Port-Hamiltonian Architectural Bias for Long-Range Propagation in Deep Graph Networks

1 INTRODUCTION

• MPNNs transmit information less effectively as node distance grows, motivating a framework for flexible long-range propagation.The introduction notes that this can hinder tasks such as predicting anti-bacterial properties of peptide molecules.Source: §1 · p. 1
• Internal damping and additional forces can move the system away from purely conservative dynamics, potentially increasing downstream-task effectiveness.Source: §1 · p. 2
• The framework can incorporate neighborhood aggregation functions, extending its properties to different deep graph networks.Source: §1 · p. 2
• PH-DGN outperforms state-of-the-art methods on synthetic and real-world tasks and can stack thousands of layers.Source: §1 · p. 2
• PH-DGN balances non-dissipative long-range propagation with non-conservative behavior, allowing the dynamics to suit the task.Source: §1 · pp. 1–2
• Under purely conservative dynamics, node states retain their past, supporting long-range propagation in both continuous and discretized message-passing flows.Source: §1 · p. 2

• PH-DGN treats the graph’s node states as a global state with position and momentum components, whose Hamiltonian gradient drives continuous message passing.Its neighborhood aggregation can implement any node- and edge-information aggregation function.Source: §2 · pp. 3–4
• Under sufficiently slow Jacobian variation and purely imaginary eigenvalues, the initial condition is propagated into the final representation, supporting stable, non-dissipative long-range propagation.The passage links this condition to sensitivity of the ODE solution to its initial condition.Source: §2 · p. 4
• The autonomous conservative system keeps its Hamiltonian constant and has a divergence-free vector field, yielding purely rotational dynamics without energy loss.These properties do not depend on slow Jacobian variation or eigenvalue location.Source: §2 · pp. 4–5
• With driving forces null, PH-DGN retains each node’s past, and this result also holds under Symplectic Euler discretization.The backward sensitivity bound indicates that gradients do not vanish during propagation.Source: §2 · pp. 4–5
• The PH-DGN upper bound for discretized message passing is at least N^ℓ times bigger than the bound computed for an MPNN.The comparison is stated for Symplectic Euler discretization.Source: §2 · p. 5
• Damping and learnable, state- and time-dependent external forces can alter the conservative trajectory, letting the system model non-conservative behavior.Positive semi-definite damping decreases energy along the flow; negative semi-definite damping implements internal acceleration.Source: §2 · p. 6

3.1 NUMERICAL SIMULATIONS

• Energy difference oscillates around zero and is more accurately preserved with smaller step sizes; backward sensitivities remain above the theoretical lower bound, indicating non-vanishing gradients.Figure 2 examines fixed-weight Carbon-60 simulations spanning 100 to 1000 layers; sensitivity increases with distance to the final layer, and the authors note it can be controlled through weight regularization or normalized aggregation.Source: §3.1 · pp. 6–7, Figure 2 · p. 7

3.2 GRAPH TRANSFER

• At distances k ≥10, PH-DGN achieves low transfer errors, outperforming classical MPNNs and averaging better performance than A-DGN.The tests use line, ring, and crossed-ring graphs and PH-DGN without driving forces; Figure 3 reports test mean-squared error and standard deviation.Source: §3.2 · p. 7, Figure 3 · p. 8

3.3 GRAPH PROPERTY PREDICTION

• PH-DGN improves on all baselines overall, while relaxing the conservative bias improves graph-property prediction beyond pure-conservative PH-DGNC.The tasks predict graph diameter, single-source shortest paths, and node eccentricity; the authors suggest that distance counting can benefit from non-purely conservative behavior.Source: §3.3 · pp. 8–9, Table 1 · p. 8

3.4 LONG-RANGE GRAPH BENCHMARK

• Both PH-DGNC and PH-DGN outperform classical MPNNs, graph transformers, most multi-hop DGNs, and recent DE-DGNs on Peptides-func and Peptides-struct benchmarks.The reported benefit does not require positional or structural encodings, global attention, or rewiring; the framework uses local message exchanges.Source: §3.4 · p. 9, Table 2 · p. 9

Differential-equation-based DGNs

• Differential-equation graph networks use diffusion, oscillatory, or interpolated dynamics; PH-DGN instead uses port-Hamiltonian dynamics to balance non-dissipative propagation and non-conservative behavior.The authors characterize PH-DGN as the first DGN to leverage port-Hamiltonian dynamics and provide theoretical guarantees of long-range propagation.Source: §4 · p. 10

Hamiltonian and temporal approaches

• Prior Hamiltonian-inspired graph methods use Hamiltonian dynamics to encode node features or improve robustness, whereas PH-DGN applies port-Hamiltonian dynamics to the graph network.The discussion also notes work studying the temporal domain of differential-equation graph networks.Source: §4 · p. 10

Long-range propagation on DGNs

• Long-range strategies include weight regularization, message filtering, graph rewiring, all-pairs transformer exchange, and non-local fractional graph dynamics; some can increase propagation complexity through denser graph operators.The cited rewiring methods alter the original edge set to densify the graph during preprocessing.Source: §4 · p. 10

5 CONCLUSIONS

• PH-DGN provides a general framework that balances non-dissipative long-range propagation and non-conservative behavior while incorporating neighborhood aggregation functions.The framework’s port-Hamiltonian dynamics are presented as a way to gauge that equilibrium.Source: §5 · p. 10
• Under purely conservative dynamics, continuous and discretized PH-DGN support long-range propagation because node states retain their past.Source: §5 · p. 10
• Across synthetic and real-world long-range benchmarks, the method outperforms state-of-the-art models, and data-driven forces often improve efficacy.The authors frame effective propagation as requiring both long-term memorization and selective forgetting when needed.Source: §5 · p. 10

ETHICS AND REPRODUCIBILITY STATEMENTS

• The authors report no identified ethics concerns, sensitive data, human subjects, crowdsourcing, or conflicts of interest.They also state that they release no datasets or models posing significant misuse risk.Source: ETHICS AND REPRODUCIBILITY STATEMENTS · p. 11
• The reproducibility statement points to experimental setups and supplementary details, and says code and data for reproducing the evaluation will be released upon acceptance.The experiments use publicly available benchmarks.Source: ETHICS AND REPRODUCIBILITY STATEMENTS · p. 11

A.1 SENSITIVITY UPPER BOUND

• The sensitivity upper bound assumes a nonlinearity with bounded derivative and a neighborhood aggregation function of the stated form.The theorem bounds the backward sensitivity matrix; the authors note that sensitivity may grow and potentially cause gradient explosion, which they did not observe in experiments.Source: §A.1 · p. 15

A.2 ARCHITECTURAL DETAILS OF DISSIPATIVE COMPONENTS

• The dampening term is a diagonal square matrix block, and energy loss requires it to be positive semidefinite.The authors use ReLU activation except for the learnable param option, which allows a learned trade-off between dampening and acceleration.Source: §A.2 · p. 16
• Dampening options include learnable vectors, an MLP, and a DGN aggregation layer, while the external force is a vector of component-wise forces.The listed dampening variants include param, param+, MLP4-ReLU, and DGN-ReLU; MLP4-Sin and DGN-tanh are listed for the external-force architecture.Source: §A.2 · p. 16

A.3 DISCRETIZATION OF PORT-HAMILTONIAN DGNS

• Symplectic integration is required to preserve conservative properties in the discretized system, and the authors also note that Strömer-Verlet can be used.The section focuses on Symplectic Euler for simplicity.Source: §A.3 · p. 16
• The Symplectic Euler update is implicit; obtaining an explicit form uses a block-diagonal structure assumption on W and V to separate p and q.Under that assumption, the component weight matrices remain unconstrained learnable matrices.Source: §A.3 · p. 16
• The explicit discretization can be understood as coupling two DGN layers in one message-passing step.With driving forces, the q update is unchanged while the other update includes dissipative components.Source: §A.3 · p. 17
• Symplectic Euler discretization preserves the lower bound on the backward sensitivity matrix, supporting conservative message passing with null driving forces.The authors also note that properties in continuous time need not hold after discretization; they state that the result holds for Symplectic Euler.Source: §A.3 · p. 17

B.1 PROOF OF THEOREM 2.1

• The proof considers an eigenpair of the Jacobian written as a product of a symmetric and an anti-symmetric matrix, and concludes its eigenvalues are purely imaginary.The argument uses the conjugate transpose of the eigenvector.Source: §B.1 · pp. 17–18

B.2 PROOF OF THEOREM 2.2

• The conservative PH-DGN shares W and V across layers, making its Hamiltonian independent of explicit time.The passage identifies this as the setting in which energy is naturally conserved.Source: §B.2 · p. 18
• The conservation result applies to the shared-weight conservative model, whose Hamiltonian has no explicit time dependence.The passage cites the symmetry of the Hessian in the proof.Source: §B.2 · p. 18
• The proof shows that the time derivative of the Hamiltonian vanishes because J is anti-symmetric, so H(y(t)) remains equal to its initial value.The stated equality is H(y(t)) = H(y(0)) = const for all t.Source: §B.2 · p. 18

B.3 PROOF OF THEOREM 2.3

• The proof of Theorem 2.3 establishes that the backward sensitivity matrix is constant over time and uses its norm to derive a gradient bound.The argument is presented for a node’s dynamics and uses a sub-multiplicative norm.Source: §B.3 · p. 20

B.4 PROOF OF THEOREM A.1

• The proof strategy for the sensitivity upper bound is to bound each column of the backward sensitivity matrix before applying a spectral-norm bound.The cited lemma bounds a matrix’s spectral norm when its columns have uniformly bounded norms.Source: §B.4 · p. 20

B.5 PROOF OF THEOREM A.2

• In the discrete proof, the backward sensitivity matrix is symplectic at arbitrary depth.The proof applies the chain rule iteratively to the discretized system.Source: §B.5 · p. 21

B.6 PROOF OF THEOREM 2.4

• The proof estimates neighboring-node sensitivity from four update-gradient blocks, bounding the block-matrix norm by the sum of its sub-block norms.It stops at first-order terms in ϵ and uses cσ = maxx |σ′(x)| in the estimate.Source: §B.6 · pp. 21–22
• The same-node update is bounded by repeating the block-norm argument, after which induction extends the one-step estimates.The proof makes the upper bound independent of the particular nodes and uses N as the maximum node degree.Source: §B.6 · p. 22
• The derived upper bound is at least N^ℓ times larger than the bound in Di Giovanni et al. (2023).The proof obtains it by bounding same-node and neighboring-node updates and then iterating the bound.Source: §B.6 · pp. 22–23

B.7 ADDITIONAL THEOREMS FOR PH-DGN WITH DRIVING FORCES

• With bounded activations and bounded, Lipschitz driving forces, the appendix gives lower and upper bounds for same-node and cross-node backward sensitivity in the full port-Hamiltonian update.The driving-force effect on the BSM norm can be controlled with a sufficiently small step size.Source: §B.7 · pp. 23–25

C.1 EMPLOYED BASELINES

• The evaluated baselines span classical MPNNs, DE-DGNs, graph transformers, and higher-order DGNs; DE-DGNs are identified as direct competitors.The listed methods include GCN, GraphSAGE, GAT, DGC, GRAND, GraphGPS, DIGL, MixHop, and DRew.Source: §C.1 · p. 25

C.2 GRAPH TRANSFER TASK

• Graph transfer tests source-label propagation to a more distant target on line, ring, and crossed-ring graphs, with source-target distances of 3, 5, 10, and 50.The task uses random intermediate-node features and 1000 training, 100 validation, and 100 test graphs.Source: §C.2 · p. 26, Figure 4 · p. 26

C.3 GRAPH PROPERTY PREDICTION

• Graph-property prediction covers single-source shortest path and eccentricity at node level, plus graph diameter at graph level, on synthetic graphs of 25–35 nodes.The described split uses 5120 training graphs and 640 validation graphs, with the remainder for testing.Source: §C.3 · p. 26

C.4 LONG-RANGE GRAPH BENCHMARK

• The LRGB evaluation uses Peptide-func classification and Peptide-struct regression on peptide graphs averaging 150 nodes and 307 edges.The tasks contain 15,535 graphs and use stratified 70%/15%/15% train/validation/test splits; model selection is subject to a 500K-parameter budget.Source: §C.4 · p. 27

C.5 EXPLORED HYPERPARAMETER SPACE

• Hyperparameter grids cover graph transfer, graph-property prediction, and LRGB, with PH-DGN layer count and step size selected according to benchmark protocol and, when relevant, training-set graph diameter.Table 3 identifies the task-specific grid and refers to Appendix A.2 for damping and external-force implementations.Source: §C.5 · p. 27, Table 3 · p. 27

D.1 GRAPH PROPERTY PREDICTION RUNTIMES

• With 20 layers and embedding dimension 30, PH-DGN runtimes are improved or comparable to MPNNs, but longer than DE-DGN baselines.Without driving forces, PH-DGN averages 5.92 seconds faster than GAT and 5.19 seconds faster than GCN per epoch; the comparison averages four initializations.Source: §D.1 · p. 28, Table 4 · p. 28

D.2 COMPLETE LRGB RESULTS

• The complete LRGB results include additional multi-hop baselines and both the original and 3-layer-MLP readout settings, but the authors cannot determine which setting’s baseline is best from validation scores.Table 5 reports Peptides-func and Peptides-struct results averaged over three training seeds.Source: §D.2 · p. 28, Table 5 · p. 29

D.3 COMPARISON TO HAMGNN AND HANG

• PH-DGN outperforms HamGNN and HANG while using less computation in the Eccentricity comparison.Table 7 reports test log10(MSE), memory consumption, and average epoch time for models configured with L = 20 and d = 30.Source: §D.3 · p. 30, Table 7 · p. 30

D.4 ABLATION ON DRIVING FORCES

• On Minesweeper, purely conservative PH-DGNC achieves new state-of-the-art performance, while adding driving forces does not improve the task’s result.The authors suggest counting-based tasks benefit from preserving all information.Source: §D.4 · p. 31, Table 8 · p. 31
• Purely conservative PH-DGNC performs better than full PH-DGN on two of three graph-transfer tasks in the long-range regime.The authors connect this result to the task’s need to retain the source-node label without dissipation.Source: §D.4 · p. 31, Figure 5 · p. 32
• The authors conclude that model selection should determine the driving-force configuration for each data setting.This follows the reported differences across Section 3 results and the ablations.Source: §D.4 · p. 31