SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The "almost" appearing here has to do with the fact that (1) the attentional numerator  $f(\tilde{a} (\mathbf{q}_i,\mathbf{q}_j))$  is generally asymmetric in  $i,j$ , and is therefore symmetrized by the divergence operator  $d_0^\top$ , (2) the activation function between layers is not included, and (3) learnable weight matrices  $\mathbf{W}^k$  in GAT are set to the identity.
Remark 4.3. The interpretation of graph attention as a combination of learnable inner products admits a direct generalization to higher-order cliques, which is discussed in Appendix A.4.
Hamiltonian case: A purely conservative system is generated by solving  $\dot{\mathbf{x}} = \mathbf{L}(\mathbf{x})\nabla E(\mathbf{x})$  , or
$$
\left( \begin{array}{c} \dot {\mathbf {q}} \\ \dot {\mathbf {p}} \end{array} \right) = \left( \begin{array}{c c} 0 & - d _ {0} ^ {*} \\ d _ {0} & 0 \end{array} \right) \left( \begin{array}{c c} \mathbf {A} _ {0} ^ {- 1} & 0 \\ 0 & \mathbf {A} _ {1} ^ {- 1} \end{array} \right) \left( \begin{array}{c} \mathbf {q} \\ \mathbf {p} \end{array} \right) = \left( \begin{array}{c} - d _ {0} ^ {*} \mathbf {A} _ {1} ^ {- 1} \mathbf {p} \\ d _ {0} \mathbf {A} _ {0} ^ {- 1} \mathbf {q} \end{array} \right).
$$
This is a noncanonical Hamiltonian system which generates a purely reversible flow. In particular, it can be shown that
$$
\dot {E} (\mathbf {x}) = (\dot {\mathbf {x}}, \nabla E (\mathbf {x})) = (\mathbf {L} (\mathbf {x}) \nabla E (\mathbf {x}), \nabla E (\mathbf {x})) = - (\nabla E (\mathbf {x}), \mathbf {L} (\mathbf {x}) \nabla E (\mathbf {x})) = 0,
$$
so that energy is conserved due to the skew-adjointness of  $\mathbf{L}$ .
Gradient case: On the opposite end of the spectrum are generalized gradient flows, which are totally dissipative. Consider solving  $\dot{\mathbf{x}} = -\mathbf{G}(\mathbf{x})\nabla E(\mathbf{x})$ , or
$$
\left( \begin{array}{c} \dot {\mathbf {q}} \\ \dot {\mathbf {p}} \end{array} \right) = - \left( \begin{array}{c c} \Delta_ {0} & 0 \\ 0 & \Delta_ {1} \end{array} \right) \left( \begin{array}{c c} \mathbf {A} _ {0} ^ {- 1} & 0 \\ 0 & \mathbf {A} _ {1} ^ {- 1} \end{array} \right) \left( \begin{array}{c} \mathbf {q} \\ \mathbf {p} \end{array} \right) = - \left( \begin{array}{c} \Delta_ {0} \mathbf {A} _ {0} ^ {- 1} \mathbf {q} \\ \Delta_ {1} \mathbf {A} _ {1} ^ {- 1} \mathbf {p} \end{array} \right).
$$
This system is a metric diffusion process on nodes and edges separately. Moreover, it corresponds to a generalized gradient flow, since
$$
\dot {E} (\mathbf {x}) = (\dot {\mathbf {x}}, \nabla E (\mathbf {x})) = - (\mathbf {G} (\mathbf {x}) \nabla E (\mathbf {x}), \nabla E (\mathbf {x})) = - | \nabla E (\mathbf {x}) | _ {\mathbf {G}} ^ {2} \leq 0,
$$
due to the self-adjoint and positive semi-definite nature of  $\mathbf{G}$ .
Remark 4.4. The architecture in GRAND [49] is almost a gradient flow, however the pre-attention mechanism lacks the requisite symmetry to formally induce a valid inner product.
Double bracket case: Another useful formulation for incomplete systems is the so-called double-bracket formalism. Consider solving  $\dot{\mathbf{x}} = \mathbf{L}\nabla E + \mathbf{L}^2\nabla E$ , or
$$
\left( \begin{array}{c} \dot {\mathbf {q}} \\ \dot {\mathbf {p}} \end{array} \right) = \left( \begin{array}{c c} 0 & - d _ {0} ^ {*} \\ d _ {0} & 0 \end{array} \right) \left( \begin{array}{c} \mathbf {A} _ {0} ^ {- 1} \mathbf {q} \\ \mathbf {A} _ {1} ^ {- 1} \mathbf {p} \end{array} \right) + \left( \begin{array}{c c} - d _ {0} ^ {*} d _ {0} & 0 \\ 0 & - d _ {0} d _ {0} ^ {*} \end{array} \right) \left( \begin{array}{c} \mathbf {A} _ {0} ^ {- 1} \mathbf {q} \\ \mathbf {A} _ {1} ^ {- 1} \mathbf {p} \end{array} \right) = \left( \begin{array}{c} - \Delta_ {0} \mathbf {A} _ {0} ^ {- 1} \mathbf {q} - d _ {0} ^ {*} \mathbf {A} _ {1} ^ {- 1} \mathbf {p} \\ d _ {0} \mathbf {A} _ {0} ^ {- 1} \mathbf {q} - d _ {0} d _ {0} ^ {*} \mathbf {A} _ {1} ^ {- 1} \mathbf {p} \end{array} \right).
$$
This provides a dissipative relationship which preserves the Casimirs of the Poisson bracket generated by  $\mathbf{L}$ , since  $\mathbf{L} \nabla C = \mathbf{0}$  implies  $\mathbf{L}^2 \nabla C = \mathbf{0}$ . In particular, it follows that
$$
\dot {E} (\mathbf {x}) = \bigl (\dot {\mathbf {x}}, \nabla E (\mathbf {x}) \bigr) = \bigl (\mathbf {L} (\mathbf {x}) \nabla E (\mathbf {x}) + \mathbf {L} ^ {2} (\mathbf {x}) \nabla E (\mathbf {x}), \nabla E (\mathbf {x}) \bigr) = 0 - \bigl | \mathbf {L} (\mathbf {x}) \nabla E (\mathbf {x}) \bigr | ^ {2} \leq 0,
$$
since  $\mathbf{L}$  is skew-adjoint and therefore  $\mathbf{L}^2$  is self-adjoint.
Remark 4.5. It is interesting to note that the matrix  $\mathbf{L}$  is essentially a Dirac operator (square root of the Hodge Laplacian  $\Delta = (d + d^{*})^{2}$ ) restricted to cliques of degree at most 1. However, here  $\mathbf{L}^2 = -\Delta$ , so that  $\mathbf{L}$  is in some sense "pure imaginary".
Metriplectic case: Metriplectic systems are expressible as  $\dot{\mathbf{x}} = \mathbf{L}\nabla E + \mathbf{M}\nabla S$  where  $E, S$  are energy resp. entropy functions which satisfy the degeneracy conditions  $\mathbf{L}\nabla S = \mathbf{M}\nabla E = \mathbf{0}$ . One way of setting this up in the present case is to define the energy and entropy functions
$$
E (\mathbf {q}, \mathbf {p}) = f _ {E} (s (\mathbf {q})) + g _ {E} (s (d _ {0} d _ {0} ^ {\intercal} \mathbf {p})),
$$
$$
S (\mathbf {q}, \mathbf {p}) = g _ {S} \left(s \left(d _ {1} ^ {\mathsf {T}} d _ {1} \mathbf {p}\right)\right),
$$
where  $s$  is sum aggregation over nodes resp. edges,  $f_{E}:\mathbb{R}^{n_{V}}\to \mathbb{R}$  acts on node features, and  $g_{E},g_{S}:\mathbb{R}^{n_{E}}\rightarrow \mathbb{R}$  act on edge features. Denoting the "all ones" vector (of variable length) by  $\mathbf{1}$ , it is shown in Appendix A.6 that the  $\mathbf{A}$ -gradients of energy and entropy can be computed as
$$
\nabla E (\mathbf {x}) = \left( \begin{array}{c} \mathbf {A} _ {0} ^ {- 1} \mathbf {1} \otimes \nabla f _ {E} \left(h (\mathbf {q})\right) \\ \mathbf {A} _ {1} ^ {- 1} d _ {0} d _ {0} ^ {\intercal} \mathbf {1} \otimes \nabla g _ {E} \left(h \left(d _ {0} d _ {0} ^ {\intercal} \mathbf {p}\right)\right) \end{array} \right), \qquad \nabla S (\mathbf {x}) = \left( \begin{array}{c} 0 \\ \mathbf {A} _ {1} ^ {- 1} d _ {1} ^ {\intercal} d _ {1} \mathbf {1} \otimes \nabla g _ {S} \left(h \left(d _ {1} ^ {\intercal} d _ {1} \mathbf {p}\right)\right) \end{array} \right).
$$
Similarly, it is shown in Appendix A.6 that the degeneracy conditions  $\mathbf{L}\nabla S = \mathbf{M}\nabla E = \mathbf{0}$  are satisfied by construction. Therefore, the governing dynamical system becomes
$$
\left( \begin{array}{c} \dot {\mathbf {q}} \\ \dot {\mathbf {p}} \end{array} \right) = \mathbf {L} \nabla E + \mathbf {M} \nabla S = \left( \begin{array}{c} - \mathbf {A} _ {0} ^ {- 1} d _ {0} ^ {\mathsf {T}} d _ {0} d _ {0} ^ {\mathsf {T}} \mathbf {1} \otimes \nabla g _ {E} \left(s \left(d _ {0} d _ {0} ^ {\mathsf {T}} \mathbf {p}\right)\right) \\ d _ {0} \mathbf {A} _ {0} ^ {- 1} \mathbf {1} \otimes \nabla f _ {E} \left(s (\mathbf {q})\right) + \mathbf {A} _ {1} d _ {1} ^ {*} d _ {1} d _ {1} ^ {\mathsf {T}} d _ {1} \mathbf {1} \otimes \nabla g _ {S} \left(s \left(d _ {1} ^ {\mathsf {T}} d _ {1} \mathbf {p}\right)\right) \end{array} \right).
$$
With this, it follows that the system obeys a version of the first and second laws of thermodynamics,
$$
\dot {E} (\mathbf {x}) = (\dot {\mathbf {x}}, \nabla E (\mathbf {x})) = (\mathbf {L} \nabla E (\mathbf {x}), \nabla E (\mathbf {x})) + (\mathbf {M} \nabla S (\mathbf {x}), \nabla E (\mathbf {x})) = (\nabla S (\mathbf {x}), \mathbf {M} \nabla E (\mathbf {x})) = 0,
$$
$$
\dot {S} (\mathbf {x}) = (\dot {\mathbf {x}}, \nabla S (\mathbf {x})) = (\mathbf {L} \nabla E (\mathbf {x}), \nabla S (\mathbf {x})) + (\mathbf {M} \nabla S (\mathbf {x}), \nabla S (\mathbf {x})) = 0 + | \nabla S (\mathbf {x}) | _ {M} ^ {2} \geq 0.
$$
Remark 4.6. As seen in the increased complexity of this formulation, enforcing the degeneracy conditions necessary for metriplectic structure is nontrivial. This is accomplished presently via an application of the exact sequence property in Theorem 3.1, which we derive in Appendix A.6.
Remark 4.7. It is worth mentioning that, similar to the other architectures presented in this Section, the metriplectic network proposed here exhibits linear  $O(N)$  scaling in the graph size. This is in notable contrast to [29, 31] which scale as  $O(N^3)$ .
This section reports results on experiments designed to probe the influence of bracket structure on trajectory prediction and nodal feature classification. Additional experimental details can be found in Appendix B. In each Table, orange indicates the best result by our models, and blue indicates the best of those compared. We consider both physical systems, where the role of structure preservation is explicit, as well as graph-analytic problems.
5.1 Damped double pendulum
As a first experiment, consider applying one of these architectures to reproduce the trajectory of a double pendulum with a damping force proportional to the angular momenta of the pendulum masses (see Appendix B.1 for details).
Double pendulum	MAE q	MAE p	Total MAE
NODE	0.0240 ± 0.015	0.0299 ± 0.0091	0.0269 ± 0.012
NODE+AE	0.0532 ± 0.029	0.0671 ± 0.043	0.0602 ± 0.035
Hamiltonian	0.00368 ± 0.0015	0.00402 ± 0.0015	0.00369 ± 0.0013
Gradient	0.00762 ± 0.0023	0.0339 ± 0.012	0.0208 ± 0.0067
Double Bracket	0.00584 ± 0.0013	0.0183 ± 0.0071	0.0120 ± 0.0037
Metriplectic	0.00364 ± 0.00064	0.00553 ± 0.00029	0.00459 ± 0.00020
Table 2: Mean absolute errors (MAEs) of the network predictions in the damped double pendulum case, reported as avg±stdev over 5 runs.
Since this system is metriplectic when expressed in position-momentum-entropy coordinates (c.f. [56]), it is useful to see if any of the brackets from Section 4 can adequately capture these dynamics without an entropic variable. The results of applying the architectures of Section 4 to reproduce a trajectory of five periods are displayed in Table 2, alongside comparisons with a black-box NODE network and a latent NODE with feature encoder/decoder. While each network is capable of producing a small mean absolute error, it is clear that the metriplectic and Hamiltonian networks produce the most accurate trajectories. It is remarkable both that the Hamiltonian bracket does so well here and that the gradient bracket does so poorly, being that the damped double pendulum system is quite dissipative. On the other hand, it is unlikely to be only the feature encoder/decoder leading to good performance here, as both the NODE and  $\mathrm{NODE + AE}$  architectures perform worse on this task by about one order of magnitude.
5.2 MuJoCo Dynamics
Next we test the proposed models on more complex physical systems that are generated by the Multi-Joint dynamics with Contact (MuJoCo) physics simulator [57]. We consider the modified versions of Open AI Gym environments [23]: HalfCheetah, Hopper, and Swimmer.
We represent an object in an environment as a fully-connected graph, where a node corresponds to a body part of the object and, thus, the nodal feature  $\mathbf{q}_i$  corresponds to a position of a body part or an angle of a joint. As the edge features, a pair of nodal velocities  $\mathbf{p}_{\alpha} = (v_{\mathrm{src}(\alpha)}, v_{\mathrm{dst}(\alpha)})$  are provided, where  $v_{\mathrm{src}(\alpha)}$  and  $v_{\mathrm{dst}(\alpha)}$  denote velocities of the source and destination nodes connected to the edge.
Since the MuJoCo environments contain an actor applying controls, additional control input is accounted for with an additive forcing term which is parameterized by a multi-layer perceptron and introduced into the bracket-based dynamics models. See Appendix B.2 for additional experimental details. The problem therefore consists of finding an optimal control MLP, and we evaluate the improvement which comes from representing the physics surrogate with bracket dynamics over NODE.
All models are trained via minimizing the MSE between the predicted positions  $\tilde{\mathbf{q}}$  and the ground truth positions  $\mathbf{q}$  and are tested on an unseen test set. Table 3 reports the errors of network predictions on the test set measured in the relative  $\ell_2$  norm,  $\| \mathbf{q} - \tilde{\mathbf{q}} \|_2 / \| \mathbf{q} \|_2 \| \tilde{\mathbf{q}} \|_2$ . Similar to the double pendulum experiments, all models are able to produce accurate predictions with around or less than  $10\%$  errors. While the gradient bracket makes little to no improvements over NODEs, the Hamiltonian, double, and metriplectic brackets produce more accurate predictions. Interestingly, the Hamiltonian bracket performs the best in this case as well, meaning that any dissipation present is effectively compensated for by the autoencoder which transforms the features.
5.3 Node classification
Moving beyond physics-based examples, it remains to see how bracket-based architectures perform on "black-box" node classification problems. Table 4 and Table 5 present results on common benchmark problems including the citation networks Cora [58], Citeseer [59], and Pubmed [60], as well as the coauthor graph, CoauthorCS [61], and the Amazon co-purchasing graphs, Computer and Photo [62]. For comparison, we report results on a standard GAT [12], a neural graph differential equation
3Results of an experiment with an alternative embedding (i.e.,  $\mathbf{q}_i = (q_i,v_i)$ ) are reported in Appendix B.2.2.
Dataset	HalfCheetah	Hopper	Swimmer
NODE+AE	0.106 ± 0.0011	0.0780 ± 0.0021	0.0297 ± 0.0036
Hamiltonian	0.0566 ± 0.013	0.0279 ± 0.0019	0.0122 ± 0.00044
Gradient	0.105 ± 0.0076	0.0848 ± 0.0011	0.0290 ± 0.0011
Double Bracket	0.0621 ± 0.0096	0.0297 ± 0.0048	0.0128 ± 0.00070
Metriplectic	0.105 ± 0.0091	0.0398 ± 0.0057	0.0179 ± 0.00059
Table 3: Relative error of network predictions for the MuJoCo environment on the test set, reported as avg±stdev over 4 runs.
Planetoid splits	CORA	CiteSeer	PubMed
GAT	82.8 ± 0.5	69.5 ± 0.9	79.0 ± 0.5
GDE	83.8 ± 0.5	72.5 ± 0.5	79.9 ± 0.3
GRAND-nl	83.6 ± 0.5	70.8 ± 1.1	79.7 ± 0.3
Hamiltonian	77.2 ± 0.7	73.0 ± 1.2	78.5 ± 0.3
Gradient	79.9 ± 0.7	71.8 ± 1.4	78.6 ± 0.7
Double Bracket	82.6 ± 0.9	74.2 ± 1.4	79.6 ± 0.6
Metriplectic	57.4 ± 1.0	60.5 ± 1.1	69.8 ± 0.7
Table 4: Test accuracy and standard deviations (averaged over 20 randomly initialized runs) using the original Planetoid train-valid-test splits. Comparisons use the numbers reported in [49].
architecture (GDE) [63], and the nonlinear GRAND architecture (GRAND-nl) from [49] which is closest to ours. Since our experimental setting is similar to that of [49], the numbers reported for GAT, GDE, and GRAND-nl are taken directly from this paper. Note that, despite the similar  $O(N)$  scaling in the metriplectic architecture, the high dimension of the node and edge features on the latter three datasets led to trainable  $E$ ,  $S$  functions which exhausted the memory on our available machines, and therefore results are not reported for these cases. A full description of experimental details is provided in Appendix B.3.
Remark 5.1. To highlight the effect of bracket structure on network performance, only minimal modifications are employed during network training. In particular, we do not include any additional regularization, positional encoding, graph rewiring, extraction of connected components, extra terms on the right-hand side, or early stopping. While it is likely that better classification performance could be achieved with some of these modifications included, it becomes very difficult to isolate the effect of structure-preservation. A complete list of tunable hyperparameters is given in Appendix B.3.
The results show different behavior produced by each bracket architecture. It is empirically clear that there is some value in full or partial reversibility, since the Hamiltonian and double bracket architectures both perform better than the corresponding gradient architecture on datasets such as Computer and Photo. Moreover, it appears that the partially reversible double bracket performs the best of the bracket architectures in every case, which is consistent with the idea that both reversible and irreversible dynamics are critical for capturing the behavior of general dynamical systems. Interestingly, the metriplectic bracket performs worse on these tasks by a large margin. We conjecture this architecture may be harder to train for larger problems despite its  $O(N)$  complexity in the graph size, suggesting that more sophisticated training strategies may be required for large problems.
This work presents a unified theoretical framework for analysis and construction of graph attention networks. The exact sequence property of graph derivatives and coercivity of Hodge Laplacians which follow from the theory allow the construction of four structure-preserving brackets, which we use to evaluate the role of irreversibility in both data-driven physics simulators and graph analytics problems. In all contexts, the pure diffusion bracket performed most poorly, with mixed results between purely reversible and partially dissipative brackets.
Random splits	CORA	CiteSeer	PubMed	Coauthor CS	Computer	Photo
GAT	81.8 ± 1.3	71.4 ± 1.9	78.7 ± 2.3	90.5 ± 0.6	78.0 ± 19.0	85.7 ± 20.3
GDE	78.7 ± 2.2	71.8 ± 1.1	73.9 ± 3.7	91.6 ± 0.1	82.9 ± 0.6	92.4 ± 2.0
GRAND-nl	82.3 ± 1.6	70.9 ± 1.0	77.5 ± 1.8	92.4 ± 0.3	82.4 ± 2.1	92.4 ± 0.8
Hamiltonian	76.2 ± 2.1	72.2 ± 1.9	76.8 ± 1.1	92.0 ± 0.2	84.0 ± 1.0	91.8 ± 0.2
Gradient	81.3 ± 1.2	72.1 ± 1.7	77.2 ± 2.1	92.2 ± 0.3	78.1 ± 1.2	88.2 ± 0.6
Double Bracket	83.0 ± 1.1	74.2 ± 2.5	78.2 ± 2.0	92.5 ± 0.2	84.8 ± 0.5	92.4 ± 0.3
Metriplectic	59.6 ± 2.0	63.1 ± 2.4	69.8 ± 2.1	-	-	-
Table 5: Test accuracy and standard deviations averaged over 20 runs with random 80/10/10 train/val/test splits. Comparisons use the numbers reported in [49].
The linear scaling achieved by the metriplectic brackets has a potential major impact for data-driven physics modeling. Metriplectic systems emerge naturally when coarse-graining multiscale systems. With increasing interest in using ML to construct digital twins, fast data-driven surrogates for complex multi-physics acting over multiple scales will become crucial. In this setting the stability encoded by metriplectic dynamics translates to robust surrogates, with linear complexity suggesting the possibility of scaling up to millions of degrees of freedom.
Limitations: All analysis holds under the assumption of modified attention mechanisms which allow interpretation of GAT networks as diffusion processes; readers should take care that the analysis is for a non-standard attention. Secondly, for all brackets we did not introduce empirical modifications (e.g. regularization, forcing, etc) to optimize performance so that we could study the role of (ir)reversibility in isolation. With this in mind, one may be able to add "tricks" to e.g. obtain a diffusion architecture which outperforms those presented here. Finally, note that the use of a feature autoencoder in the bracket architectures means that structure is enforced in the transformed space. This allows for applicability to more general systems, and can be easily removed when appropriate features are known.
Broader impacts: The work performed here is strictly foundational mathematics and is intended to improve the performance of GNNs in the context of graph analysis and data-driven physics modeling. Subsequent application of the theory may have societal impact, but the current work anticipated to improve the performance of machine learning in graph settings only at a foundational level.
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Glossary of Notation and Symbols
The next list describes several symbols that will be later used within the body of the document
$\{\{\cdot ,\cdot \} \}$  (Irreversible) double bracket on functions with generator  $\mathbf{L}^2$
$[\cdot ,\cdot ]$  Degenerate (irreversible) metric bracket on functions with generator  $\mathbf{M}^{\intercal} = \mathbf{M}$
$\{\cdot ,\cdot \}$  Poisson (reversible) bracket on functions with generator  $\mathbf{L}^{\intercal} = -\mathbf{L}$
$\mathcal{N}(i), \overline{\mathcal{N}}(i)$  Neighbors of node  $i \in \mathcal{V}$ , neighbors of node  $i \in \mathcal{V}$  including  $i$
[S] Indicator function of the statement  $S$
$\delta f, \nabla f$  Adjoint of  $df$  with respect to  $\langle \cdot, \cdot \rangle$ , adjoint of  $df$  with respect to  $(\cdot, \cdot)$
$\Delta_{k}$  Hodge Laplacian  $d_k d_k^* + d_k^* d_k$
$\delta_{ij}$  Kronecker delta
$\dot{f}$  Derivative of  $f$  with respect to time
$(\cdot ,\cdot)_{k}$  Learnable metric inner product on  $k$  -cliques with matrix representation  $\mathbf{A}_k$
$\langle \cdot ,\cdot \rangle_{k}$  Euclidean  $\ell^2$  inner product on  $k$  -cliques
$\mathcal{G}, \mathcal{V}, \mathcal{E}$  Oriented graph, set of nodes, set of edges
$\mathcal{G}_k, \Omega_k$  Set of  $k$ -cliques, vector space of real-valued functions on  $k$ -cliques
$d, d_{k}$  Exterior derivative operator on functions, exterior derivative operator on  $k$ -cliques
$d_k^{\mathsf{T}}, d_k^*$  Adjoint of  $d_k$  with respect to  $\langle \cdot, \cdot \rangle_k$ , adjoint of  $d_k$  with respect to  $(\cdot, \cdot)_k$
A Mathematical foundations
This Appendix provides the following: (1) an introduction to the ideas of graph exterior calculus, A.1, and bracket-based dynamical systems, A.2, necessary for understanding the results in the body, (2) additional explanation regarding adjoints with respect to generic inner products and associated computations, A.3, (3) a mechanism for higher-order attention expressed in terms of learnable inner products, A.4, (4) a discussion of GATs in the context of exterior calculus, A.5, and (5) proofs which are deferred from Section 4, A.6.
A.1 Graph exterior calculus