Graph Neural Networks for Edge Signals: Orientation Equivariance and Invariance
Authors: Dominik Fuchsgruber
Tim Postuvan
Stephan Günnemann
Simon Geisler
Venue: ICLR 2025 Poster (2025)
Keywords: Graph Neural Network
Graph
Edge
Equivariance
Invariance
Topology
Directed Graphs
Primary area: learning on graphs and other geometries & topologies
TL;DR: We study edge-level problems with directed and undirected inputs and targets, and develop a GNN that satisfies novel theoretical desiderata.

Abstract: Many applications in traffic, civil engineering, or electrical engineering revolve around edge-level signals. Such signals can be categorized as inherently directed, for example, the water flow in a pipe network, and undirected, like the diameter of a pipe. Topological methods model edge signals with inherent direction by representing them relative to a so-called *orientation* assigned to each edge. 
They can neither model undirected edge signals nor distinguish if an edge itself is directed or undirected. We address these shortcomings by (i) revising the notion of *orientation equivariance* to enable edge direction-aware topological models, (ii) proposing *orientation invariance* as an additional requirement to describe signals without inherent direction, and (iii) developing EIGN, an architecture composed of novel direction-aware edge-level graph shift operators, that provably fulfils the aforementioned desiderata. It is the first work that discusses modeling directed and undirected signals while distinguishing between directed and undirected edges. A comprehensive evaluation shows that EIGN outperforms prior work in edge-level tasks, improving in RMSE on flow simulation tasks by up to 23.5%.

Published as a conference paper at ICLR 2025
GRAPH NEURAL NETWORKS FOR EDGE SIGNALS: ORIENTATION EQUIVARIANCE AND INVARIANCE
Dominik Fuchsgruber', Tim Poštuvan', Stephan Günnemann', Simon Geisler! Depparmeent of Computer Science & Munich Data Science Institute, TU Munich EPPL {d. fuchsqruber, s. guennemann, gewslerrerrrrreee @ de, tim. wwt.posuuanneepf..ct
ABSTRACT
Many applications traffic, civil engineering, or electrical engineering revolve around edge-level signals. Such signals can be categorized as inherently directed, for example, the water flow in a pipe network, and undirected, like the diameter of pipe. Topological methods model edge signals with inherent direction by representing them relative to a so-called orientation assigned to each edge. They can neither model undirected edge signals nor distinguish if edge itself is directed or undirected. We address these shortcomings by (i) revising the notion of orientation equivariance enable edge direction-awaad topological models, (ii) proposing orientation invariance as an additional requirement to describe signals without inherent direction, and (iii) developing EIGN, an architecture composed of novel direction- aware edge-level graph shift operators, that provably fulfills the aforementioned desiderata. is the first work that discusses modeling directed and undirected signals while distinguishing between directed and undirected edges. A comprehensive evaluation shows that EIGN outperforms prior work edge-level tasks, improving in RMSE on flow simulation tasks by up 23.5%.
INTRODUCTION
Most research on Graph Neural Networks (GNN) research has focused on node- or graph-level tasks
(Wu et al. 2021 Kipf & Welling 2017: Hamilton et al.) 2017; Velickovic et al. 2018 Gasteiger al. 2020; AlQuraishi][222 2001) while edge-level problems remain underexplored. Edge-level signals describe the properties of existing edges and are either the features, hidden representations of GNN, or the targets. They naturally arise in applications involving traffic and many areas of engineering (e.g. electric circuits (Dörfler et al. 2018) hydraulics (Grzzón et al.) 2022 Herrera et al. 2016
Signals on edges come in two different modalities: (a) They can have an inherent orientation, like the water flow in pipe network or traffic flow streets. We call them orientation -etetaatioonequivaiiinn signals as they are naturally expressed as scalar values relative to chosen orientation as reference (see Section 2 (b) The other signal modality has intrinsic direction, like pipe diameter or speed limits. We refer to such signals as ceentation-invariann Similar concepts also exist in continuous domains, c.g. a scalar field assigns single value to each point in space, while a vector field describes magnitude and direction.
At the same time, edges themselves may be directed or undirected. Examples that induce directed edges include valves in water networks, one-way roads in street networks, or diodes in electrical circuits. Often, these applications. directed edges prohibit (to a large extent) signal that is oriented against the edge direction. Such constraints can have a big impact on the targets. For example, imagine a one-way street that forces cars take detour.
There are two relevant streams of work that, however, fall short of either representing orientation- cquivariant or orientation-invariant signals. (a) One strategy is to map between orientation-invarianl edge-level signals using Line Graphs. (b) The other line of work relies on Algebraic Topology (Schaub et al. 2021 Ebli et al., 2020; Bunch et al. 2020) to cope with orientation-equivarian signals. Topological models define an arbitrary reference direction for each edge, so-called orientation, and use positive values to indicate that an orientation-equiviii
signal aligns with this reference orientation whereas negative values indicate misalignment (see how in Figures Ta

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Equivariant
60
100
1.1
0.3
Orientation D
Inva fant
Orientation
4.1 m
-1.3 -0---
EIGN
-0.2
0.6
Directed Edge
LOMER
100
80
-2.0
60
100
1.1
0.3
Orientation S
Seeed
Spaid
EIGN
-2.0
1.5
(a) Application: Traffic (b) Graph representations
(c) EIGN
(d) EIGN Predictions
Figure I: EIGN models arbitrary combination of orientation-cquuvarian and -invariant edge-level inputs or targets. In this example, the car flow is equivariant and represented relative to two different (top and bottom) arbitrary direction-consiisent orientations D and D (notice the sign of equivariant signals), while speed limits are invariant. EIGN makes consistent predictions for D and Đ: It outputs the same invariant signals while the sign of equivariant outputs is determined by the orientation.
and Tb the signal 1.3 is represented as 1.3). For undirected edges, there is no preferred reference orientation- signal must change
orientation. Consequently, common requirement for topological models is equivariant (Roddenberry et al. 202I):: The sign of the (orientation-equiiaait)) together with the orientation of the corresponding edge (see Figure [ Orientation equivariance is a property of topological model that allows it to deal with orientationneequivrrinl signals.
Many applications have both orientation eeuiiiarinivvii and invariant features with an arbitrary com- bination of orientation-equiiariaee and invariant targets. Moreover, real scenarios are often only accurately modeled if using both undirected and directed edges. However, Line Graph approaches signals entirely, while topological approaches treat
ignore the properties of orientation-equivarian and can not distinguish edge direction. Arguably, the inductive
every signal as orientation-equivarinnn biases of prior methods render them ineffective large range of applications, which consistent with our experimental findings.
We address these shortcomings and the first to model edge-level tasks with orientation-squivarian and invariant signals on graphs with directed and undirected edges. Our key contributions are:
a) Desiderata. We formalize suitable desiderata: (i) Joint orientation equivariance for orientation- cquivariant signals, and (ii) joint orientation invariance for grientation-invaiiann signals. b) EIGN: A general-purpose edge-level topological GNN. EIGN provably fulfills these desider- ata, leveraging novel direction-awe aware convolution operators. and a fusion operation between both modalities to model their interactions see Section 4.2).
Benchmarking, We devise suite of challenging and diverse benchmarks covering synthetic and real-words tasks, including novel dataset for electric circuits that requires dealing with and iecentationinvariinn signals on graphs with directed and undirected edges. EIGN outperforms prior work by reducing the RMSE up to 23.5% in our experiments
BACKGROUND
Algebraic Topology is a principled framework for representing orientation -uquaiiiinnnuiviiiiann edge sig- nals (Roddenberry & Segarra 2019; Roddenberry et al.) 2021 Ebli al. 2020). In the interest of
notational clarity, we simplify many concepts by directly applying them to the edge domain and omit general topological definitions. For comprehensive introduction, refer to Schaub al. (202T).
Notation. We consider edge-level problems on graph & = (V,E) with n = [V] nodes and m [|] edges. We distinguish between: (i) directed edges, represented by ordered tuples E V V. and undirected edges & C u[,,v) u, u E V}, represented by unordered sets. (ii) edge-signals
We provide code uss.ci.uum.deddaml//ig.
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(inputs, hidden representations, outputs) that come with inherent direction orientationn-equiiariial Xequ E - Xdee and orientation-ivvaiiant invariant Xuv E C Xdle otherwise.
Orientation.
While arientation-invariant edge signals can be represented by scalar values as they are, ihicntation-equivriann edge signals need to be defined relative to the orientation of the associated edge. To that end, topological methods define an arbitrarily chosen orientation D E ->>>> V V for signal x aligns with
every edge, also represented by ordered tuples. the orientation of its edge, we represent it with æ and -z otherwise. Changing from orientation D to another orientation D induces sign flips for all prientation-equivariae orientation was flipped. We can represent this orientation change for signals with a diagonal matrix D,S E mm with entries Ap,sle. = 1 if (e)) = D(e) and -1 otherwise.
an inientationeequiaaat
signals of edges whose
Edge direction and orientation are different concepts. While the former is part of the given problem topology and may influence model predictions, the latter can be thought of as basis in which ceenntation-qqivariin
edge signals are represented. For undirected edges, which orientation is chosen must not impact the problem itself. However, unlike previous work, we fix the orientation of directed edges to match their direction (D(en) = ep) and, consequently, represent their orientation- equivariant signals relative to their direction. We refer to such orientations as direction-comsisteni They can encode information about the direction of directed edges.
Boundary and Laplace Operators. For given orientation D. one can define boundary operator that maps signals on m edges defined with respect to orientation D to the domain of n nodes. Reusing the analogy of water flow, the boundary operator sums all flow coming into node and subtracts all outgoing flow. It can be represented by a matrix Bequ
E R x
{
if (ee) (v. if (f) - (:, otherwise
{
Besul n
(1)
The boundary operator can be used define Laplace operator the edges of the graph, the so-called (Equivariant) Edge Laplacian (Schaub et al., (022]: Equ = B Bequ. This operator can be understood as message passing between edges that are incident shared node (see Section 4.3)
RELATED WORK
Methods grounded in Algebraic Topology often represent iiientation-qqiiariin signals on undirected graphs relative to an arbitrary orientation (Schaub et al. 2021) Some of these approaches also utilize higher-order structures composed of edges that, however, often need to be handcrafted (Bunch et al., 2020, Grusti et al.] 2022). HodgeGNN (Roddenberry & Segarra (219)) and similar architectures (Park et al. 2023) Redddebedyyyy et al. (02I) satisfy a more limited notion of orientation equivariance compared to pro- posal. This results in major shortcomings of these mod- els in practice: (i) They treat all input and output sig- nals as orientation-equiiariinn and, therefore, can not model orientation-invariant edge signals appropriately. In the example of Figure T they do not predict the same iantatiion-ivariinn output under different orientations D and D but instead induce sign flips as if they were orientation-equivarian signals leading to inconsistent predictions for different (arbitrary) orientations of the same topology. (ii) These approaches are equivariant re- garding the orientation of all edges, whereas our desider- ata relax this requirement to hold for undirected edges only. As depicted in Figure 2 edge direction often alters the nature of the problem: In one scenario cars go against the direction of one-way street while in the
Topological Models.
Equivar lent Inveriant Orientation
Speod
100
4.1
1.3
80
SPEED
Speed UiFt 80
IMMI
60
Lini 100
100
1.3
SPEED UMMT 60
Speee Link
80 -2.8
80
(a) Traffic Scenarios
(b) Representations
Figure 2: that differ the direction of one edge but model different situations (flame in bottom left). Their representations are indistin- guishable for models that are orientation- equivariant for directed edges.
Two scenarios (top. bottom)
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other they do not. For fully orientation equivariant models, both representations are indistinguishable: The edge orientation serves only as a representation basis and can not encode directionality. An exception among topologiaal methods is concurrent work (Lecha al. 2024) that implicitly distin- guishes between directed and undirected edges by representing undirected edges with two directed yet antiparallel edges in directed graph that is augmented with higher-order structures. However, they neither discuss the combination of equivariant and invariant features/targets nor is their model applicable to this setting without modifications.
Flow Interpolation. An orthogonal line of research studies flow interpolation problems (Ford & Fulkerson 1956 Lippi et al., 2013). Even though both features and targets are technically orientation- equivariant, many methods do not approach this task from a topological perspective. Instead, they utilize specialized inductive biases such as flow conservation (Jia et al., 2019) or physics-inforne informed constraints (Smith et al.) 2022). da Silva et al. (2021) frame interpolation as a bi-level optimization problem where edge-level models serve as learned regularizers. Such approaches are limited to flow interpolation problems while our model is a general framework that can be applied to broader range of edge-level tasks. As we find in Section 5.3 EIGN can learn the physical properties of a problem without explicitly encoding physics- inyfcmd inductive biases.
Edge-Level GNNs, Beyond bow-based problems, GNNs have been applied to (orientation snrentattooniivaaiaat)) edge-level tasks as well. One family of approaches employs node-level GNNs and a successive readout function ZZha et al.] 2023). Such approaches have been particularly popular for link prediction (Zhou 2021). where target edges are not present in the input data. Another paradigm is to apply node-level GNNs to the dual Line Graph (Jiang et al., 2019: Jo et al. (021) of the topology. Only a few approaches rely on edge-specific methods (Zhang al. 2020). Models that are not grounded in a topological framework lack appropriate inductive biases to model orientation- equivariant signals with inherent direction. These approaches can be categorized as treating both input and target as orientation-invenn
signals. In the context of Figure | they would not represent diienatioo-equivariinn signals relative to the respective orientation.
GNNs for Directed Graphs. Directed graphs have received lot of attention for node-level problems. Many approaches discriminate between adjacent in-neighbors and out-neighbors (Li et al. 2016) Rossi et al., 2023). Also, spectral convolutions, that are based on the Node Laplacian, have been generalized to directed settings (Ma et al., 2019; Monti et al. 2018). From the different possible direction- aware Laplacians (Tong et al. (020), our work takes inspiration from the Magnetic Node Laplacian (Zhang et al.] (021)) which can be defined using a complex-valuse boundary operator. was recently utilized to compute direction-aware positional node encodings (Geisler al. 2023) The Laplace operators of our approach generalize this concept to edge-level problems.
METHOD
4.1 DESTRABLE PROPERTIES FOR EDGE-LEVEL, GNNS
At the core of our work stand novel desiderata which enforce model to make consistent predictions for both erientation-equivant our novel constraints to undirected edges by requiring equivariance/ivvrrincee among direction- consistent orientations only. We additionally prove that EIGN is also equivariant with respect edge permutations, which we defer to Appendix A.2]
and -invariant edge signals among different orientations. We restrict
Definition 4.1 (Joint Orientation Equivariance). Let Đ, be arbitrary direction-consisteet ori- entations of edges on G. We say that mapping f is jointly orientation-equivariann if for any orientation-equivarii input Xequ E Cmmxdap and any orientation-
input Ximv E C xmm:
A X X f(Ag,0 X.sxxeuu Xin, G,D).
Definition 4.2 (Joint Orientation Invariance). Let D, Ô be arbitrary direction-consisten orientations of edges on G. We say that mapping g is jointly orientation- ihentation-nnvariant if for any orientation- equivariant input Xoqu E C coue and any iisentationinvvaiian
input X E C xdw:
g(Xeuu egu XIn. G. = g( A Xequ Xin, G,D).
Both definitions ensure that a model predicts the same output signal when the orientation is changed from D to D: While Definition 4.1 ensures that the orientation-equuiariin output is represented
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relative to the new orientation D Definition 4.2 requires the same orientation incentatioonivvriannl predictions which are not relative to the new orientation. In both desiderata, the orientation -quiiiiriaiivviaaann input Xequ needs to be represented relative to the respective orientation as well. Restricting these properties to direction-consisten orientations means that both properties only need to hold regarding the arbitrary orientation of undirected edges. This allows models to break both desiderata for the orientation of directed edges, which enables using their orientation to encode direction.
4.2 EIGN: AN ORIENTATION
AND ORIENTATIONNINVARII MODEL
We propose EIGN (Figure 3). a model that satisfies these desiderata. It consists of L layers each of which takes orientation- equivariant and -invariant input edge sig- nals Hequ H (1-1) and transforms them into outputs of the corresponding modal- ity Heq and H Its message passing between edge signals is based on different graph shift operators (sce "Convolutions" in Figure 3 and Section 4.3 for details) both
Convolutions
Interactions
H ogu
Fegus
Fusion
(1) HOPK
[1) "iop
(0) Feep
Fusion
within and between edge signal modalities.
Zinu
We set Hequ Xequ. H = Xinv and
X
H and H H are the equivariant and invariant output signals, respectively. In (.) model
Figure 3: EIGN architecture: In each layer, message passing using novel Laplacians is performed within and between rrientation-equvaane invariant signals. The two aggregated modalities Zequ
the following, we denote with W parameters of appropriate dimensions, with
and orientation- and then 2000 are then fused.
Equ an element-wise sign equivariant acti- vation function, i.e. Tequl (-x) -euu (x)), and with din an arbitrary non-lina oon-linearity. We detail the implementation of EIGN in Appendix
D.3
= = Jequ (Fequl equ
(((l) (1- W
1 mn (H (1-1)
W ()) inq- + H
(w (()) equ
(2)
equ (1-1)
z (1) Oim (1)
( w (1)
S iin (H wll) (I) +
((-) ( inv
inv
The intra-modality message passing schemes feg and f update the orientation-cquiiarraat and and -invariant
innvariant signal representation of an edge by aggregating orientation-cquivariinn signals of adjacent edges. Similarly, the mnter-modality schemes f Jan-voqu and f() f aggregate messages from adjacent edges of one modality and transform it into the other. While this enables information exchange between directed and undirected edge signals, restricts their interaction to local aggregates: The edge orientation-equivarian orientation-invart signals and vice versa. This makes modeling interactions between orientation- equivariant and -invariant edge signals of the same edge difficult. Therefore, EIGN uses second fusion operation that does not use Laplacians (depicted as "Fusion" in Figure
signal only depends on the average of adjacent
3)
Tequ (Z Wi( ( equ
inv Fiiny ttequ + 2Z(1)
((()()
equ
Feeuu-equu
equ
HD inv = Jinv Z imv
()(()
O abs( (1) (!()) equ E,equ (eeu->inn)
) + Z(0)
(5)
Fin
Here, @ and abs denote element-wise multiplication and ab- solute value. In general. the fusion operation can be realized arbitrarily. We chose point-wise multiplication and absolute values as they are jointly orientation -equvvrriiniviiiiant....aa / -invariant fusion operations respectively (see Appendix A)..
Table I: Convolution and self- interac- tion operators for both input and out- put signal modalities.
Output
Equ.
Inv.
Input
EIGN models all possible types of interactions between edge signal modalities []] The Laplacian operators of Equations ( and B describe interactions of orientation- equivariant and -invariant signals with local aggregates of both the same and different modality. The residual connec- tions in Equations (2) and (3) and the fusion operation of
(q)
L (9)
Equ.
Lequ
agu-a
W
Wigggiin
I (g)
I!()
Inv.
equ
W WF.ine Sego
W in
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Equations (4) and model interactions between orientation-equivanii and -invariant signals of the same edge. The design choices of the fusion operation in Equations (4) and (5) and the definition of the message passing schemes fequ u f pin- and f() to conform to all desiderata proposed in Section Jequ-inn> which we detail next, enforce EIGN
4.3 ORIENTATION
EEUIIARIIIIIN ORIIENTATTON-EQUVAIAAN
AND ORIENTATION ORIENTATION-INVARAN INVRIIANT LAPLACIANS
Equivariant and Invariant Edge Laplacians. The Equivariant Edge Laplacian (see Section 2 arises
from an (cquivariant) boundary operator as Legu B B equ. Its sparsity pattern corresponds to the adjacency matrix of the dual Line Graph @Jian et al. 2019 Jo et al. 2021). D. i.e. edges are adjacent they are incident to the same node. It is jointly thentationeequivarian -qquaaaran mapping (see Lemma 4.1) and therefore suitable to convole orientation-equiiariann edge signals. Intuitively, Lequ performs signals of adjacent
message passing: A target edge & aggregates the orientation-equivrrian edges, Additionally, it re-orients the signal of an adjacent edge e' E & by multiplying with -1 if the orientations of and e' misalign. The orientations of two edges misalign if they are consecutive, i.e. the endpoint of one is the starting point of the other (see Figure 4).
ife if (e), ((e") consecutive
Lequleje
(6)
if (e)), D(ee) not consecutive, but e, e/ adjacent otherwise
We propose novel Laplacian for orientation-invann edge signals by using the orientation- independent unsigned boundary operator (Bodnar et al. 2021; Papillon et al. (023)):
{
if Ee otherwise
B Bunee
It induces the Invariant Edge Laplacian Liny BT In particular, the Invariant Edge Laplacian can be obtained by taking the element- absolute value Lim = abs( Le Its message passing scheme similar to Lequ as well: It, too, aggregates messages of adjacent edges but does not re-orient them if their orientations are consecutive. Thus, it is a jointly iitentationinvariian mapping and suitable to convolve orientation-invariant edge signals.
ins Bin which has the same sparsity pattern as Lequ-
Direction-awaat Edge Laplacians. Neither Lequ nor Limy can distinguish directed from undirected edges as they do not ex- plicitly model them differently. However, following recent
Orientation
Directed Undirected
πQ
line of work on Magnetic Node Laplacians (Forman 1993
Re
Shubin 1994. Colin de Verdière 2013) Furutani et al. 2019
Geisler et al. 2023), we generalize both operators to make them direction-aware Intuitively, the Magnetic Node Lapla- cian represents edge direction through complex phase shifts of magnitude æQ for some fixed hyperparamter q E IR. It can be computed using a complex-valued boundary operator (Fanuel Bardenet 2024):
Re
Im
Re
Figure 4: The Equivariant Mag- netic Edge Laplacian L induces complex phase shift of mq for sig- nals of directed edges that are ag- gregated by the black undirected edge. Signals of misaligned edges re-oriented.
exp(inq) exp(-ing
if e = (u. .) and e E Ep
if = (., v) and e &p if (ee) (v.-) and e Eu
B u,e
1
if De) (s,e) v) and e E Eu otherwise
0
(8)
This boundary operator (9) extends Begu by inducing complex
phase shifts of лQ along directed edges. Consequently, q = 0 reçovers the direction aagection-agnosticc boundary operator of Equation (T). Using the boundary operator B.g), we can define direction- aware Equivariant Magnetic Edge Laplacian analogously to its direction-agnent L [B (qu) y B with (.)" denoting the conjugate transposed. Its sparsity pattern is the same as L Legu and for undirected edges both operators coincide. Thus, its aggregation scheme is similar as well: The key difference is that L applies complex phase shift to the signals of directed edges
counterpart as

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before aggregating them instead of just re-orienting them by flipping their sign. The direction of the phase shift is determined by how the edges relate to their shared incident node. The phases of ingoing edges will be shifted in a different direction than the phases of outgoing edges. Figure 4 depicts this mechanism for adjacent edges that are (i) undirected and aligned in orientation (ii) undirected and misaligned in orientation (iii) directed and aligned in orientation. Lequ is a jointly orientation- equivariant operator as per Definition 4.1 for the (arbitrary) orientation of undirected edges (even they are adjacent directed edges). however, specifically breaks orientation equivariance for directed edges by inducing complex phase shifts:: Changing the direction of a directed edge flips the sign of the complex phase shift that is applied before aggregation (see Table
Defining Laplace operators through boundary maps also enables a different interpretation of the induced message passing schemes. First, each node aggregates information from incident edges. Then each edge computes its representation from the information at its endpoints. In the case of equivariant signals, the node representations are analogous to potentials and the edge representations to the flow they induce. We utilize this through learnable node feature transformations and realize the message passing in Equations (2) and (3) as ( (X) =
(-) 0 ()) instead of directly using
(!)
the corresponding Laplacian (B()) ( graph shift operator.
Lemma 4.1. The Magnetic Equivariant Edge Laplacian implies a jointly orientationneequivariaat
mapping fXXeu: Xim, D) (B(@)) (Beg)) X eqq
Analogously, we use complex phase shifts to encode direction into the boundary operator Bin (Equation (7) that induces the Laplacian Liny for orientation-invarian signals:
eep((ing) exp(-ing)
if (v, and E &D
if (, v) and e E &D if u E e and e E Eu otherwise
B(aa) inv
1
0
It induces a Magnetic Invariant Edge Laplacian [ = (2)) )dd " B that performs message passing analogous the Magnetic Equivariant Edge Laplacian. Like its direction-agnostic counterpart Lime, it is defined independently of orientation and does not re-orient its inputs. Instead, it generalizes Lins by only encoding edge direction through the direction of complex phase shift similar to L()) It is jointly orientation-invarian mapping as per Definition 4.2
Lemma 4.2. The Magnetic Invariant Edge Laplacian implies a jointly prientation-invariine mapping g( X equ Xin: G, D) = [ (4) h( B (4) bny X.m)
Fusing Invariant and Equivariant Signals. Both L Loj and ((()) are convolutions within edge signals of the same modality, i.c. orientation-equuivriann equivariant or orientation-invarian signals. We enable information exchange between both by combining the boundary operators of Equations (8) and (9). First, we define Laplacian to transform rrientation-cquivrrinn
edge signals into irrientationinvariii equ- inv = (B mz Big This operator allows modeling
edge signals as (
eeeentationniivarinnt
outputs even if no orientation- invariant inputs are available. Since it is constructed from direction-
aware boundary operators it, too, induces complex phase shifts to encode directed edges.
Analogously, fusion operator that transforms orientation-invariann equivariant edge signals can be constructed as = equ invariant inputs into an cennntattiooeqqvariiaine edge signal. Again, directed edges are encoded with complex phase shifts. The outputs of both operators satisfy joint orientation equivariance/invaiie (Definitions 4.1 and 4.2p for undirected edges for which no complex phase shift is applied.
edge signals into orientation- (4) =( (Bequ) ) Bin- It transforms orientation-
Lemma 4.3. The Invariant and Equivariant Fusion Magentic Edge Laplacians implies and jointly orientation-equivariaa mapping g g(Xequ- Xim, G. =
jointly
(B (4) H ((BB)) equ) and s(X X im X 9) (B H B (4) in Xiw) respectively.
Since EIGN is composed of Laplacians that are jointly orientation orientation-equiiariaaaee iquivariant/-nvvriiat mappings respectively, it satisfies the desiderata for edge-level GNNs stated in Section 4.1 The proof follows
Lemmata 4.1 to 4.3 and is supplied in Appendix
Theorem 4.1. (i) egui is jointly orientation -qqiivarineuvvviiint mapping. (ii) (())
in is a jointly orientation- invariant mapping. (ii) Both H and H Hi"l) are permutation equivariant mappings.
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Table 2: Modelling capabilities of all archi- tectures. "_" denotes that the modality is modeled without satisfying orientation invari- ance/equivariance.
Table 3: Datasets in terms of edge direction and input/target feature modality.
Edges
Features
Dataset
Targets
Model
Edges
Features
Targets
Dir.
Undir.
Equ.
Inv.
Dir.
Undir.
Equ.
Inv.
Equu.
Inv.
RW Comp
Inv.
MLP
X
LD Cycles
Inv.
LINEGRAPH
X
Tri-Flow
Equ.
HODGEGNN
Anaheim
4
(
Equ.
HODGE+INV
HODGE+DIR
X
x
X
Barcclona
(
Equ.
LINU-MAGNET
Chicago
(
Equ.
Dir-GNN
Winnipeg
(
Equ.
EIGN*
Circuits
v
Equ.
5 EXPERIMENTS
We showcase the efficacy of EIGN on three synthetic problems and three tasks five real-world datasets. We devise the synthetic tasks to require proper handling of directionality as well as orientation-equiveaa -qquivaiion-quivaiiiantt and infentationninvariinn
information. We categorize all datasets in terms of whether the graphs are (partially) directed and if features and/or targets are arientation-equivarrant or orientation-invariant in Table (see details in Appendix D
Baselines. We compare EIGN to five baselines: (i) An MLP. (ii) LINEGRAPHGNN, node- level spectral GNN applied to the line graph of the problem (Bandyopadhyay
al., 2020)). (iii) HODGEGNN (Roddenberry et al. 2021: Park et al. 2023). based on the Edge Laplacian Legu (see Section 2 It assumes inputs to be orientation-equiiarinn and edges to be undirected. (iv) HODGE+INY a variant of HODGE+INV that models orientation-invarian features as orientation- cquivariant. (v) HODGE+DIR, variant of HODGEGNN that breaks orientation equivariance, and, thus, treats all edges directed. We also adapt two node-level GNNs for directed graphs (see Appendix D.3): (vi) MAGNE, graph transformer similar to Geisler et al. (2023) and (vii) DIR-GNN- edge-level GNN that represents directed edges through separate message-passiine operations (Rosii et al. 2023) Battaglia et al. 2018). As depicted in Table 2 EIGN is applicable in every possible scenario. While Dir-GNN can be adapted to all modalities using our proposed Laplacians, this comes with significant drawbacks (see Appendix D.3). To mitigate side-effects from the cyclical nature of complex phase shifts, choose q 1/m (see Appendix
5.1 SYNTHETIC TASKS
Random Walk Completion (RW Comp). The first synthetic problem we devise is completing random walks directed graph. We input the (orientation-invariann) transition probabili- ties and subset of the edges that were traversed. The task is to classify if an edge is part of the ran- dom walk (details in Appendix D.I. Therefore, this problem tests if model can distinguish different edge directions.
Table 4: Average performance of different models on synthetic tasks (best and runner-up), EIGN is particularly effective on the hard Tri-Flow problem with interactions between orientation-equivarti and deeetatton-invariii inputs.
RW Comp
LD Cycles
Tri-Flow
Model
ACC-BOC11
AUC-BOCm
RMSB/4)
MLP
0.720
0.500
0.547
Longest Directed Cycle Prediction (LD Cy- cles). We increase the problem difficulty by including both directed and undirected edges: We generate different graphs that contain cycles entirely composed of each respective edge type. The task is to predict which edges belong to the largest cycle of only directed edges (details in Appendix D.I.. Since there are no additional inputs, this task tests a model's ' ability to distinguish directed and undirected edges.
LINEGRAPH
0.758
0.683
0.497
HODGEGNN
0.500
0.500
0.458
HODGE+INV
0.811
(1.754
0.293
HODGE+DIR
0.819
0.799
0.293
LINE-MAGNET
0.729
0.502
0.542
Dir-GNN
0.757
0.768
0.453
EIGN
0.864
0.996
0.022
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Published as a conference paper at ICLR 2025
Triangle Flow Orientation (Tri-Flow). This is the most challenging synthetic task as it requires combining orientation orienatton-eqqiivariant equivariant and orientation-ine -nvvarian features a partially directed graph. We create multiple graphs containing disjoint triangles and provide an inventation-equivartt input. The task is to reorient the flow such that for triangles satisfying certain conditions there is no iientation-invariinn attribute assigned to
flow
excess flux. These constraints are based on direction and every edge, thus introducing relationship between direction and both edge feature modalities.
Results. T shows that on all three synthetic tasks, EIGN is the superior architecture. Its high efficacy on the RW Comp and LD Cycles problems confirm its merits in modeling edge direction: It distinguishes between edges of different directions as well as undirected edges, As it satisfies our desiderata, it can model the relationship between orientation orientation-equivarian and orientation-invant features and relate them to edge direction. This is reflected in its impressive performance on the Tri-Flow task while the direction-aware HODGEGNN as well as HODGE+DITS which also combines both cenitaatioooeqivariiiin and -invariant inputs, struggle to achieve comparable results.
5.2 REAL-WORLD PROBLEMS
Datasets. We also apply EIGN to real-world traffic networks and electrical circuits. (i) We select four transportation networks (Stabler et al. 2016) (Ana- heim, Barcelona, Chicago, Winnipeg) where the targets are the best-known flow solutions in terms of lowest Average Excess Cost (Boyce al. 2004). While these datasets only con- tain ihentation-invvriiant features, two tasks use information from the inrentationequiivali targets as additional inputs. (ii) We generate different electrical circuits consisting of resistances, diodes, and one power outlet with given voltage. We then simulate currents and voltages using LTSpice (Asadi 2022). Each circuit has orientation-cqqiivriintt (voltage at the source) and orientation-invariant features (resistance, component type), while the target current also orientation -quivvarian (details in Appendix D.I)
Table 5: Average RMSE (L) of different models for the simulation task on real-world datasets (best and runner-up). EIGN achieves substantial improvements over all baselines.
Model
Anaheim
Barcelona
Chicago
Winnipeg
Circuits
MLP
0.105
0.149
0.109
0.167
1.030
LINEGRAPH
0.101
0.149
0.109
0.164
1.037
HODGEGNN
0.280
0.170
0.107
0.173
1.016
HODGE+INY
0.098
0.146
0.108
0.151
0.828
HODGE+DIR
0.091
0.144
0.109
0.132
0.760
LINE-MAGNET
0.119
0.151
0.105
0.170
1.027
Dir-GNN
0.278
0.170
0.106
0.173
1.029
EIGN
0.090
0.133
0.078
0.101
0.696
Tasks. For each dataset, study three problems of increasing difficulty for each of which the edge flow needs to be predicted: (i) Denoising: We noise the target flow and provide as an (additional) equivariant input. (ii) Interpolation: We supply the tar- get flow on subset of edges as an (ad-
Table 6: Ablation of different components of EIGN syn- thetic tasks and simulation on real data (best and runner-up). We omit (i) direction-awaremt (q 0), (ii) the fusion op- eration of Equations and (5), (iii) the fusion operators
fequ-inv and Jin--equ- and (iv) the node embedding h.
EIGN
EIGN
BIGN
BIGN
Dataset
EIGN
wes Diratiion
No
No Norrniz-Com
No
ditional) equivariant input. (iii) Sim-
RW Comp
0.762
0.853
0.845
0.862
0.864
ulation: The target flow is to be pre- dicted without any additional inputs.
LD Cycles
0.689
0.987
0.926
0.996
0.996
Tri-Flow 0.362
0.088
0.074
0.034
0.022
Anaheim
0.289
0.097
0.283
0.099
0.090
We report the RMSE (L) of all models for the simulation task in aad defer results for the easier denoising and interpolation problems to Appendix E EIGN substantially improves over all baselines in interpolation. On the challenging simulation task, EIGN shows the
Results.
Barcelona
0.172
0.139
0.177
0.163
0.133
Chicago
0.079
0.093
0.110
0.082
0.078
Winnipeg
0.132
0.170
0.175
0.138
0.101
Circuits
0.957
0.974
0.727
0.707
0.696
merit of treating orientation-equiiarian and orientation-invariant signals in principled way.

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Published as a conference paper at ICLR 2025
5.3 ABLATIONS
Fusion Operators. In Section 4.3 we introduce Laplacian operators to fuse orientation-eequivriaan and orientation-invariant features by transforming signal from one modality into the other. This is particularly useful as it allows EIGN to output prientation-equivan orientation-equivariann imputs and vice versa. Omitting them and only relying on Equations (4) and for interaction between the two modalities is problematic for equivariant signals:
signals in the absence of
The only choice for an equivariant input signal in the absence of such features Intuitively, this is because the sign of an orientation. However, the "absence" of an input has no inherent direction. Setting the equivariant input to zero while at the same time omitting the Sinv-requ term in Equation
cettattioonnqivaiiiain input indicates its direction relative to the chosen
Zep for undirected edges (see Appendix (A.4), and, consequently, H 2 results embeddings for Equation (4) as well. Therefore, omitting the Laplacian fusion operation from Equation 2 prohibits the model from learning any con-zero srientatio-equiivrian output in the absence of orientation orientation-equivaraan equivariiut inputs. Table 6 confirms that omitting these fusion operators results in considerably worse performance in simulation tasks where no orientation- -quutviinn-uivaiiintt features are available.
At the same time, omitting the fusion operation of Equations 4 and (5) and only relying the Laplacian fusion operators fequ-sinvs Jinv-vequ restricts the information exchange between orientation- equivariant and orientation-invariant features only exchanging aggregated information through the convolutions Sequ-sime and finv-requ- Therefore, it is more difficult to learn direct interactions between the signal modalities of the same edge. Both on Tri-Flow and real data, where these interactions are relevant, omitting the operators of Equations 44) and (5) leads to worse performance, see Table 6
Direction-Awareness. We also study the im- pact of using complex-valuee valued Laplace operators to make EIGN direction-avee aware by ablating a direction-
0.10
Oriented Flow
63.3%
99.0%
0.05
agnostic version of EIGN in Tabble 6 (q = O).
First, the RW Comp and LD Cycles tasks that test firretiionaawarene benefit greatly from encoding
0.00
w/o Direction
direction with complex phase shifts. Second, many
-0.05
w/ Direction
real-world datasets contain directed edges that con- strain flows. Accurately modeling the system requires distinguishing between directed and undirected edges which is reflected in the superior performance of the direction-aware EIGN. Finally, Figure 5 shows that the direction aware EIGN yields more plausible pre- dictions than the perection-annostte variant for one- way roads. It learns the problem constraints without encoding them into the architecture explicitly.
-0.10
One- One-Way
All Streets
Figure 5: Distribution of simulated traffic flow on Anaheim for EIGN and direction- agnostic variant. Direction-awarenes. makes predictions that obey flow constraints of one- way streets more likely.
LIMITATIONS
We address lack of benchmarks for edge-level tasks by designing three synthetic tasks and propose novel task involving real-world electric circuits. Even though EIGN is grounded in Algebraic Topology, we limit our study to edge-level problems and do not model higher-order structures. W use Magnetic Laplacians to encode edge direction but other frameworks satisfying our formal desiderata may be effective as well in practice. Lastly, since our Laplacians are normal matrices, they enable global propagation similar to Geisler al. (2024) to extend upon the local scheme used in our work.
7 CONCLUSION
We propose EIGN, framework for edge-level problems that allows modeling crientation-qquivant and
features and can encode edge direction. It relies on novel graph shift operators that provably preserve novel notions of joint orientatiion-qquivrranne and -invariance for undirected edges while also being sensitive to flipping directed edges. On a benchmark of synthetic tasks and real-world flow modeling problems from two domains, we show the high efficacy of the inductive biases encoded by EIGN.
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Published as a conference paper at ICLR 2025
ETHICS STATEMENT
We acknowledge that we thoroughly read and adhere to the code of ethics. Since our work can be categorized as foundational research, do not see any immediate implications beyond the risk of advancing Machine Learning, in general. We, nonetheless, encourage readers and practitioners building on our work keep in mind the potential risks the context of reliability, interpretabilltyy fairness, and privacy for which we do explicitly account.
REPRODUCABILITY STATEMENT
We detail assumptions and proofs for all claims made in our work clearly in Appendix Furthermore, we describe in detail how the data and models are tuned in Appendix D Additionally, we provide our code and the optimal hyperparameter configurations we found in the supplementary material.
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A PROOFS
A. JOINT ORIENTATION EQUIVARIANCE AND JOINT ORIENTATION INVARIANCE
Proposition A.I. Let D, S be two direction-consisteri orientations of edges on G. Let Xequ E Cmxd
Then:
B (G, (G,D)A Bigg (G.
Proof. By assumption, for any E &D we have that Ö(eD) = D(eD) = DeD) Now consider the case eU E Ev. ((uu) = D(v)), then B (g. D) D) = B (9. D),ee for all v E V. Similarly, if ((v)) + ((v)). then [Begi (G, D)l B (9, D),,,oo for v E V.
Lemma 4.1. The Magnetic Equivariant Edge Laplacian implies jointly orientation-eqquivrrian mapping (XX Xim, tm, 9 D) H B X eqq)-
Proof. Let D, D be direccion-cooniiseet orientations of edges G. Let Xequ E Cmxxda and Xinv Cmxd Then:
1(AD,6 Xequ, Xinv, =
(G, (
D) Xaqu) (1)
(B (9, D) )
(
(g, D A Xeuu)
equ
(A " (B (4)
D)) ((B
equ ()
equ
= inv D)
Here, we used Proposition A. and the facts that A = Im by definition and A (A,,s) as it is diagonal real matrix. The proof holds for arbitrary node feature transformations
Proposition A.2. Let D, be two direction-coosisteent orrrrr/cooccssstteeooreennttttt of edges on G. Let X E Cax Then:
B (G.
( = ((() (G,
Proof. By assumption, for any eD E Ep we have that D(e)) = (ep)) = - and con-
sequentially, [B]]] (G, D) p.eD = [B("] (G, D) for all v e V. For eu E Eu, the claim follows directly from Equation
Lemma 4.2. The Magnetic Invariant Edge Laplacian implies jointly orientation-invariann mapping (XXegu Xin: D) (B(() (4) hh(
( Xim).
Proof. Let D. S be directiinn-consistent orientations of edges G. Let Xoq E Cm and
X inv Cmxdm Then:
g egn Xinv G, -B (9) (G. 5)) B (4) (G. 5) Xim)
Inv
(B (1) inv (g. D) Hh( inv (G, D)
(1)
g(Xequ- Xmm, G.
Here, we directly applied Proposition A.2 The proof holds for arbitrary node feature transformations h.
Lemma 4.3. The Invariant and Equivariant Fusion Magentic Edge Laplacians implies
jointly rrientationinvaaiann and jointly prientation-equiivrrian mapping g(Xegu X, G, D) (B()) h B(9)) xogu) and f(X X mm 9
( h1 B (4)
in Xmw) respectively.
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Proof. Let D, be two direction-consisseri orientations of edges g. Let Xequ E Cmx and Xinv @mxdw Then:
g(A ( Xiv, -D =B lov (G, (G,
equ (G, D,
(B I (G. (g. D) ( D.S Xeuu
(B (g, h (G, D Xqqu))
g(Xequ X i 9,
Here, used Propositions A.1 and A.2 and the fact that A s ف. Im-
Let D. S be two direction-ccosiiseen orrprio--ocosssseeetorrnnntttoo orientations of edges on G. Let Xequ E Coxda and Xin
Cmxdu Then:
S'A Xequ Xim, 9. (Big) (9, (T, N( 9 (G, (im)
(B (G, D D H h (1) (G, )) X
(A
(B G 5)) ( (9, D) Xin))
equ
-o, (Xequ: Xins 9, D)
Here, we used Propositions A.1 and A.2 and the fact that (A D as a diagonal real matrix.
We now prove that the composition of orientation-equuiarrani / invariant functions preserves ori- entation equivariance / invariance, which allows us to prove Theorem 4.1 by using Lemmata 4.1
to 4.3
Proposition A.3. Let fi and f2 be jointly orientation--equivarioe mappings according Def-
inition 4.1
Then fi Oegu f2 = fi fi(f((emp) Xim, 9, D), Xime D) also jointly orientation- equivariant.
Proof.
(f2 eegu f2)(A Xoqu X G. = =i X 9, Xinv 9,
12.2222Xaa =fi(A,,0/(Xeag X G, X xinv: 9 D) = =f(A
Ao.0fi (fo.oo((2(X.... (f2( X G, D) X inv: 9)
-Ao (fi Gequ (X 9)
Next we discuss the relationship between sign-equivariant and jointly coonaaation-eqiiariain map- pings.
Proposition A... Let a be a sign-equivariant mapping, i.e. o(-2.-1) -0 (x v) and fi and fz be jointly orientation-equivvrian mappings. Then f(X Xmw. 9,D) o[JiXX equs X G,D), S2l X Xim, G, D)) is jointly orientation-equivariin mapping as well.
Proof.
Xo = oo(i(A S Xequ Xin, G, X equ, Xino G, CD))) = =o Si (Xequs Xmv. 9, 5.2 X equ Xinv 9, 1)) A0,00 (X Xiv, 9, D), fz ( Xin, G,
= = (( Xim.gg. 9,9)
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Note that Proposition A.4 generalizes to sign equivariant functions with argument by defining o' (a = o (2x 0).
We can now prove that H as defined in Equation 2 is jointly equivariant mapping.
Lemma A.5. Heju as defined in Equation 2 is a jointly orientation-equivarii mapping. H (1) defined in Equation is a jointly orientation
bn as
iitetaatonn-mmarian invariant mapping.
Proof. The proof is done inductively. Therefore, assume Heq (1-1) be jointly orientation-
equivariant mapping and H to be a jointly orientation-invann invvriian mapping. For notational simplicitly, absorb the dependency Xequ inv. and D into the mapping itself: That is, for mapping F, we denote F( (xeoq mwv g = F and F(As, Xequ Xiv: G, G F.
We first show that to be a jointly iitentatioo-equivariann mapping.
2 equ
Jequ (Fequ
Ar equ
(ill)
+ + (H (H
W
inq- + (1111) inv
equ
inv
equ
equ
equu (fcp
(1-1)
(t
cqu + 1 mm (H
11!111) w ieq + A و
H
(1
W (1)
D.
equ
inv
oqu
oqu
Tequ (AD,5
{(1)
(1-1)
(t)
equ + A.S inv Ilr (H (1 will
+ A D.S
H (1-1) equ
equ
equ
lav
inq
equ
(1)
(I-1)
W/
equ-tequ + mn (H w
H (1-11
W (1)
equ
equ
inv
inv-
-A, (
Here, we first used the induction assumption, then applied the joint orientation equivariance property of Ligu and ( inv+ equ proven in Lemmata 4.1 and 4.3 and lastly used the fact that we assume degu to be sign-invariant function apply Proposition A.4
Similarily, we can show that (!)
be jointly orientation-equiiarinn mapping.
2000 an (Fim (H (HI!1-1)
(1-1)
((())
inv- f!ll Sequ-inn (H w(() equ-tin + HH H inv
any (1 H (1-1)
w(l) 1( -inu-inv (Ap.s ( equ H ((--))
inv
inv
i (f (H (1-1) w (1)
iqu + H H w! [10] sin (H W
inv
equ
Here, we first used the induction assumption, then applied the joint orientation invariance property of ( and L (9) inw
equ-+ inv proven in Lemmata 4.2 and 4.3
Next, we can show that ( an orientation erientation-equiivriaan mapping.
A (1)
equ
Jegu
equ
W
Euqu
!I)
Zin
(4)
w
F.equ.inv (
(1)
Jequ (A. D
(1)
w (1)
z
(
(1())
A . 2!
equ
oqqqu.oqu
inv
F.equ,inv
Dagu (
((())
(1)
(1)
W
+ A A Z
equ
oqqqu.qqu
inv
F Feequiin>
Ag, Tequ
(l) equ
(t())
(1)
(())
eeqqu.qqu
inv
F Fnn
A
(1) equ
We first use the previously proven joint orientation equivariance of ( and then the fact both Ap.s and @ are element-wise multiplications and, therefore, are associative operations. Lastly, we again make of Proposition A.4 for the sign-equivariant function Tequ-
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Lastly, we show that H is orientation-invariint mapping.
H
inv
ainy
inv
Finny inv
abs(
(1) w equ
Fiin, eqw.
+ 2(1)
inv
Finy
(2) inv
(
F,inv,im inv
abs( (bbs(As
D
oqu
w
Tiimv,equ/ + 2
W (t)
abs( Z
W Fii F,inv,equ + 2 inv
Juv
F,inv, inv
equ
H
Here, first use the previously proven joint orientation invariance of - Then, we notice that DD corresponds to element-wise multiplication with +:1 and is therefore canceled out by the element-wise absolute value function.
L
A.2 PERMUTATION EQUIVARIANCE
common requirement for GNNs is equivariance with respect to the ordering of input to enable generalization (Maron et al. 2018 Wu et al. (02I). While node-level GNNs are equivariant with respect to node permutations, we require edge-level permutation equivariance.
Definition A.1 (Permutation Equivariance). Let D be an orientation of edges on G and P E m be permutation matrix the edges. Let Dp p and Gp be its application to D and respectively. We say that mapping / satisfies permutation equivariance if for any orientation-cquuiarian signal X Xequ E Cmmxda and any orientation-iinvaiinnt signal X iny Cmxdd:
Pf(Xequ- X G, = f(P PX 9p.Dp)
We now proceed to prove that EIGN is a permutation equivariant mapping according to Definition A. Again, we first concern boundary operators and show results for the generalized, directed case and recover Equations and with O.
Proposition A.6. Lequ induces permutation equivariant mapping according to Definition A.
Proof. Let D be orientation of edges 9 and P be permutation matrix. Let Dp and Gp its application to D and G respectively.
f(P PXins, Gp, Dp) = (B (Gp,Dp)) (B Pp)
equ
(B(g) (9, D) h (G, pd PX.
P(Blg) (g)
equ H G)
P Xinv 9)
Proposition A.7. [!() induces a permutation equivariant mapping according to Definition A.I
Proof. Let D be an orientation of edges on G and P be permutation matrix. Let D and Gp its application to O and & respectively.
PXiin PXinv 9p, Dp) (B(9) (Gp, Dp)) hh(B"h(BB
im PP)
(B(g) (G, D) H (G, PX
P(B!9) (G ( ( (G,
Pg(Ximu Xinv, G, D)
Proposition A.8. I [())
equ-+ induces a permutation equivariant mapping according Definition A.
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Proof. Let D be orientation of edges 9 and P be permutation matrix. Let Dp and Gp its application to D and & respectively.
f(PX equ PXimv 9p, Dp) = ( (B.
(B (g, D) () (C PX
P( lav (g, H| X
Pf X 9D
Proposition A.9. L in equ induces permutation equivariant mapping according to Definition A.J
Proof. Let D be orientation of edges 9 and P be permutation matrix. Let D P and Gp its application to D and G respectively.
f( f(P PXmv, Dp) (B (Gp, (G), hhB (Gp,D))
(B(g) () ( (G,D)P* PXim)
P( (9, D) H M( inv (g, Dix)n))
= Pf(Xxqu Xim, G. D)
Lemma A.10. H and H defined in Equation are permutation equivariant mappings.
Proof. The proof is done inductively. Therefore, assume (l-1) and H (1-1) to be a permutation equivariant mappings. For simplicitly, we, again, absorb the dependency on X Xinv and D into the mapping itself: That is, for a mapping F, we denote F F(X Xinv GD F and (q PX Op)
We first show that Zuu to be permutation equivariant mapping.
( equ
Jequ (equ(Sequ (1-1)
(!)
+ iin-bequ (
+ (1-11) ())
equ +
equ
inv
in rqqu
equ
equ
T (S (HH
(1
(!)
equ + (1()) -equ (PH
w(()
PH
(1-
w(() equ
equ
my
inv tequ
equ
Jogu (PquPP (()))) (H
(1-1)
W
tqqu + P
(1)
p egu (H (1-1)
((()
PH (1-1)) w
equ
inv
inv -cqu
equ
equ
PPo
(()) H
(1)
+ 2()) H (1-1)
+
H wwll) equi
equ
eqqu
toqu
lav
iny inq
equ
P!!!)
equ
Here, we first used the induction assumption, then applied the permutation equivariance property of (9)
L and _ equ proven Propositions and A.9 and lastly used the fact that we assume doqu to be element-wise function which, hence, commutes with permutations.
Similarily, we can show that
Z to be a permutation equivariant mapping.
Z (1) inx own ( (E
(E
(1-1) (! inv
p(1)
Sequ-vim (1-1)
(1)
H (1 w(l)) (1)
equ
equ-xin
Own (f ( (1-
(!
inv-iin + f111) fequ-rin (PH
(1-1) (!
PH (1- will)
equ
equ-+inv
Dim (PS())
Hiv
(1-1) (l(l) inv-sinv
+ +P p P
inm (H (1-1) (I
equ equ-sinv
PH (H!l1-1) w!l))
Poine
(f(()) (H
(1-1)
()())
inv-siny + f()) fou (H((11)) ( (1) iqu
+ .. (1-1) w(())
inv
-PZ())
Here, we first used the induction assumption, then applied the permutation equivariance property of (g) and - equ proven in Propositions A.7 and A.8 and lastly used the fact that assume Tequ be element-wise function which, hence, commutes with permutations.
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Next, can show that H (1)
an permutation equivariant mapping.
H (0)
W
(1)
W(O
Z (1)
equi
eqqu
E.oqu.oqu
ijequ,inv
equu ((P
equ
W (1)
F,qqu.qu
(PZ ins
(1)
W
F,equ,inv) + P Z
(1)
Jequl (P( (Z
W
(1)
Z ! wil
F,qqu,inv) PZ
Foqqu.oqu
Po (Z()
W
7 W inqqu.ivv
equ
E,qqueequ
PH (U)
We first use the previously proven joint permutation equivariance of Zepu and then the fact both cqu and @ are element-wist operations and, therefore, commute with permutations.
Lastly, we can show that H (1)
is an orientation invarrann mapping.
H 0 Jinv
2 abs(
! W equ
Fiiny + (1)
inv
inv
Finviinv
inv
Jinv (( ( mm
w
+ (abs(
(!) w
Finnv.equ) )) + P PZ
(1)
Fiin, inv
iun ((P Z tn
W
(1)
3 Pabs( Z ( W Fiin cqw.coqu/ + PZ
Fiinv inv
Onnv (P( ( Lnv
W (1)
abs( Z (1)
Fiin equ )) + PZ
Finnv inv
Poin / W (t)
abs ( W (1)
Finnvequ) + k
F,inx,im inv
PH! (1)
Here, we first use the previously proven joint permutation equivariance of / and then the fact dines abs and @ are element-wise operations and, therefore, commute with permutations.
A.3 THE MAIN RESULT
We can now plug Lemmas
A.5 and A.10 together and prove Theorem 4.1
Theorem 4.1. (i) Hog is jointly printattniiinnttiineeeeivvviai
(L)
mapping, (ii) H (L)
in is a jointly orientation-
invariant mapping. (iti) Both Higl and H (L) are permutation equivariant mappings.
im
Proof. The proof of (i) and (ii) directly given by Lemma A.5 The proof of (iii) is directly given by Lemma A.10
A.4 INTER-MODALITY CONVOLUTIONS TO LEARN ORIENTATION-EQUVVAIINYY REPRESENTATIONS FROM ORIENTATION INVARIANT REPRESENTATIONS
Here, we show that not using the inter-modality convolutions fiv results in the model outputting for the orientation-equivariiie
edge representations in case there are orientation-equivariall inputs
available (X (Xem = 0). Therefore, omitting these convolutions prevents the model from predicting any non-zero orientation-cquivriinn output for undirected edges if orientation-equivarii inputs unavailable.
Lemma A.11. Computing ((I) as per Equation 2) but without using f(0) Hegu
Jim- implies that if
O, then also Z Eg 0 for undirected edges.
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Proof.
(Z (Zqu) Eu
equu ( (fe())
(H
W
equ + H -
W equ Ee
equ
Jeuu
(0)
W
equ + OW
equ 5o
Tequl
Tequ
(B
(B4))
equ
Blo)
equ
W
equ equ eu
Jequ
((B h(0) W
equ Eu
equ
Jquu (BB (g)
h(O) W
EL
rqu
Jequ (B
H W
equ
equ equ-requ>
Jequi OW
[1)
equ equ
Tequ (0))
Here, have used that the node feature mapping h will get zeros input and therefore produce the same constant embedding с for all nodes. Along undirected edges, the flow that is induced by (B()))" " corresponds to the differences between its endpoints, which consequently will be zero well. Note that similarly each directed edge will be assigned the representation (exp(i=q) - exp(-irq)c. While being non-zero, this layer architecture is only able to produce constant feature for both directed and undirected edges which heavily limits its expressivity. This is also well reflected in Table 15
B CONSIDERATIONS FOR MODELLING ORIENTATION EQUIVARIANT AND ORIENTATION INVARIANT SIGNALS
Here, we provide additional considerations for how orientation equivariant and invariant signals should be modeled on graphs that contain both directed and undirected edges. This discussion supplements Figure 2 We begin by highlighting the limitations of previous architectures that treat all edge signals as fully orientation invariant or orientation equivariant respectively.
Orientation Invariant Architectures. Most architectures that are not grounded in Algebraic Topology treat all inputs and targets as orientation invariant. That is, the input features and the output features are assumed to not have inherent direction. This is, however, limiting, as in many real-world applications (traffic, electrical engineering, water flow networks, etc.) accounting for the direction of a signal is erucial. Therefore, these architectures not applicable to these settings.
Orientation Equivariant Architectures. Note that it is impossible to represent the direction of signal through scalar value without defining reference orientation for the associated edge: Either the signal direction matches the reference orientation or it does not. For example, topological methods use the sign of signal indicate if its direction matches the reference orientation. In the case of directed edges, there is a clear choice for defining this reference. For undirected edges, however, an arbitrary reference must be fixed which is typically done in topological approaches. Since the reference orientation is used only to represent direction in signals and can be seen representational basis should not affect the predictions of a model. This leads to an equivariance condition for these architectures: If the reference orientation changes, inputs, and outputs of a model should be represented with respect to this new orientation but not change apart from that. Topological models typically satisfy this notion (Roddenberry & Segarra 2019 Roddenberry al. 2021).
They, however, treat all inputs and outputs as signals that have inherent direction, i.e. represent them with respect to orientation. Therefore, they can not model information that does not come with inherent direction like the number of lanes a street, resistance, or pipe diameter. Enforcing orientation equivariance for all inputs and outputs of model does not apply to settings where also orientation invariant signals that are irrespective a reference orientation are available. In particular, these models require representing input and output signals with respect orientation and it is
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edge. If the two directions are aligned, these phase shifts cancel out (كككك)) ). Otherwise, they add to a total phase shift of 2xq. One can also see that which direction an edge has (relative to the Published as a conference paper at ICLR 2025
ill-defined to define how an orientation invariant signal should be represented with respect reference orientation, e.g. through its sign.
Edge Direction. In addition to the inherent direction to the signal of an edge, the edge itself can be directed. In practice, the direction of an edge implies constraints on the solutions admissible to a problem: One-way streets in traffic networks prohibit certain traffic flow direction, and diodes and valves play a similar role in electrical circuits or pipe networks. However, orientation equivariant models are insensitive to the (arbitrarily chosen) orientation of edge edge. Therefore, even if the orientation of directed edges is fixed to coincide with their direction, this information can not be utilized by a fully orientation equivariant architecture. Figure exxmplifies this issue: Figures 2a and 2b] differ only in the orientation chosen to represent the scenario and, thus, any orientation equivariant model will output the same value for both. In Figure 2a] however, the flow orientation defined opposite to the direction of directed edge. In traffic applications, this drastically affects the scenario: one case, one-way constraints are violated while in the other they are not.
This motivates our adopted notions of orientation equivariance and invariance: The concept of orientation is needed only for undirected edges. Directed edges already provide a reference orientation for representing signals through their direction. Furthermore, models should not be equivariant with respect to the direction of directed edges. In our work, we represent equivariant signals of directed edges with respect to their inherent direction and use an arbitrary orientation only for undirected edges. Consequently, the notions of orientation equivariance and orientation invariance only need to hold for undirected edges.
C THE LAPLACE OPERATORS OF EIGN
Here, we explicitly list the definitions of all Laplace operators and briefly describe some intuition. This is supplementary to the description for the Equivariant Edge Laplacian Leq in Section 4.3 Intuitively speaking, each Laplacian is a composition of two boundary operators. Therefore, the message passing from an adjacent edge e' & to a target edge e can be understood as a two-step procedure: 1. The signal of e' is expressed relative to the node both e and e' are incident to. This corresponds the right-hand term in the construction of the Laplacian. Expressing the signal on the node level makes it invariant to the orientation (if the suitable boundary operator was used, i.e. the
boundary for an iiienatiionequivaann signal of e', currently expressed in reference to the shared node (ingoing versus outgoing), is then expressed relative to e. This corresponds to the left-hand term in the construction of the Laplacian from composing two boundaries. Depending on which boundary operator is used, this signal will be ozientation-equivarian or grientationinvvriiant.
signal and vice versa). 2. The
The values in Table assume values based on if (i) and e' align (i.e. are not consecutive). If they align, the first value is assumed, if they misalign, the second. (ii) The direction of e relative to the node. For the direction depicted in Table 7 the first value is assumed. Flipping the orientation / direction of both edges gives the second value.
Based on this intuition, we interpret how the Laplacians of EIGN materialize in Table 7 We can see that only the Magnetic Laplacians are direction-aware and Lequ and Liny do not depend on the direction. Furthermore, messages between two directed edges experience phase shifts twice: Once when being emitted from the directed source edge and once when being aggregated by the target
shared incident node) is encoded in the sign (i.e. direction) of the complex phase shift for both (g) and L (4) lav The key difference between the orientation-equiveee equivariant gg) and the ioannaation-ivvaiinn L inv is that the former also re-orients signals depending on if the edges align, similar to Loqu The Fusion Laplacians L (9) intuitively as easily. equ-> imv and L (99 mix both modalities and therefore can not be categorized
Since all Laplacians can be constructed without materializing any boundary operator by following Table our approach scales in the number of edge pairs connected by a shared node. Put differently, EIGN can be seen as node-level GNN on the line graph of the problem with sparse convolution
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Table 7: Realization of different Laplacian operators for directed edges - and undirected (oriented) edges =. Equivariant operators encode if two edge orientatiooss/direciion align, i.e. whether they are non-consecutive through the sign of the real part. The sign of the complex phase determines the direction relative to the reference node both edges are incident to, i.e. whether they are in- or outgoing. The relative orientation (alignment) of two edges is ignored by irientationninvariann operators. We omit the cases for which the direecion//rientaiionn of both and e' are flipped (this realizes a different sign in the complex phase).
Adjacency
Lequle.e'
Dquuee.'
L le.e'
[T eap-+ inle.e
[L (o) aqule.e'
e e'
2
2
2
2
2
2
exp( 2mgg)
exp(+2
# exp(++2rig)
+ expex=xt (
1
1
=1
+1
- Fexp(+
+ exp(+mig)
exp(+rig)
exp(#Trig)
exp(#mmg)
i cxp(#mig)
+ expp
+ exp(
1
#1
operator that has non-zero entries only when two of its nodes (i.e. edges) are adjacent. EIGN, therefore, has the same runtime complexity all line-graph-like methods.
D EXPERIMENTAL DETAILS
D.I DATASETS
Here, we detail statistics for datasets and the exact generation process for synthetic data. An overview of the statistics of each dataset is given in Table
Table 8: Average performance of models synthetic tasks (best and runner-up) with 95% confi- dence intervals of the mean.
#Edgges
#Feeaturss
Dataset
#Graphs
#Nodes
Dir.
Total
Equ.
Inv.
RW Comp
1000
50
161-249
161-249
0
2
LD Cycles
1000
12-16
16-31
25-33
0
1
Tri- Fri-Flow
100
300
245-282
400
3
Anahcim
416
354
634
Bareelona
930
1074
1798
Chicago
933
0)
1475
0
10
Winnepeg
1
1040
354
1595
0
8
Circuits
591
8-12
0-7
12-20
1
4
RW Comp. In this inductive setting, we generate 1, 000 Erdös- Rényi graphs Erdos et al. (1960) with n 50 nodes and choose the edge probability such that the expected number of edges 200. All edges are directed. Each graph is assigned transition probability sampled uniformly from [0. 1]. For each graph, we sample a random walk starting from randomly selected node (uniform probabilities) up to length 100. We then randomly select 20% of the sampled transitions and provide them as orientation ireetatioo-invariant features as one-hot encoding and the transition probabilities. The task is to predict if node is part of the 80% nodes in the random walk that are not provided as input, i.c. binary classification problem. We use a 70/ 10/20 split to assign graphs to train, validation, and test sets respectively.
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LD Cycles. In this inductive setting, we generate 1, 000 graphs and sample cycle size uniformly from [6.8]. We then generate two cycles, each consisting of с nodes: The first one contains only directed and consecutive edges. The second one is similar but contains one undirected edge. To each cycle, we then add additional e random edges between its nodes, each of which is directed with probability of 25%. We ensure that second consecutive, purely directed cycle of length is manually formed in this process. Lastly, we connect both cycles with an edge that, too, is directed with probability of 25%. Therefore, each graph can be seen as consisting of two components: One contains one consecutive, purely directed cycle of length c, while the other contains non-purely directed, consecutive cycle of the same length. The task is to classify edges as being part of the longest purely-directed cycle or not. We provide the constant 1 vector as a surrogate irientation-invarian input, and, therefore, this task is purely topological. Models need to understand the concept of edge direction to perform well in this setting. Again, we use 70/10 10/20 split for the graphs.
Tri-Flow. In this inductive setting.
we construct 100 graphs with n = 300 nodes and m = 400 edges. We want to induce a relationship between orientation-invare
and orientation-equivarinn features. We assign each edge one of = 3 colors as an iirentationninvariinl feature. We then plant t t= 100 disjoint triangles in the graph. Each triangle is constructed according to one of three patterns: (i) Triangles with at least one directed edge, all edges of the same color. (ii) Triangles with edges of the same color, but least two edges directed and ensure that the triangle can not be consecutive. (iii) Triangles with edges of different colors. We generate 50% of triangles with mechanism (i), and 25% with mechanisms (ii) and (iii) respectively. We then add 100 random edges connect the triangles and ensure that no new triangles are formed. For the task of this problem, we introduce an ireetttion-eqiivaiaan flow at each edge and the task to re-orient these flow labels such that flows within triangles are closed. To that end, in the solution to this task, flow along directed edges must follow the edge direction while undirected edges permit flow both directions. We introduce a dependency to orientation-invarinn the same color. That is, the target for triangles of type (i) is to re-orient the flow correctly such that is closed within the monochromatic triangle. Triangles of type (ii) contain two directed edges that prevent consecutiveness and therefore, the target is O as no flow is permitted through such triangles. The same holds for triangles of type (iii) that prohibit flow their edges are not monochromatic. We then add 100 edges, while ensuring no new triangles of type (i) are formed, that, consequentially, also have target of 0. This task, therefore, involves understanding and relating the concepts of direction, orientation, color (which is orientation-invariand and flow (which is arientation-equivarian)) It probes all possible modalities an edge-level GNN can be exposed to. We again split each of the graphs into train, validation, and test sets using 10/10/20 split.
features by requiring that flow can only pass through triangles of
Traffic Datasets. We collect the Anaheim. Barcelona. Chicago, and Winnipeg datasets from the TNTP transportation network repository (Stabler et al., 2016)) This project studies the Traffic Assignment Problem (Patriksson] 2015) and provides flows that correspond to the best-known solutions in terms of lowest Average Excess Cost (Boyce et al.] 2004). We select these graphs based on that they are decently sized, provide both orientation-invarian features, and contain directed edges. Nodes correspond to intersections while edges represent links, e.g. streets. We use the following inputs (if available): (i) capacity, (ii) length, (iii) free flow time (i.e. travel
as orientation-invarin time with no congestion), (iv) B factor and power, which are calibration parameters for the Traffic Assignment Problem, (v) there is a toll on the link, and (vi) the link type, e.g. highway. (vii) if the edge corresponds to excess flow on source / sink node The optimal flow solution is computed with respect to pre-defined demand for given source-target node pairs. To represent this as edge features, we identify all edges incident to edges that correspond to sources or sinks in terms of the transportation problem, add them to the training set and identify these edges with an invariant feature. By keeping these edges in the training set, the model can infer the problem constraints and predict flow on all edges that are not incident to sources or targets. The problem topology is preprocessed as follows: The input data is entirely directed, and, therefore, we aggregate edges (u, v) and (u, u) into one undirected edge if both are present. The flow labels are combined using subtraction (i.e. we subtract the from the edge direction that used as a reference orientation to represent the flow). We then normalize the orientation invvarinn features to follow a standard normal and normalize the target flows to [0, 1].
Electrical Circuits. We generate random topologies with the following procedure: First, cycle of length Canit = 3 is generated. Then, we iteratively attach new nodes v to the graph by selecting random source and sink nodes s, t and adding the edges (s, r) and t). We then randomly assign different
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component types to each edge: (i) exactly one power outlet, (ii) resistors with resistance uniformly sampled from [10052, 10, 0002], (iii) diodes with saturation current 18. 18.8.A, parasitic resistance 0f2, reverse breakdown voltage 0.5μV, zero-bias junction capacitance 30F, a linearly graded junction, emission coefficient of 2.0 and transit time of Os. Diodes only permit current to flow in one direction. The ratio of resistors and diodes is 80/20. We then use LTSSpice (Asadi 2022) simulate the currents at each edge and voltages at each node until a steady state is reached. We use the component type and resistance at resistors as orientation-invariint features and normalize the latter to follow a standard normal distribution. The voltage along the power outlet is the only orientation -equiiiioinn input signal, and we assign 0 to all the other edges. For each graph we individually normalize the currents: First, we divide by the voltage at the power outlet. We then also normalize the currents again by dividing by the standard deviation of all flows on all graphs, i.e. a constant.
D.2 TASKS
Denoising. The first task is denoising the flow signal of traffic data and circuits. To that end, for each graph, we compute the standard deviation By all flows and then add noise sampled uniformly from (-0y, Tyl to all edges. The task is to recover the original signal, which we measure with the Root Mean Squared Error (RMSE) (J). Since the additional urientationiivvariant input, the noisy flow provides a strong signal for the direction of the target, it is, arguably, considered to be the casiest of the three tasks.
The second task is predicting the traffic flow current from only a few ground-truth values. To that end, for each graph, we provide 10% of the true flow labels as an additional orientation- equivariant input and set the rest to O. The task is predict the orientation-equivarian for which flow was not provided. Again, we use RMSE as the main evaluation metric. Similar to the denoising task, the auxiliary orientation-invariant input signal can provide information about the true direction of the orientation-equiva -qquivarinnn target. Since this information is now only available at some edges, we consider the interpolation task to be harder.
Interpolation.
flow at edges
The third task is predicting the orientation orientation-cquuvaaiine equivariant target (traffic flow / electrical current) from just the topology and available orientation orientation-equivariian equivariant and orientation-invann invariant features. This task can be seen as learning to simulate the traffic flow / electrical current. An effective model has to learn the underlying physical processes the data is derived from. Since we do not provide any input regarding the direction of the orientation-equivarian hardest problem among the three. Again, we can measure the performance of model using RMSE but also report the Mean Absolute Error (MAE) (L) and R² value (1).
Simulation.
target, we consider this task to be the
D.3 MODEL ARCHITECTURE
MLP. We use a standard MLP with ReLU activations. We concatenate both orientation-equivarre and intentation-invariae inputs model input. This, effectively, mistreats dientation-eqqiiarian input signals as incentationinvariinnn The same is true for the iventation-equivarinn model out- put/target.
LINEGRAPH. Similar to the MLP, we concatenate orientation-equivarian and iitentation-ivaaiiinn inputs, thereby mistreating the equivariant modality, and successively apply convolutions with the Line Graph Laplacian LLINEGRAMI DLANEGRAPH AlinegrapHi:
ALINEGRAPule.. = {
if + e' and e, el adjacent otherwise
(10)
Here, DLINEGRAPH is diagonal matrix of edge degrees, i.e. [DLINEGRAPH]C.Cc Lel [ALNEGRAPH]]... A LINEGRAPI convolution is then realized as:
H (1) = a a (1)) + bD))
(1))
Here, is the ReLU non-linearity, and W and (()) are the trainable model parameters.
HODGEGNN. We use the ODDGGGNNNrchhtetttr of Roddenberry & Segarra (019), which is a (jointly) orientation-eqquivriaan architecture. It, however, not model orientation-invarie inputs,
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as it treats all input signals as equivariant. We therefore do not input this modality at all. Each layer uses the Equivariant Edge Laplacian Lequ as a convolution operator.
H = Deuu (L (1-1) w()).
(12)
Here, Jequ a sign-equivariant activation function, which we initialize as tanh. To not break (joint) orientation equivariance, we can not use a bias term.
HODGE+INY. We use the same architecture for HODGEGNN, but also concatenate the orientation- invariant inputs to the orientation-equii equivariant inputs. This incorrectly models arinttatinninvariian signals as orientation-ect -equaaiiiiiniiviiiatt...
We use the same architecture as for HODGEGNN, but use the ReLU as an activation function instead of the sign-equivariant tanh. This breaks (joint) orientation equivariance: The model is now sensitive to the chosen orientation and, therefore, can be seen as direction-aware. However, this holds for all edges, and hence HODGE+ can also not model undirected edges anymore.
HODGE+DIR,
LINE-MAGNET. We adopt the node-level graph transformer of Geisler et al. (2023) for edge- level problems on directed graphs as follows: We first compute the spectral decomposition of the orientation equivariant and invariant Laplacians (g) and L (9)
inv as they are both normal matrices:
(9) (I Jequ equ
H
(4) (()
L (4) (T H (9) (())( inv in
inv
Jin
(13)
We then concatenate orientation equivariant and invariant edge features together with the k 32 (complex-valued) eigenvectors rigi and r(9) associated with the corresponding eigenvalues of smallest magnitude. We flatten real and complex parts. Similar to Geisler et al. (2023), these can be understood as direction-aware positional encodings of the edges in the graph. We then feed this as an input to a 4-layer transformer with hidden dimension d = 32, mirroring the hyperparameter choices for EIGN, Note that while this model can distinguish directed and undirected edges is not a jointly orientation equivariant or invariant model according to Definitions 4.1 and 4.2 despite utilizing Lig and L
lav
Dir-GNN. We also ablate alternative mechanism to represent directed edges while correctly modeling orientation equivariant and invariant signals. The core idea closely follows the node-level GNN proposed by Rossi et al. (2023) and distinguishes directed from undirected edges through two different message-passing schemes. To that end, we separate the boundary operators Bagu and Bin
into three boundary maps each:
if)) (u. and E En if) = (v,)) and e E Ep- otherwise
B epplm
(14)
0
{
-11
if (v. and e E ED otherwise
[ equlii
(15)
0
if = (su) and e E ED
B
equueee
(16)
0
otherwise
1
if E (ee) and e E Ey otherwise
B
inv]
(17)
0
if e = (v.-) and e Ep
B inv |u,e
(18)
0
otherwise
if = (f.)) and e ED
Bimcuuee
(19)
(0
otherwise
They imply three separate Edge Laplacians each set corresponding to one of the Magnetic operators proposed in our work through a similar construction. Instead of relying complex numbers to represent directionality, this approach represents directed edges through three separate message- passing operations: One between undirected edges, between directed edges that are both incoming
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edges concerning shared node, and one between edges that outgoing concerning shared node. One natural draw of this approach is that there is no direct message passing between all three types of edges (like in EIGN). We construct the Dir- GNNarchitecture similar to EIGN: In Equations (2) and (3). we replace our Magnetic Laplacian convolutions with the aforementioned three Laplacians and associate separate set of learnable weight matrices for each of the three. Summation serves to aggregate the result of the three distinct convolutions. All remaining design choices of EIGN remain. Therefore, this baseline closely ablates an alternative representation of edge directionality, Importantly, since the boundary operators are adapted from the ones proposed in our work, Dir-GNN, too, is a jointy orientation equivariant and invariant model according Definitions 4. and 4.2
EIGN. We realize EIGN according to Equations
to We realize the sign-equivariant
activation eeu as tanh to ensure joint orientation equivariance (see Appendix [A] In the main text, we omitted biases for brevity. However, to not violate joint orientation equivariance, biases can only used when the ouiput of the affine transformation not orientation -etitattion-euvviiinnnnn i.e. orientation-invarian Consequently, the following linear transformations are supplied with biases: W
w inv inv-+ W equ-time? inv Fiine-bequ" Fiinv-inn" For each message passing operation, we realize the node transformation h as a simple 1-layer MLP with hidden dimension 32 for each of the inter- <(1)
W W!(
modality convolutions Jinv-oqqu and ((t)
Joqu-inv and as identities for the intra-modality convolutions Jqqu and (!)) f which we empirically find to perform better.
second caveat is that EIGN's convolution operators output complex-valued signals. While one could, in general, backpropagate through such signals as well, we instead flatten the output signal of each convolution operation by concatenating real and imaginary parts. Notice that this operation is sign-invariant:
(LATTEN(-X:) = (- (-)e(X)) (1 (-Im(X) (Re((X) Im(X))
(20)
(21)
--FLATTENX))
(22)
By Proposition A.4. Orientation invariance is trivially preserved as there are no restrictions on the mappings applied this signal modality long as they are orientation-independdet This way, all model weights act on real-valued signals. Due to the concatenation of real and imaginary values, we keep hidden dimensions consistent by letting the affine/ linear transformations project into a space of half the desired dimension. We fix the complex phase shift to q = m as this is the longest cycle length possible in any graph. Consequently, potential side-effects due to the cyclical nature of complex phase shifts are avoided.
applying this flattening operation, therefore, preserves orientation-equivarranee.
Third, we apply a final linear/affine transformation to the representation of the last layer (for the respective signal modality) H(! In the case of orientation coentaation-ivariian outputs, we can use an affine transformation with biases ineentation-indpenddee transformations do not affect joint orientation invariance. For eecentttion-qqivariiia outputs, we have to omit biases and use linear transformation only. This linear transformation is sign-equivariant function and, again, preserves joint orientation equivariance as per Proposition A.4
Lastly, we also omit the normalization of the Laplacians in the main text. Each Laplacian x (B()) B is normalized symmetrically by normalizing the corresponding boundary maps:
= (D
(23)
Here, = Ee [abs( [abs(L (2)eee is a diagonal matrix of degrees of the Laplacian.
Since the normalization is independent of the orientation (due to the absolute value), it does not affect joint orientation equivariance invariance.
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D.4 TRAINING SETUP
For all models, including the baselines, we perform an extensive hyperparameter search to select the most suitable configuration. To that end, perform 20 different data splits for each hyperparameter configuration and compare the average performance to select the best hyperparameter configuration for each model. We search cartesian grid of the following hyperparameter options:
Learning Rate: {0.03, 0.01, 0.003, 0.001}
Hidden Dimension: {8, 16, 16,32}
Number of Layers: (2,3, 3,
For the LD Cycles, we always fix the number of layers to the cycle size e since our convolution operators are local and eycles can not be detected if the receptive field of an edge not access the entire cycle.
In all reported results here, compute average metrics over 50 different dataset splits and also provide the 95% confidence interval for the mean.
All models trained with the ADAM optimizer with weight decay, mean- eeean-squared error objective for regression problems, and cross-entropy for classification tasks. We clip gradients to norm 1 and apply dropout (Srivastava al. 2014) with probability 0.1 after every layer. We the following configurations for each dataset:
Electrical Circuits: Batch size of 10 graphs. 200 epochs.
Traffic Datasets: Batch size of 1, 500 epochs.
RW Comp, LD Cycles: Batch size of 10 graphs, 50 epochs.
Tri-Flow: Batch size of graph, 50 epochs.
For the denoising, interpolation and simulation tasks we use dataset split of 80/10/10 for traffic and 50/25/2 25 for the circuits dataset. For synthetic tasks, we use 70/ 70/10/20. We track the validation models in PyTorch (Paszke et al. 2017) and Py Torch Geometric (Fey & Lenssen, 2019) and train on these types of GPÜs: NVIDIA GTX 1080TI GPU (ii) NVIDIA A10O GPU (m NVIDIA H100 GPU.
RMSE over all epochs and select the model with the best validation metric. We implement
E ADDITIONAL RESULTS
Here, report average results over 50 runs with the best-found hyperparameter configuration for every model. We also report the 95% confidence interval for the mean.
Synthetic Tasks and Simulation. We supply the confidence intervals for tables Tables 4 and 5 in
Tables 9 and 12 Additionally, we report Mean Absolute Error (MAE) (L) and I22 score (1) for the
simulation task Tables 13 and 14 The findings are consistent with the RMSE metric.
Denoising and Interpolation. We show the RMSE for the denoising task in Table 10 EIGN achieves the best results on four out five datasets. Because the noisy flow input makes this problem significantly easier, some baselines achieve satisfactory performance and the gap to EIGN is not as pronounced in the challenging simulation task. We make similar observations for the interpolation problem in Table [I]
Ablation. We supply the confidence intervals for the mean RMSE of Table Table Additionally, we visualize the distribution of simulated electrical currents criienatiio-equivariin targets) in Figure Similar to Figure 5 we find that the direction-aware EIGN provides physically more plausible predictions. The non-negativity constraint imposed by diodes is learned to lesser extent than on the Anaheim dataset (Figure 5).
Phase shift g. Figure showcases the performance of EIGN and two variants without inter-modality convolutions and fusion respectively for different relative phase shifts q m. EIGN suffers from phase shifts that picked too small such shifts become too small to notice. In contrast, if the phase shift is picked too large, accumulated phase shifts may overshoot 2T. For example, choosing q 2 for & E N is equivalent to applying phase shift of i.e. phase shift. However, even
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Table Average performance of models on synthetic tasks (best and runner-up) with 95% confi- dence intervals of the mean.
RW Comp ROO11)
Cycles
Tri-Flow
Model
ADC-ROOY1Y
RMSB:11
MLP
0.720-o.021
0.500
0..277
LINEGRAPH
0.758
0..83
0,497
HODGEGNN
0.500
0. 0.500za.oor
0,458
HODGE+INY INV
0.81 =0
0.75 gg oes
0.293
HODGE+DIR
0.819 0.001
0.799 +o.00s
0.293
LINE-MAGNET
20.00t
0. 20.000
0.54222 2011
Dir-GNN
g0.001
0. 0.768go.m o0
0,453 2022
RIGN
0.864 001
0.9962o
0.022 000
Table 10: Average RMSE (L) of different for the denoising task on real-world datasets (best and runner-up) with 95% confidence intervals of the mean.
Model
Anaheim
Barcelona
Chicago
Winnipeg
Circuits
MLP
0.076
0.028 0.063.o.cot
0.053 one
0.462.o.00 26622..oom
LINEGRAPPH
0.070
0.059gn cmn
0.030 mon
0.054 om
0l.43
HODGEGNN
0.084
0.063an.cm
0.048
0.069
0.360
HODGE+INV
0.062
0.056
0,030
0.05 a:
0.346
HODGE+DIR
0.053 10.002
0.048 c.000
0.025 00,00
0.041 a.,0co
0.262 po.os
LINE-MAGNET MAGNET
LINE
a.0
0.006
0.034 20.000
0.062 10022x..oo
0.508 co.o
GNN
0.0G9
0.066g...o
0.036 moo
0.061 Le
0.422
EIGN
0.050
0.053 o00
0.039
0.050 om
0.266
smaller values of q may cause problems as discussed in (Geisler et al.) 2023): Consider any pair of directed edges connected through a sequence of L other directed edges (e.g. in cycle). After repeatedly applying the Magnetic Laplacian operators we propose the relative phase shift of these two edges will be * q. Since these relative phases are, however, taken modulo 2x this introduces ambiguities if L q exceeds 2T.
To mitigate this issue, we choose q 1/m which ensures such issues can not arise. In practice, this may be a conservative value and we, in fact, also observe in Figure that slightly larger values improve performance in interpolation and simulation problems. Interestingly, the pattern slightly deviates for the easier denoising task: We conjecture that there identifying directed edges may not play as large of a role as much of the information is already encoded the noisy input signal.
Hidden Size and Number of Layers. Table 16 ablates different hyperparamter configurations of EIGN on the electrical circuits simulation task. In particular, we vary the learning rate, number of hidden dimensions and number of layers as described in Appendix D.4 In general, we observe that both under-pant parametrized models (i.c. small number of layers and / or hidden dimension) and too larger learning rate lead to worse performance. However, models that are sufficiently deep and wide enough are consistently able to outperform the baselines in terms of RMSE.
E. VARIATIONS ON EIGN AND BASELINES
GCN-like Convolutions. Instead of using our proposed (Magnetic) Laplacians as graph-shift operators according to Equations and (3), we also ablate an operator A that is defined as
- [ akin GCNs Kipf & Welling 2017 in Table 17 In particular, replace each Laplacian with its corresponding GCN-like counterpart A (.) and leave all other components of EIGN unaffected.
Similarly, following recent line of work using polynomials of the Node Laplacian as graph-shift operator Defferrard et al. (011)). In particular, for each Magnetic Laplacian operator L ( we instead convole an input signal H of each respective modality with
Chebyshev Convolutions.
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Table 11: Average RMSE (4) of different for the interpolation task on real-world datasets (best and runner-up) with 95% confidence intervals of the mean.
Model
Anaheim
Barcelona
Chicago
Winnipeg
Circuits
MI.
0.103
0.1
0.110go
0.166aa
1.030
LINEGRAPH
0.103
0.1 0.148g
0.109
0.165aa
1.025
HODGEGNN
0.2502o..rr oco
0. 16
0.105 20.00
0.166 oo1
0,997 =0.00
HOnGE+INV
0.005
0.144
oooon
0. 00.002
0.145
0.778 co.0a
HODGE+DIR
0,088
0.1 41
0.112
0.133
0.753
LINE-MAGNET
0.116
0,151 20.002
0.106
0.171 1
1.031 an
DirGNN
0.277
6.1699cs
0. 1.1099g
0.160 mra
0.945
EIGN
0.081 20.002
0.125 pe
0.081 20.000
0.100
0.600 go..o
Table 12: Average RMSE (L) of different for the simulation task on real-world datasets (best and runner-up) with 95% confidence intervals of the mean.
Model
Anabeim
Barcelona
Chicago
Winnipeg
Circuits
0.1055:..cc
0.149:o..0
0. 0.109.co.a
0.167
1.030
LINEGRAPH
0.101
0.149 oo.
0. 0.109
0.164
1.037
HODGEEGN
0.280
0.1 70
0.107 20.00
0.173
1.016go.co
HONGE+INV
0.098 occ
0.1 1446g oon
0. 0.108ga oas
0.151 as on
0.828 no
HOnGE+DIR
20
44 ooa
0.109 non
0.132 on
0.760
LING-MAGNBT
0.1192a.0r
0.151
0.105
0.170
1.027
Dir-GNX
0.278 00.002
0.170
0.106 00.00
0.173
1.029
EIGN
0.090
0.133
0.078
0.101
2.696
Chebyshev polynomial of order k = 5. The resulting filter can be defined recursively as follows:
c H())
(24)
c0 [ H
(25)
(k) 21 (0) (l--)) c(k-2)
( -
(26)
For the convolution operators within a signal modality Ligu and (g) we set
L
L and
i (9) For the inter-modality Laplacians, however, using higher powers would break joint inv imv
orientation equivariance and invariance respectively. That is, because the operators transform one signal modality into the other and repeated application would apply the wrong Laplacian to different modalities. We therefore only apply the inter-modality Laplacians for the first term
((3 and use
the Laplacians of the target modality for higher order terms, i.c. we set L (9) equ->inv
L (4)
Live
and
i (4)
(g) Leq respectively.
iny tuq
Large Baselines. While the search space over which optimize the hyperparameters for each baseline is shared different architectures can result in models with different parameter counts even for the same hyperparameter settings. To that end, we evaluate our baselines for a hidden dimension of dl = 80 for four layers. These models roughly have the same number of parameters EIGN. In Table [7 we find that even for comparable parameter counts EIGN outperforms the competitors. This underlines that it is the inductive biases that carefully follow from our theoretical considerations which are responsible for the high efficacy of EIGN and not its larger number of parameters.
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Table 13: Average MAE (4) of different for the simulation task on real-world datasets (best and runner-up) with 95% confidence intervals of the mean.
Model
Anahcim
Barcclona
Chicago
Winnipog
Circuits
MLP
0.0G9 2o.0ca
0.098
0.000.o.003
0.106
0.542
LINEGRAPH
g.008
0.095
0.006gm..0
0.104ge.oo
0.549
HODGEGNN
0.183 oem
0.102g
0.065 ow
0.109 oo
0.514go
HODGE+INV
0.067
0.098
0.067
0.100as
0.459
HODGE+DIR
0.06222.cm 001
0.094
0,066 0006...00
0.089 40.0c1
0.420an.c1:
LINE-MAGNRT
0.079 oc
0.097 oon
0.065 cwn
0. 0.108ge oos
0.536go nr
Dir-GNN
0.181
0..03
0.065....oz
0. 0.109kp +0.000
0.518
EIGN
0.061 g0.00
0.088 o01
0.049 2m c01
0.069 am
0.373gmor1
Table 14: Average R2 (1) of different for the simulation task on real-world datasets (best and runner- up) with 95% confidence intervals of the mean.
Model
Anaheim
Barcelona
Chicago
Winnipeg
Circuits
MI.P
0.89022
0.2998 one
0.003gn
0.2
0.1 16
LINEGRAPH
0.897 -0.00
0.319
n.a.
n.a.
0.097 s0.ous
HODGEGNN
0.000..o.oe
0.000
0.000...00
L..00
0. 0.039...
HOnGEeINy
0.905
0.3399
one
-0.025
0.444
go
0.509 ons
HODGE+DIR
0.919
0.402g
0.033 AMM
0.640
0.656
LINE-MAGNET
0.855 0.004
0.347 10.021
0.006
0.213
0.090 rou0ss
Dir-GNN
0.000
2.000
0.0
0.000
0.00 0.0144g
EIGN
0.
0.5
0.679gmmm
0.806
0.744g
Table 15: Ablation of different components of EIGN on synthetic tasks and simulation on real data (best and runner-up) with 95% confidence intervals of the mean. We omit (i) direction-awareness by setting q = 0, (ii) the fusion operation of Equations and (5). (iii) the fusion operators using convolutions L (9) equ inv and L (9)
equ
BIGN
EIGN
EIGN
EIGN
Dulaset
EIGN
RW Comp
0.76220.00t
0.853
0.845
0.86220.on
0.8642...m
LD Cyeles
0.689_omm
0.987
0.926
0.9962n.0m
0.996
Tri- Floow
0.362
0.074
0.034-20.00
0.022.s..0
Anaheim
0.289
0.097
0.283
0.099
0.090gm.cm
Barcelona
0.172 0.004
0.139
+0.005
0,177 s.o0s
0.163 $0.007
0.133 10.002
Chicago
0.079
0.093
0.1
0.082 20.22
0.078
Winnipeg
0.132
0.170
0.175
0.138
0.101
Circuits
0.957 20
0.974
1.974gnoes
0.727 am
0.707
0.696
1.0
Oriented Flow
65.8%
84.8%
0.5
0.0
-0.5
w/o Direction
w/ Direction
-1.0
Diodes
All Components
Figure 6: Distribution of simulated traffic on the electrical circuits dataset for EIGN and a direction- agnostic variant that sets = O. Direction-awaree leads to more more plausible predictions, i.c. current that follows the constraints of diode components that only permit one flow direction.
32

---

Published as a conference paper at ICLR 2025
Denoising
Interpolation
Simulation
1.1
0.5
No mix
1.0
No fusion
RMSE
0.4
0.9
0.9
EIGN
0.8
0.8
0.3
0.7
0.6
100
102
100
102
100
102
q /
q m
q / m
Figure 7: Ablation of the relative phase shift strength * m on the performance for all three tasks on the Circuits dataset. Phase shifts that are small become hard notice while large phase shifts may accumulate to total phase shifts larger than 2n.
Table 16: Average RMSE (L) of the simluation task on the circuits dataset for EIGN at different hyperparameter configurations with 95% confidence intervals of the mean. Bold numbers indicate improvements over all baselines. We vary the learning rate (LR), hidden dimension d and number of layers.
LR
Hidden Dim.
2-layers
3-layers
4-layers
0.001
8
1.003
0.94820.010
0.799 036
0.001
16
0.974 40.003
0.91 20.00
0.001
32
0.983
0.931
0.746#0.04z 00.042
0.003
8
0.977 20.037
0.827 20.04
0.707 20.033
0.003
16
1.0024
0.834
0.701 1..0110.0a5
0.003
32
0.993 030
0.837 2 034
0.67 1 10.033
0.01
8
0.944
0.799 o.00
0.675 0.000
0.01
16
0.965
0.776
0.05
0.716 000.030
0.01
32
0.98611.02 031
0.828 100000000
0.896 p38
0.03
8
0.909 00094..am
0.939
0.966
0.03
16
0.971 20.024
0.994 o32
1.023 1:0 p33
0.03
32
0.96811.om
1.01620 o33
0.99441..0m
33

---

Published as a conference paper at ICLR 2025
Table 17: Average RMSE ()) of different for the simulation task real-world datasets (best and runner-up) with 95% confidence intervals the mean for baselines with paramater counts comparable EIGN as well as variations on EIGN.
Model
# params
Anaheim
Barcelona
Chicago
Winnipeg
Circuits
MLP
209
0.105
0.149
0.109
0.167 20.001
1.030
LINEGRAPH
753
0.101 1: 001
0.149 002
0.109 20.000 0.002
0. 0.16422.000
1.037 10.008
HODGEGNN
2k
0.280 nos
0.170 oo1
0.107 .011
0.173
1.016
HODGE+INV
2k
0.098 to.
0.146
0.108 go
0.151 20.001
0.828 $0.026
HODGE+DIR
2k
0.091 to 001
0.144+to 002
0.109 .001
0.132 + 0.001
0.760 t0.000
LINE-MAGNET
554k
0.119 or
0.15 a:
0.105
0.170
1.027 10.
Dir-GNN
42k
0.278 +0.002
0.170+a.co 000000
0.106++o -0.001
0.173++oool
L.029
MIP--
20k
0.109 1.0 005
0.14619 oos
0.115 11152o ooo
0.16620.00g
1.008
LINEGRAPH -L
40k
0.103++ 005
0.152+a Dos
0.112 .077
0.163 00.66=+0.008
0.994++..88
HODGEGNN-L
39k
0.279 +0. 007
0.172 .04
0.109 =0 .000
20.004
1.035
HODGE+INV-L
39k
0.099 too 004
0.146 2:0 bos
0.109 .008
0.157 200.00
0.847 100000
HODGE+DIR-L
40k
0.098
0.152
0.115
0.135
0. 10.922ss
EIGN-GCN
36k
0.104
0.146
0.111 +o
0.139 20.004
0.869
EIGN-Cheb
134k
0.078 10.006
0.159 1.0 007
0,068 1 004
0.101 1 .o.
0.705 20.037
EIGN
36k
0.090
0.133 1:
0.078
0.101 1 0000
0.69620-098
34


Reviews:
[{"review_id": "ubJ51lHB2n", "reviewer": "Reviewer_X95w", "summary": "This paper distinguishes between orientation invariant (i.e. those with fixed sign regardless of orientation) and equivariant (those that change sign depending on orientation) features. It highlights the importance of being able to model both types of features in a graph learning scenario and proposes a new model that is able to do so.", "questions": "- How many trainable parameters do the models use respectively?\n- How sensitive to hyperparameters is your proposed model?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "The paper is well-written, bringing across the main point nicely. The theoretical claims are sound and the proofs seem correct.", "weaknesses": "The main idea of the paper brings little novelty. Both invariant and equivariant edge features have been explored previously - just not together. The performance gain could well be attributed to a more complex model. \nThe theoretical contributions are very simple. Essentially Theorem 4.1 follows by design.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mMjeUDgsWz", "reviewer": "Reviewer_s2oj", "summary": "This paper focuses on dealing with edge values on graphs that can only be defined up to an orientation assigned to that edge. Such values flip their sign if the orientation direction of that edge is changed. Conversely, there are edge values that are scalars and do not require an orientation to be defined. Functions that aggregate and update edge values can be either equivariant (the predicted value changes with orientation) or invariant (the predicted value remains the same) to changing the orientation of edges. The authors propose GNN model that at each layer takes in as input orientation-aware equivariant and invariant edge values and outputs updated orientation-aware equivariant and invariant edge values. The authors use a clever design to ensure that the model keeps the equivariant and invariant edge values consistent across the layers. They also introduce a principled way to aggregate across equivariant and invariant edge values. Finally, they extend they approach to graphs with directed edges by using a magnetic Laplacian. Directed edges always have a well defined orientation given by the edge direction. The magnetic Laplacian incorporates a complex phase shift to values of directed edges during the aggregation step. The authors use synthetic data and some real world data sets that their proposed method outperform existing methods on graphs with orientation-aware edge values.", "questions": "Please see comment above regarding weaknesses. Can the authors comment on whether it is possible to do away with magnetic Laplacian and construct their method using the most general framework for aggregating information in directed graphs from Rossi et al.? If so, can the authors implement this approach and do an empirical comparison to their current approach \nusing the magnetic Laplacian?\n\nTo address this reviewer's concern about the niche nature of the problem tackled in the paper, could the authors elaborate on the broader applications of their work? What sort of real world problems can be solved using the proposed approach? Are there more general implications of the proposed approach for solving related problems?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "This reviewer generally liked the paper. The reviewer was not aware of the problem of graphs whose edge can take on both orientation-aware equivariant and invariant values. However, the author's real world data sets seem convincing and not too contrived. It seems like there is a problem here to solve. The way the authors construct their framework, for example how the equivariant and invariant graph Laplacians are constructed using boundary operators and stitched together, is clever and principled. There is no argument that the proposed GNN respects the symmetries of the problem when it comes to equivariance and invariance of edge values with respect to edge orientations. The empirical evidence is also strong. It seems that respecting the symmetries does help with performance. Finally, the authors do a reasonable job with their ablation studies to highlight where the improvements form their method is coming from.", "weaknesses": "A minor weakness is potentially the nuanced nature of this problem. To this reviewer the problem of applying GNNs to graphs with mixed undirected and directed edges and edge values on undirected edges that can be both equivariant and invariant to their orientation could be viewed as too much of a niche problem and perhaps not enough of a significant advance for the field of learning of graphs more broadly. Therefore, this paper may lack broad interest at ICLR.\n\nThe main weakness of the proposed method is the use of magnetic Laplacian and phase shifts in the complex plane to account for directed edges. Although magnetic Laplacian is a reasonable way to aggregate information across neighboring nodes in a directed graph, it is not the most general way to do so and has limited expressivity. The reason for this is simple. The phase used to weigh features across directed edges can result in information loss when the complex features are aggregated at each node. For example, a phase angle of q aggregated with a phase angle of -q results in real value that is indistinguishable from a value aggregated from undirected edges. Similarly if q = 1/2 then 4 hops along directed edges will result in a purely real value indistinguishable from messages aggregated from undirected edges. Similarly, after multiple hops where the edge values may become complex, it will not be possible to determine whether a complex value arriving at a node came from a real value propagated along a directed edge or a complex edge value sent along an undirected edge. Therefore, the use of a phase shift and complex numbers is not the most general way to aggregated information in directed graphs and results in loss of information and thereby lower expressivity of GNNs.\n\nArguably, the most general way to do aggregation of features of neighboring nodes in a directed graph is proposed by Rossi et al. There, the features from neighbors connected by outgoing edges is aggregated separately from that of neighbors connected by incoming edges. This can be easily extended to include undirected edges as a separate category. During the update step the aggregated features of outgoing and incoming neighbors are concatenated alongside the self feature of the node. The authors should consider such a scheme and construct their boundary operator this way (a vector now that separately aggregates features from outgoing and incoming directed edges and undirected edges). The Laplacian operator can be constructed using this boundary operator and the same constraints applied to ensure that equivariance and invariance requirements are kept. Given how principled the author's approach has been to respecting the symmetries with respect to edge orientation, it would be interesting to see if the most principled way to incorporate edge directions will further improve the performance of their model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6oVlgUY0at", "reviewer": "Reviewer_mjjQ", "summary": "The author proposes a novel model that decouples direction and orientation. The framework explicitly models equivariant/invariant orientation constraints for undirected and directed edges in graphs via different types of Laplacians. The model is comprehensively evaluated across a wide range of tasks and datasets, and outperforms prior works by a large margin.", "questions": "1. On line 200-203, the authors state that “We restrict our novel constraints to undirected edges by requiring equivariance/invariance among direction-consistent orientations only. This allows representing the direction of directed edges through their orientation which, consequentially, must not be arbitrary.” Could you please explain the meaning behind this statement in a simpler term? If I understand correctly, orientation is generated arbitrarily for undirected edges, and direction-consistency only matters for directed edges. Also why does the representation of undirected edges allow representing the direction of directed edges?\n2. Figure 2 should be explained more instead of just stating that the representations are indistinguishable for models that are orientation-equivariant for directed edges.\n3. What is the time complexity to form the incidence and laplacian matrices in the paper?\n4. I don’t think $\\mathbf{B}_{\\text{inv}}$ should be claimed to be a novel Laplacian as it is already standard definition in many works.\n5. How sensitive the model is with respect to the hyperparameter $q$? It seems to me that if an edge has $k$ incoming edge messages such that $kq=2\\pi m$ where $m$ is a natural number, the information regarding direction will be wiped out (assuming edge features are the same across edges).", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper defines a clear problem to solve, and the solution is well-motivated. Specifically, decoupling direction from orientation and differentiating directed equivariant edge signals in different scenarios make the work unique compared to prior works. I believe the setup for controlling edges’ signals can be set as a new standard for the field. The theoretical analysis is also comprehensive and sheds light on the design choice. The experiments are also well-designed. They cover different tasks related to directed graphs with no parallel edges. The proposed novel set of benchmarks, which includes 3 synthetic datasets and 5 real-world datasets with 3 different tasks, is a significant contribution to the field.", "weaknesses": "One weakness of the paper is on the presentation side. The paper, even displays a lot of figures, to demonstrate different concepts, is still vague in many arguments. Please refer to the Questions section for detailed concerns. Another problem is model comparison. I think the paper will be more comprehensive if the model is compared with the more recent relevant work such as [1].  The next problem is the scalability problem. It seems to me that the novel way of different Laplacian constructions is a fairly computationally expensive process due to the unconventional way of defining incidence matrices. The benchmark itself also only contains small graphs, so it is unclear if the model can scale. Also, the method itself seems to not work on directed multigraphs. Intuition for using Magnetic Laplacian isn’t clear to me; do we really need to introduce complex phase shift to represent directed edges, or can we just pick an arbitrary value and separate undirected and directed Laplacian? Also please refer to question 5 for my concern.\n\n[1] Geisler, S. et al. Transformers Meet Directed Graphs. (2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RjOKgN7Pvf", "reviewer": "Reviewer_fGAT", "summary": "This paper seeks to extend equivariant machine learning to signals (functions) defined on the vertices of a graph, including signals which are inherently directed (as well as undirected signals). It aims to produce representations which are either invariant or equivariant to changes in edge direction as appropriate, which is non-trivial since standard methods require a boundary operator which chooses an arbitrary edge direction. \n\nIt achieves its goals by formulating the notion of direction consistent orientations and uses this notation to formulate joint orientation equivariance / invariance and then applying several distinct convolution operations (with various generalized magnetic Laplacians ) together with a fusion module and showcases its numerical effectiveness on real and synthetic data sets including data sets from traffic, which was one of the motivating applications.\n\nOverall: I am rating this paper a six, but would have rated this paper a seven if that were an option.", "questions": "The condition that \\sigma_{equ}(-x)=-\\simga_{\\equ}(x) (line 242) seems stange. Are there common activation functions that satisfy this?\n\nWhy is there multiplication by the Laplacians in (2) and (3)? Based on standard GCNs it seems that left multiplication by low-pass filters (I-L/2) would be more natural?\n\nCan this framework be extended to settings where there are both edge signals and node-signals?\n\nSimilarly, can it be extended to signed and directed graphs? The reference Zhang et al (2021), which the authors take setting from was extended to this setting in subsequent work ``MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian” (He et al.)", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The tasks are well motivated by realworld scenarios – figure 2 is quite nice!\n\nThe proposed network is an elegant solution for achieving the desired properties\n\nExperiments are well sought out and feature a thorough ablation", "weaknesses": "The use of separate equivariant/invariant Laplacains L_equ, L_{inv-> equ}, etc is quite interesting. However, it is unclear how the proposed model compares to other networks defined in these Lapacians. Could one instead constructed a ChebNet style architecture as was done in (Zhang et al 2021) with the magnetic Laplacian\n\nI am unsure if the baselines adequately capture the current SotA\n\n\nMinor:\n\nMany equations, e.g., (6) and (7) are missing punctuation marks. (This does not affect my score but should be fixed)\n\nThe notation B^H is used without definition. I understand that this is the Hermitian transpose, but it should still be defined. (This does not affect my score but should be fixed)\n\nCapitilization in the references e.g., ``laplacian” vs ``Laplacian”, should be fixed (This does not affect my score but should be fixed)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)