAR, with components that can be selectively activated, provides high flexibility and allows SONAR to adapt to a wide range of scenarios while maintaining computational efficiency, without relying global attention mechanisms or graph rewiring.
This analysis also suggests a practical guideline: for propagation- heavy tasks, foreing (and often dissipation) is beneficial; dissipation is particularly useful under high noise (e.g., graph transfer tasks); and adaptive resistance tends help particularly with heterophilic tasks. These trends narrow the hyperparameter search space, though a small hyperparameter search is still necessary for optimal performance.
The Effect of the Step-Size h. Since the step size controls the numerical accuracy of the wave equation, its careful selection is essential for preserving SONAR's properties and performance. As with classical numerical solvers for physical equations, a high number of iterations requires a smaller step size, and vice versa. In practice, we observed that small variations in the step size have little impact on performance, and exploring different orders of magnitude is sufficient to identify npar-optimal values. For a practical example, consider the ring- 10 task described in Section 4.] and train an identical configuration of SONAR with 4 different step sizes. The results, reported in Table [0] indicate that close step sizes produce comparable performance, with noticeable differences arising only for large changes. This demonstrates that SONAR's 's behavior is robust to small changes in the step size.
Runtimes. Additionally, we report runtime analyses to evaluate the computational efficiency of SONAR with respect state-of-the-art methods in Table 11 Specifically, we report the training and inference times in milliseconds, as well as the memory usage obtained on the Roman-Empire dataset, fixing the model trainable parameters to 100k. As can be seen from the results in the Table, our
33

---

SONAR maintains similar runtime and memory consumption to GCN, which has linear complexity with respect to the graph size. All runtimes are measured on an NVIDIA HI00 GPU.
Table 11:
Training and inference time (in milliseconds) and memory usage on the Roman- Emmpre dataset measured on a NVIDIA H100 GPU. All models have approximately 100k trainable parameters; GPS uses 2 attention heads.
Depth
Method
Metries
4
8
16
32
Training (ms)
3.32
5,08
7.99
13.04
Inference (ms)
1.98 4-002
3.24 08
5.0+0.07
8.05
GCN
Training Mem (GB)
0.25
0.25
0.28
0.34
Inference Mem (GB)
0.21
0.18
0.16
0.14
Training (ms)
101. 101.21 20.25
321.33
642.53
OOM
Inference (ms)
88.51
157.31
287. 28..27+0.15
OOM
GPS
Training Mem (GB)
0.35
43.59
75.13
0OM
Inference Mem (GB)
15.79
15.78
15.78
OOM
Training (ms)
10. 10.11
18.01-1.11
32.62+0.12
63.18
Inference (ms)
+0.04
6.59
12.23
23.45=0.09
SONAR (1 black)
Training Mem (GB)
0.71
1.24
2.29
4.41
Inference Mem (GB)
0.27
0.28
0.28
0.28
Training (ms)
9.63
16,21
29.62
56.56
Inference (ms)
4.03
5.87
10.47
19.7440.07
SONAR (2 blocks)
Training Mem (GB)
0.57
0.97
1.77
3.37
Inference Mem (GB)
0.23
0.23
0,23
0.23
D
Limitations
This paper contributes to the field of machine learning by advancing (differential equation-inspired) graph neural networks, with a focus on enhancing long-range information propagation. Specifically, we introduce a novel methodology that enables provable long-range propagation on graphs, grounded in physically interpretable model based on wave dynamics and their underlying properties.
While we believe work offers meaningful contribution, it is important clarify that SONAR designed to address the long-range propagation problem in GNNs through the lens of DE-GNNs. Therefore, our model prioritizes on problems that require effective modeling long-range interaction while maintaining low computational overhead. However, if the goal is to maximize downstream performance regardless of computational complexity, alternative approaches such as multi-hop GNNs or graph transformers may also be appropriate.
A second limitation arises from SONAR's formulation discrete approximation of continuous dynamical system. Its behavior depends on the discretization scheme, making the choice of the step size h crucial. As shown in Theorem 3.4| the sensitivity bound can grow exponentially with the number steps, potentially leading to exploding gradients h is not appropriately tuned. We emphasize, however, that this issue can be effectively mitigated through careful step size selection and was never encountered in our experiments.
34


Reviews:
[{"review_id": "MRqYbEWib3", "reviewer": "Reviewer_gMHH", "summary": "This paper introduces SONAR, a novel GNN architecture that models information flow using the graph wave equation. The goal is to address the over-squashing problem and enable effective long-range information propagation. The model incorporates adaptive resistances and external forces to balance conservative, energy-preserving dynamics with more flexible, non-conservative behaviors. The authors provide comprehensive theoretical analysis and well-grounded empirical results on several benchmarks to support their claims.", "questions": "- For disconnected nodes, is the adaptive resistance term a_uv always kept at 0, or can the model learn a non-zero weight, effectively creating a new edge?\n\n- There appears to be a typo in Equation (6). Shouldn't the second line be  V˙(t)=−L_aX(t)W−D(X(t))⊙V(t)+F(X(t)) to align with Equation (5) and the discretized version in Equation (7)?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- The motivation for using wave-based dynamics is very clear and well-articulated.\n- The paper includes a complete theoretical derivation regarding over-squashing and information propagation, which is commendable.\n- The writing is exceptionally clear and the paper is well-structured.\n- The claims are supported by multiple experiments across a range of tasks.\n\n**Weaknesses**\n\n- **Lack of Qualitative Insight**: While the motivation is strong, the paper lacks insights into why the wave-like message passing works, and how is different than the exisitng GNN designed for long-range. The results section is simply a description of performance metrics rather than an analysis of the model's internal dynamics. I would appreciate if author could include some examples or case study to show how the proposed wave-like messages help improve the long-range, for example, in some synthetic datasets. This would be my major concern.\n\n - **Outdated/Questionable Benchmarks**: LRGB datasets, particularly Peptides-func and Peptides-struct, have been shown by recent work [1] to not be truly long-range tasks. The authors should consider including more convincing real-world experiments that truely require long-range, such as other datasets in LRGB. For heterophilic dataset, the comparison is missing more recent and relevant baselines designed for heterophily, such as CoGNN [2] or adaptive message passing neural network [3]. I would appreciate if the author could add additional experiments over these to support the claim that SONAR are working as expected in real-world experiments, but due to the benchmarking issues I still have some doubts over the performances of SONAR.\n\n[1] Bamberger, Jacob, et al. \"On Measuring Long-Range Interactions in Graph Neural Networks.\" ICML (2025).\n\n[2] Finkelshtein, Ben, et al. \"Cooperative graph neural networks.\" ICML (2024).\n\n[3] Errica, Federico, et al. \"Adaptive message passing: A general framework to mitigate oversmoothing, oversquashing, and underreaching.\" ICML (2025).", "was_revised": true, "final_justification": "The author has successfully address the concerns I have for the rebuttal, mostly regarding the behaviour of SONAR over synthetic long-range task. I find the sensitivity analysis very useful. I would recommend acceptance of this paper providing that there are strong theoretical and empirical evidence of this paper.", "rebuttal": "We thank the Reviewer for the useful feedback and for recognizing the soundness of our work and our efforts to provide a theoretically grounded methodology while addressing information propagation and over-squashing problems in GNNs. We also appreciate their comments on the exceptional quality of our writing.\n\nWe hope our clarifications and revisions will address their concerns and prompt a reconsideration of their evaluation and score.\n\n1) Regarding more insights. We thank the Reviewer for the thoughtful comment. Compared to other GNNs designed for long-range propagation, SONAR achieves a non-dissipative behavior by design without requiring any weight regularization (as for ADGN or SWAN). Furthermore, SONAR propagates information long-range without rewiring techniques (as for DRew) or attention mechanisms (as for GAT or GPS), which are inefficient and computationally intensive. We would like to highlight that GCN-like models are the most comparable baselines to our proposed SONAR, as both leverage the graph Laplacian in their message-passing mechanisms. However, GCNs use the Laplacian directly, thus leading to the dissipative dynamics described by the equation $\\dot{X} = LXW$. This formulation lacks guarantees for long-range information propagation. In contrast, SONAR is grounded in the wave equation on graphs [4]; thus, it follows the explicit non-dissipative dynamics defined by $\\ddot{X} = -L_aXW$. This formulation ensures energy conservation and long-range propagation, as formally established by our theorems. Moreover, by integrating adaptive edge resistances and external forces, our method balances conservative and non-conservative behaviors, improving the ability to learn more complex dynamics. Regarding the inclusion of case studies, we would like to note that the synthetic tasks in Sections 4.1 and 4.2 already serve this purpose. Specifically, the tasks in Sec 4.1 are based on shortest path computations; thus, GNNs must heavily rely on global information and effectively learn to transport information across the entire graph. Tasks in Section 4.2 challenge models to transfer information from a source to a target node located at an increased distance. As shown in our results, SONAR consistently outperforms state-of-the-art models. To provide further insights into the long-range capabilities of SONAR in these controlled and synthetic tasks, we analysed in depth the long-range effects on the “line 50” graph in the transfer task. Here, the topology and the task require the model to perform effective long-range communication up to 50 hops. To verify long-term propagation of information, we studied the norm of the sensitivity matrix (Theorem 3.4) between node 0 and nodes at different distances in the graph (10, 20, 30, 40, 50). We compare the results of SONAR ( with step size h=0.1) with those of a standard GCN.\n| distance  | **SONAR sensitivity** | **GCN sensitivity** |\n|---|:---:|:---:|\n| 10 | 0.0413 | 1.15 $\\times 10^{-5}$ |\n| 20 | 0.0334 | 1.04 $\\times 10^{-5}$ |\n| 30 | 0.0616 | 0.526 $\\times 10^{-5}$ |\n| 40 | 0.0253 | 0.0 |\n| 50 | 0.0324 | 0.0 |\n\n   As we can see, the sensitivity matrix norm for SONAR is stable across distances, thus it propagates more information consistently. GCN, on the other hand, struggles to propagate information effectively, as the sensitivity norm is 2 orders of magnitude lower than SONAR, and it becomes null at larger distances, indicating that distant nodes fail to communicate.\n   We believe this supports that the wave-like propagation implemented by SONAR helps in improving long-range propagation and that it provides deeper empirical insights and discussions, along the lines of what was highlighted by the reviewer. We included this discussion in the revised paper.\n\n2) Regarding benchmarks: \nWe thank the reviewer for the references. For our submission, we focused on the most relevant and widely used benchmarks for long-range propagation that were available at the time. Specifically, the work in  [1] became publicly available after the submission deadline. Nevertheless, to accommodate the Reviewer’s comment, we are now including results on the Pascal-VOC superpixel task, which, according to [1], is considered a long-range benchmark.\n| Model | PascalVOC (F1 score $\\uparrow$) |\n|---|:---:|\n| **GPS** + LapPE | $0.451_{\\pm 0.01}$ |\n| **GT** + LapPE | $0.392_{\\pm 0.00}$ |\n| **Gated GCN** | $0.378_{\\pm 0.00}$ |\n| **GINE** | $0.275_{\\pm 0.00}$ |\n| **GCN+VN** + RWSE | $0.245_{\\pm 0.00}$ |\n| **GCN** + RWSE | $0.185_{\\pm 0.00}$ |\n| **DREW** (Gated GCN) + LapPE | $0.331_{\\pm 0.00}$ |\n| **SONAR** | $0.376_{\\pm 0.01}$ |\n\n\n   We remark that the results obtained by SONAR are without Positional Encoding.\n\n   Due to the limited time and the difficulty of the task, we were not able to perform a complete grid search for the optimal parameters of our model now (but this is running and will be included in the final version of the paper). However, our results are in line with the most expressive and long-range models, falling very close to Transformer models. We argue that this further shows the long-range capabilities of SONAR.\n\n   We would also like to emphasize that our paper already includes results on at least six other tasks requiring strong long-range capabilities, as discussed in our previous response. For example, the graph transfer tasks require propagating information up to 50 hops, providing further evidence of SONAR’s ability to propagate information over long distances. Finally, following up on the Reviewer’s comment, we have also included results from [2] and [3] in the revised version of the paper.\n\n[1] Bamberger, Jacob, et al. \"On Measuring Long-Range Interactions in Graph Neural Networks.\" ICML (2025).\n\n[2] Finkelshtein, Ben, et al. \"Cooperative graph neural networks.\" ICML (2024).\n\n[3] Errica, Federico, et al. \"Adaptive message passing: A general framework to mitigate oversmoothing, oversquashing, and underreaching.\" ICML (2025).\n\n[4] Joel Friedman and Jean-Pierre Tillich. Wave equations for graphs and the edge-based Laplacian. Pacific J. Math., 216(2):229–266, October 2004\n\nAbout the questions:\n- Yes, the Reviewer is correct when saying that disconnected nodes always have a zero adaptive resistance. In particular, the resistance a_uv is calculated only between connected nodes. We clarified this aspect in the revised paper.\n- We thank the reviewer for noticing the typo; we fixed it in the revised paper.\n\n---------------------------------\nIn light of these clarifications and the additional experiments, we hope this addresses the Reviewer’s concerns and they can consider revising their score."}, {"review_id": "97BF1Hu65x", "reviewer": "Reviewer_jhtz", "summary": "SONAR is a differential-equation GNN that treats message passing as a wave equation on graphs. Each edge carries an adaptive resistance, and state-dependent damping/forcing terms let the network interpolate between fully conservative (energy-preserving) and dissipative dynamics. The authors prove that, in the conservative limit, SONAR maintains a constant total feature energy and its node-to-node sensitivity doesn't vanish, thereby guaranteeing non-degenerate long-range propagation. A finite-difference discretisation is wrapped in learnable MLP blocks. Empirically, SONAR achieves strong results across synthetic datasets, long-range tasks, and heterophilic datasets.", "questions": "Please see Weaknesses", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Pros:\n- The wave-equation view and accompanying energy/sensitivity theorems provide good theoretical guarantees for long-range information flow in GNNs\n- The model achieves strong empirical results across various benchmarks\n- The complexity is sub-quadratic and more manageable compared with graph transformers\n\nCons:\n- According to ablation studies, the introduced Adaptive Resistance and Dissipative/External Forcing terms do not seem to have a consistent effect on different datasets, making them almost pure grid search.\n- While this paper derives bounds on the sensitivity between nodes and argues it is \"well-controlled\", to support the long-range modeling capacity, it would be better to analyze how the sensitivity changes with the distance between nodes (or the relative ratio between distant node's influence and close node's influence). This would reflect whether distant nodes are downplayed compared to closer nodes.\n- This paper lacks a comparison/discussion of a relevant work GRED [1]. GRED also interleaves linear recurrence with MLPs, and seems to perform better on LRGB\n    - [1] Recurrent Distance Filtering for Graph Representation Learning. ICML 2024.\n- How would the recurrence length $L$ affect the performance?", "was_revised": true, "final_justification": "In general, my concerns have been addressed by the rebuttal. Given the new additions during discussion, I would recommend acceptance.", "rebuttal": "We thank the reviewer for the insightful review and for highlighting that our SONAR has good theoretical guarantees for long-range propagation in GNNs and that it achieves strong empirical results with manageable complexity. Below, we provide detailed responses to each of the comments. We hope that these clarifications and revisions are satisfactory, and that the Reviewer will consider revising their score. \n\n- Regarding the effects of additional components. The fact that the influence of the introduced additional components varies across tasks is indeed expected, as the specific nature of each task determines how information must be propagated through the graph to achieve optimal performance. For example, the graph transfer task requires propagating information between distant nodes while filtering out irrelevant information from other nodes. This demands strong long-range capabilities and the ability to selectively retain or discard information; therefore, all of SONAR’s additional components become essential. By contrast, in the SSSP task, adaptive resistance is not needed, as the goal is to compute distances where each edge contributes equally, effectively requiring uniform propagation across the graph. These results highlight SONAR’s ability to balance conservative and non-conservative behaviors, enhancing its ability to learn more complex and task-specific dynamics. We believe that the modular design of SONAR, with components that can be selectively activated, provides high flexibility and allows SONAR to adapt to a wide range of scenarios while maintaining computational efficiency, without relying on global attention mechanisms or graph rewiring.\n- Regarding node sensitivity: \nTheorems 3.3 and 3.4 show that, compared to a standard MPNN-like architecture as studied in [2], our SONAR has improved capabilities of long-range propagation between nodes. We note that these results are not specific to a particular distance and are true for any two nodes with a viable path in the graph.  To the best of our knowledge, no published work explicitly studies the sensitivity between two distant nodes as suggested by the Reviewer, and in a way different from [2,3], as this sensitivity is also inherently dependent on the specific topology of the graph.\n   Nonetheless, to accommodate the Reviewer’s comment, below we provide an empirical validation of our findings along the lines of the requested empirical sensitivity analysis. We considered “line 50” in the graph transfer task, where the topology and the task require the model to perform effective long-range communication, and where the conditions of the experiment can be fully controlled. We computed the norm of the sensitivity matrix between node 0 and nodes at different distances in the graph (10, 20, 30, 40, 50). We compare the results of SONAR ( with step size h=0.1) with those of a standard GCN\n| distance  | **SONAR sensitivity** | **GCN sensitivity** |\n|---|:---:|:---:|\n| 10 | 0.0413 | 1.15 $\\times 10^{-5}$ |\n| 20 | 0.0334 | 1.04 $\\times 10^{-5}$ |\n| 30 | 0.0616 | 0.526 $\\times 10^{-5}$ |\n| 40 | 0.0253 | 0.0 |\n| 50 | 0.0324 | 0.0 |\n\n   As we can see, the sensitivity matrix norm for SONAR is stable across distances, thus it propagates more information consistently, confirming our theoretical results. GCN, on the other hand, struggles to propagate information effectively, as the sensitivity norm is 2 orders of magnitude lower than SONAR, and it becomes null at larger distances, indicating that distant nodes fail to communicate.\n\n- Regarding GRED. We thank the Reviewer for the reference that we included in our revised paper. We note that while the results from GRED [1] are marginally higher, they are still comparable to the ones from our SONAR. Nonetheless, we would like to recall that GRED is a higher-order method, as, at each iteration, each node is updated with the information coming from different hops of distance based on shortest path computation, which are then aggregated using additional models such as Linear Recurrent Networks. On the other hand, SONAR aggregates from a single-hop neighborhood, retaining MPNN-like complexity and achieving comparable performance.\n\n- Regarding L. The influence of the recurrence length L on performance is task-dependent, and in our experiments, L is treated as a hyperparameter. The underlying intuition is that L should match the distance between the nodes that need to communicate to solve the task; otherwise, the model will suffer from underreaching. In tasks like graph transfer, where the communication distance is known, L can be set directly without the need for extensive tuning. In contrast, when this distance is not known in advance, tuning L is necessary to optimize performance. Figure 2 clearly shows that the performance of SONAR does not degrade by increasing the distance between such nodes, and instead, when L is big, SONAR is the best performing.\n\n---------------------------------\nIn light of these clarifications and the additional experiments, we hope this addresses the Reviewer’s concerns and they can consider revising their score.\n\n\n[1] Recurrent Distance Filtering for Graph Representation Learning. ICML 2024.\n\n[2] Topping et al. Understanding over-squashing and bottlenecks on graphs via curvature. In ICLR 2023\n\n[3] Di Giovanni et al. On over-squashing in message passing neural networks: the impact of width, depth, and topology. In ICML 2023"}, {"review_id": "ReF7jwYXKO", "reviewer": "Reviewer_byZL", "summary": "This paper introduces SONAR, a GNN architecture designed to address the challenge of long-range information propagation. SONAR models information flow on graphs using the wave equation, which allows signals to travel over long distances as oscillations with minimal energy loss. The authors provide the theoretical backing, proving energy conservation and non-vanishing sensitivity between distant nodes. Extensive experiments on synthetic, long-range, and heterophilic benchmarks demonstrate that SONAR achieves state-of-the-art performance, particularly on tasks that require modeling distant interactions.", "questions": "1. The theoretical guarantees rely on specific discretization schemes and parameters like step size $h$. How sensitive is the model's performance to these choices?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- Strengths\n1. Grounding the GNN in the wave equation provides a principled foundation for long-range propagation that is both physically intuitive and mathematically sound. \n2. SONAR shows dominant performance on synthetic long-range tasks and is highly competitive on challenging real-world benchmarks.\n\n- Weaknesses\n1. SONAR relies heavily on discretization schemes and parameter tuning, such as step size $h$, which may influence stability and sensitivity bounds.", "was_revised": true, "final_justification": "Comprehensively considering the paper's contribution and the rebuttal, I think my original score is reasonable.", "rebuttal": "We thank the reviewer for highlighting the quality of our model and its soundness both in physical and mathematical terms. We now address the comments in the review.\n\nWe agree with the Reviewer that the kind of discretization and the integration scheme used to solve the wave equation numerically are important in the effectiveness of SONAR. Nonetheless, in our work, we use a simple Euler scheme, which straightforwardly showed very good performance with low effort. We believe that more complex schemes might increase the performance of SONAR even more, but we left those for future work.\n\nHowever, we argue that our theoretical results on the long-range capabilities and sensitivity remain true with other discretization schemes, with some minor corrections. Most importantly, the energy conservation Theorem 3.1 does not depend on a specific discretization and remains true as long as the node features evolve following our defined dynamics. The sensitivity results in Theorems 3.3 and 3.4 are cast on our specific choice for discretization, but could be easily adapted to other explicit schemes, following our proofs.\n\nIn any case, the step size acts as a hyperparameter to be tuned, and our experimental results show that it is easy to control.\n\nAbout the sensitivity to $h$. Since the step size is a hyperparameter of our model and the quality of the solution to the wave equation depends on it, choosing a good step size is important to guarantee the properties and performance of our model. In general, a high number of iterations requires a lower step size, and vice versa, like classical numerical solvers for physical equations. In our practical tests, we saw that the performance does not change significantly with small variations of the step size, while it is enough to try different orders of magnitude to have optimal results. For a practical example, we considered “ring 10” within the graph transfer task. We trained an identical configuration of SONAR with 4 different step sizes, and we report the results in the table below.\n| step size $h$ | mean train loss | mean test loss |\n|---|:---:|:---:|\n| 0.05 | 0.0020273 | 0.0019732 |\n| 0.10 | 0.0019947 | 0.0019512 |\n| 0.50 | 0.0039039 | 0.00394583 |\n| 1.00 | 0.0085834 | 0.0086960 |\n\nAs we can see, similar values produce similar results, without drops in performance, with differences appearing only with large changes in the magnitude of $h$."}]

Author rebuttal:

Meta-review: SONAR is a novel GNN architecture that models information flow on graphs as oscillations governed by the wave equation, addressing long-range information propagation. It integrates adaptive edge resistances and state-dependent external forces, theoretically ensuring energy conservation and stable information propagation over long distances. Strengths include its clear wave-based motivation, comprehensive theoretical derivation, strong empirical results, and sub-quadratic complexity. Initial weaknesses involved a perceived lack of qualitative insight into wave-like dynamics, the use of some benchmarks that were questioned regarding their long-range nature, and the observation that certain components' effects appeared inconsistent across datasets. 

The paper was accepted because of its principled and mathematically sound approach to a fundamental problem in graph representation learning, supported by robust theoretical guarantees and compelling experimental evidence. 

During the rebuttal, reviewer gMHH’s concerns about qualitative insight and benchmarks were addressed with sensitivity analysis demonstrating SONAR's consistent information propagation and the inclusion of new experiments. Reviewer jhtz’s queries about component consistency and comparison with relevant prior work were met with explanations of task-dependent utility and discussions on model complexity. Reviewer byZL’s concerns regarding sensitivity to discretization schemes and step size were resolved by confirming theoretical robustness and demonstrating empirical stability. The authors' comprehensive responses and additional analyses significantly strengthened the paper, leading to a favorable assessment by the reviewers.
Decision: Accept (poster)