Sheaves Reloaded: A Direction Awakening
Authors: Stefano Fiorini
Hakan Aktas
Iulia Duta
Pietro Morerio
Alessio Del Bue
Pietro Lio
Stefano Coniglio
Venue: ICLR 2026 Poster (2026)
Keywords: directed sheaf neural network
directed graphs
directed cellular sheaves
Primary area: learning on graphs and other geometries & topologies

Abstract: Sheaf Neural Networks (SNNs) are a powerful algebraic-topology generalization of Graph Neural Networks (GNNs), and have been shown to significantly improve our ability to model complex relational data. While the GNN literature proved that incorporating directionality can substantially boost performance in many real-world applications, no SNNs approaches are known with such a capability. To address this limitation, we introduce the Directed Cellular Sheaf, a generalized cellular sheaf designed to explicitly account for edge orientations. Building on it, we define a corresponding sheaf Laplacian, the Directed Sheaf Laplacian $L^{\widetilde{\mathcal{F}}}$, which exploits the sheaf's structure to capture both the graph’s topology and its directions. $L^{\widetilde{\mathcal{F}}}$ serves as the backbone of the Directed Sheaf Neural Network (DSNN), the first SNN model to embed a directional bias into its architecture. Extensive experiments on twelve real-world benchmarks show that DSNN consistently outperforms many baseline methods. The source
code can be found at https://github.com/hakanaktas0/DSNN.

Published as a conference paper at ICLR 2026
SHEAVES RELOADED: A DIRECTIONAL AWAKENING
Stefano Fiorini 11
Hakan Aktas² Tulia Duta²
Pietro Morerio
Alessio Del Bue²
Pietro Lio Stefano Coniglio
Istituto Italiano di Tecnologia, Genoa, Italy
2 University of Cambridge, Cambridge, United Kingdom University of Bergamo, Bergamo, Italy
ABSTRACT
Sheaf Neural Networks (SNNs) are powerful algebraic-topology generalization of Graph Neural Networks (GNNs), and have been shown to significantly improve our ability to model complex relational data. While the GNN literature proved that incorporating directionality can substantially boost performance in many real-world applications, SNN approaches are known with such capability. To address this limitation, we introduce the Directed Cellular Sheaf, a generalized cellular sheaf designed to explicitly account for edge orientations. Building it, we define corresponding sheaf Laplacian, the Directed Sheaf Laplacian L which exploits the sheaf's s structure to capture both the graph's 's topology and its directions. I, serves as the backbone of the Directed Sheaf Neural Network (DSNN), the first SNN model to embed a directional bias into its architecture. Extensive experiments show that DSNN consistently outperforms many baseline methods. The source code can be found at htth//ss//tthhbbcomm///// httpg/hhhbb.ooomccaaaaaa//////S/
INTRODUCTION
The rapid advances in neural networks and deep learning have provided powerful tools for capturing complex relationships in structured data. Rooted in algebraic topology, Sheaf Neural Networks (SNNs) (Hansen & Ghrist 2019 Bodnar et al. 2022) have recently emerged as principled extension of traditional Graph Neural Networks (GNNs). They leverage the notion of cellular sheaf, which equips graph with vector spaces associated to its nodes and edges, together with restriction maps that relate the two when they are incident. This framework enables learning in higher dimeiioaiini feature spaces, naturally mitigates oversmoothing, and improves performance in heterophilic graphs (where neighboring nodes may carry dissimilar features) (Bodnar al. 2022).
Despite their strength, the SNNs proposed so far are limited to undirected graphs and overlook edge orientations, therefore failing to fully capture the graph topology of many real-world (naturally directed) applications. Indeed, directionality plays a central role in complex networks (Bianconi al. 2008). underpinning topological and dynamical phenomena that can strongly influence system behavior (Harush & Barzel 2017] Asllani et al., 2018). To amend this, we introduce principled extension of SNNs which explicitly incorporates edge directionality. We do so by designing complex- valued direction- direction-aware aware restriction maps and a corresponding directed sheaf Laplacian. With this, our method brings the expressive power of SNNs the domain of directed graphs, enabling a richer message passing which respects any asymmetries in the graph relationships. By explicitly modeling edge orientation, our framework combines the benefits of SNNs' bigher-dimension learning and robustness to heterophily with the advantages offered by directed GNNs (Zhang et al.) 2021bb. This makes our approach both theoretically principled and practically impactful for a wide range of applications, from social and biological networks to causal and flow-based systems.
We enhance the representational power of SNNs in settings where edge directionality crucial by introducing the notion of Directed Cellular Sheaves. Unlike traditional cellular sheaves employed in
"Equal contribution.
Corresponding author; stefano. fiorini 1994 @ com

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ofte-offthhearr SNNS, which assign vector spaces (or more general algebraic structures) to the cells of a complex without retaining a notion of direction, our framework incorporates direction explicitly into the sheaf"s restriction maps. In it, we define the Directed Coboundary Operator ă associated with the Directed Cellular Sheaf, which we use construct the Directed Sheaf Laplacian (DSL)
operator T,²,, capturing both the graph's 's topological structure and the orientation of its edges.
Our main contributions are the following ones:
We introduce the Directed Cellular Sheaf. a mathematical construct that enriches directed graphs by enabling principled representation of directional interactions between its nodes, This structure assigns linear maps between the vector spaces associated with the graph's s edges and vertices such a way that the edge directions are explicitly represented.
We propose the Directed Sheaf Neural Network (DSNN) SNN architecture explicitly designed to include an inductive bias that reflects the directional structure of the graph.
We conduct extensive experiments on real-world and synthetic datasets, demonstrating the advantages of our proposal to incorporate directionality in an SNNs via the Directed Cellular Sheaf and its Laplacian operator LT
BACKGROUND & RELATED WORK
2.1 CELLULAR SHEAVES
In the classical setting, sheaf assigns data (such as sets, groups, or vector spaces) to open sets of topological space (such points, open segments, and open disks), together with restriction maps that propagate this data to open subsets within them. A cellular sheaf (Shepard 1985 Curry 2014) modifies this perspective by replacing open sets with cells of cell complex (where O-cells are points, 1 1-cells edges, 2-cells faces, etc.). It assigns vector space to each cell and a linear restriction map from each higher-dimensional cell to each of its faces, reflecting the hierarchical structure of the complex. In line with recent works on SNNs (Hansen & Ghrist 2019 Bodnar et al., 2022), we focus on cell complexes consisting only of O-cells and 1-cells, which coincide with the nodes and edges of graph, and on lower-to-hipee to-hhhhrr dimensional mappings from nodes to edges. In such models, the sheaf structure enables a principled generalization of message wressage-passing architectures by allowing node features to propagate through edge-level transformations governed by linear restriction maps.
Following Hansen & Ghrist (2019), we define the cellular sheaf of an undirected graph G = (V, with n V and m [ as the triple ((F(u) Jwvv, {F(e)) (Fuge)eer(()) ). containing V, vector space F(e) associated with each edge e E E, and a linear map Fuste : F(u) + F(e) for each edge e E ((u), where T(u) is the subset of edges incident cellular sheaf, the vector spaces are referred to as stalks, while the linear maps are called restriction maps. In this framework, the vertex stalks F(u) represent the node feature vectors (traditionally denoted as v in the graph-learning literature). The space formed by all the spaces associated with the nodes (resp., edges) of the graph is called the space of O-cochains C (G; (resp., the space of I-cochains C' FGFF) = F(e)). The inter-vertex constraints are captured by the coboundary operator the edges (where, for each e = (u, E F, either Fuse or Fuge is multiplied by -1), is defined as 6(x)) = Fuse Tu Fuste Zy. From the coboundary operator, one can define the sheaf Laplacian as L 8 which, for a given x E (G; reads:
vector space F(u) associated with each vertex
assumed be real. In the In line with the SNN literature, all vector spaces
uuev F(u)
(G; - (G; which, given an arbitrary orientation on
LF(x),
Σ Fige - FoSEtt) c=(ur]
Vu V.
Both L and its normalized version L are positive semidefinite operators on the space of O-cochains Co (G) and are independent of the chosen edge orientation, mirroring similar property that holds for the standard graph Laplacian L (Chung 1997).
Several approaches have explored the use of sheaves in the context of graph-based learning. The first SNN was introduced by /Hansen& & Ghrist (2019), and later extended by Bodnar et al. (2022), who proposed the Neural Sheaf Diffusion (NSD) model. More recent SNN models build upon the NSD
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framework, incorporating attention mechanisms (Barbero et al. 2022). extending the architecture to hypergraph data (Duta et al. 2023). and introducing nonlinearities (Zaghen et al. 2024).
The SNN literature assumes that all node and edge stalks are unite-dimensional vector spaces of dimension d, all of which are isomorphic to IR. In this way, every restriction map coincides with d x d matrix. As result, the sheaf Laplacian is block-matrix of size nd x nd with blocks of size d x d which operates on an nd-dimensional vector- valued signal obtained by stacking the d-dimensional node signals Zu E F(u) for all u V associated with the graph's vertices (the O-cochain). When considering multi-feature vertex signals with / > 1 features (or channels), SNN operates on matrix-van valued graph signal of size nd x f. For any u, V, the block of indices u, of L is equal to the d x d matrix -J ude Fuse. The sheaf Laplacian generalizes the classical graph Laplacian on an undirected and unweighted graph G. This is because, in the special case of a trivial sheaf- sheaf where each stalk is isomorphic to R and each restriction map is the identity map over we recover the standard n X n graph Laplacian I. D A, where A {0, 1]"xX the adjacency matrix, and D == diag(1, (1,, .) where 1,, is the all-one vector.
To the best of our knowledge, no SNNs, including those introduced the above-mentioned papers, have been proposed to incorporate the edge directions directly. We set out to do so in this paper.
2.2 DISCRETE LAPLACIAN MATRICES FOR UNDIRECTED AND DIRECTED GRAPHS
In the literature, GNNs are typically classified into two categories: spectral-based and spatial-based (Wu et al.] 2020). Spatial-based GNNs define the convolution as localized- iegregation/nessae-aasstt
operator (Wang et al. 2019). For example, GatedGCN (Li et al. 2016) handles directed graphs by aggregating information from out-neighbors (ignoring, though, potentially valuable signals from in-neighbors) and, more recently, Dir-GNN (Rossi et al.] (2024) employs separate aggregation schemes with distinct weights for in-neighbors and out-neighbors. In contrast, spectral- -bsse GNNs define the convolution operator rigorously as a function of the eigenvalue decomposition of the graph Laplacian (Kipf & Welling 2017). Over the past few years, several approaches have been proposed to generalize spectral convolutions to directed graphs. In particular, DGCN (Tong et al.) 2020b) introduces first-order proximity matrix along with two second-orde proximity matrices to describe both the neighborhood of each vertex and the vertices that are reachable from a given vertex one hop. DiGCN (Tong et al. (2020a) adopts the Personalized PageRank matrix and incorporates k-hop diffusion matrices. Finally, several methods generalized the classical Laplacian matrix to suitably defined complex-valued, Hermitian matrices such as the Magnetic Laplacian (Lieb & Loss 1993) and the Sign- Laplacian (Fiorini et al. 2023).
The Magnetic Laplacian ( originally introduced by Lieb & Loss 1993) in the study of electro- magnetic fields and later employed in spectral GNNs by Zhang et al. (202Tb) a). is complex-valle valuue valued Hermitian matrix that captures directional information in graphs while admitting an eigenvalue decomposition with a real, nonnegative spectrum. Letting A, = (A + AT) be the symmetrized version of A and letting D, := diag(1,, 1, A,), the Magnetic Laplacian and its normalized version are defined as follows:
T,(1) D. H()) and I := I D2 H D with ((4) = Ag O exp (1 212 (4- AT), where is the imaginary unit and q E [0. 1].
The Sign- Sign-MManetic Laplacian L", introduced by Fiorini et al. (003). is a Hermitian matrix that is well-defined even for graphs with negative edge weights and possesses several additional desirable properties. When = 4. L and [(() coincide if the latter is first computed on the unweighted version of the graph and then element-wise multiplied by Ag. Thus, L" is invariant to positive weight scaling which could otherwise alter the sign pattern of [(() and, thus, the edge direction. Letting D, := diag(1, 1,, (A,.) and sign RRax - {-1.0. 0, 1]axn be the component-we wise signum function, 1.0 and its normalized version are defined as follows:
I. D.-H" H and Eg D. & Hd D. $ with Hº A. (L - +i (A)-IA7)))
3 DIRECTED CELLULAR SHEAVES AND THE DIRECTED SHEAF LAPLACIAN
In this paper, we introduce the notion of a Directed Cellular Sheaf, a special type of cellular sheaf where the node and edge stalks are vector spaces defined over the complex field and in which,
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assuming finite- dinte-ddmeesional vector spaces, the restriction maps either real-valued complex- valued matrices where the latter encode the graph's is direction. For clarity, we also include a notation table in Appendix A to help readers navigate the symbols we use throughout the paper.
3.1 DIRECTED CELLULAR SHEAF
For the ease of notation, we now introduce the Directed Cellular Sheaf for the case of finite- dimensional stalks (the definition can be easily extended to the defiite-dimensionall case).
Definition 1. The Directed Cellular Sheaf of a directed graph G = with adjacency matrix
A E {0, n is the tuple (T(() (F(u))wev, {F(e)}eEE, {F (Fugeleer(w)) consisting of: (Fee))eee.F())eeeee.
(A-AT)) 1. A Hermitian matrix T == cxp(i2mg (A - A)), parametric in q E R.
2. A vector space F(u)= E C associated with each vertex u E V;
3. A vector space F(e) E Ed associated with each edge e E E;
4. Two linear maps FugerFuss that map F(u).F(v) to F(e) for each edge e E with u <e where Fuge Raxdd and Fuge = To E C with F E Raxd is a real-valued restriction map, and u ~e v indicates that is incident to both u and o regardless of whether it is directed or not.
An illustration is provided in Figure [I The rationale of our definition is to encode the direction of each edge in the imaginary part of the restriction map of the tail node.
Puge
F T
T
Fush
F
Fisk
F n
F(E)
F(u)
f
F(e)
F(v)
go
FT
usde
u
(b)
An illustration of the complex- -aapue valued restriction maps of the Directed Cellular Sheaf showing how they encode the graph's directionality for (a) a directed edge and (b) an undirected edge.
Figure 1:
The core idea behind our Directed Sheaf Laplacian is that it is complex Hermitian operator whose magnitude captures undirected geometry while its phase encodes edge directions, with the parameter q modulating the strength of this directional component. Maintaining a PSD/Hermitian structure is crucial for spectral GNNs: it guarantees real, non-negative eigenvalues, enabling stable Fourier bases and well-defined spectral filters. Classical spectral GNNs (e.g., GCN (Kipf & Welling (017)) also rely on PSD) operators with bounded real spectra, but they cannot represent directionality. Our complex Hermitian formulation preserves the necessary PSD spectral properties while introducing direction-aware phase information, yielding a stable and expressive spectral operator for directed graphs. If we do not preserve the PSD/Hermitian structure, the spectral interpretation of graph convolutions breaks down. Specifically, a non-PSD matrix can have negative or complex eigenvalues, which makes the graph Fourier transform ill-defined for filtering purposes: the "frequencies" may no longer correspond to real oscillations on the graph, and spectral multipliers can produce unstable or non-convergent outputs. For instance, using purely real -remerymmetric matrix (antisymmetric) yields purely imaginary cigenvalues. applying spectral filter results in oscillatory or diverging behavior rather than meaningful smoothing or directional propagation. For example, in the undirected
case where = {u, v), we have Aun = A,n 1, T(g) cos(O) + sin = and, thus, Foge F0 Fg. are purely real. In the directed case where e (u, v) and assuming q = 1.- we have A.e and A,n and 7:19) cos( - 7 + isin(- -T i:: thus Fr - and the sign of the imaginary part indicates the edge's 's direction. Our proposed Directed Cellular Sheaf generalizes the classical Cellular Sheaf since, if G is undirected, (( 1 for all (u, v) E F. for any
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choice of and, thus, the two sheaves coincide. If G' is directed, but q 0, obtain the classical Cellular Sheaf associated with the undirected version of (see Appendix D for an additional example).
Let E UE' be a partition of the edge set into undirected edges (E") and directed edges (E'). We define the Directed Coboundary Operator associated with the Directed Cellular Sheaf as 5(x)) := Fuse It FozeTI, Tv, e. E E, where x is cochain of the Directed Cellular Sheaf. Thanks our definition of F Fuse:Fueer we have:
8(x)e {
FugeFu Tu - Foge Zy FusoNu- F 0gg Tue Zy
ife E if E E
define the Directed Sheaf Laplacian (DSL) Le associated to Directed Cellular Sheaf as I 8-5, where is the conjugate transpose operator. Each d x d block of I reads:
Fi Figet uše =-FT Fo T()) use esge
if e (u, 8)
FoggFose Fuge -IFu use
F T(g) )
if e (v, M)
uge use --T
Fuee Fuge
if {u, v} otherwise
0
F
Σ
F Fige uge
(3)
cET(u)
where [(u) is the set of edges incident regardless of their direction. Notice that, since with (Te2) y -Td9), for a directed edge e (u, v) or = (u, L and L only differ by the sign of their imaginary part.
As one can see (the full derivation is reported in the appendix), when applied to O-cochain x, the Directed Sheaf Laplacian operator reads follows for each u E V:
1 1²²x))= = Σ
(Fuse T(g))))))))))))))) (Fugetu Fognt.))
(
c= (1,,)EE
inflow
Fr Fr (Fuget F0 Fo x() +
Σ
Fr
Free (F Fugeta -oaaaFFogctt Fogett))
(1,))E
e= -u,0)EE
outllow
undirected
We define the normalized Directed Sheaf Laplacian
(5)
where D = diag(D,, DD..., D.)) and, for all u E V, Dr = Ever(y) Fo FisoF Fuge.
3.2 SPECTRAL PROPERTIES OF THE DIRECTED SHEAF LAPLACIAN
The Directed Sheaf Laplacian enjoys several key spectral properties, which we now illustrate. The proofs of the theorems of this section and the next can found in Appendix [ First, we show that both L and y are diagonalizable with a real spectrum and that their spectra are nonnegative:
Theorem 1. L is Hermitian and L > O, and the same holds for L
Next, we show that the spectrum of the Normalized Sheaf Laplacian is upper-bounded by 2:
Theorem 2. I. FN < 2T.
These theorems show that _ and L enjoy the same spectral properties as the classical Laplacian matrix L defined for undirected graphs. These essential to define principled convolutional operator by approximating the graph-Fourics transform of a graph signal with Chebyshev polynomials of the first kind of degree 1, proposed by Kipf & Welling (2017) for the undirected case.

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3.3 GENERALIZATION PROPERTIES OF THE DIRECTED SHEAF LAPLACIAN
First, we show that the Directed Sheaf Laplacian generalizes both the Sheaf Laplacian and the classical graph Laplacian:
Theorem 3. If G is undirected, LL coincides with the classical sheaf Laplacian L for any choice of R. Also, if the sheaf trivial and G is undirected and unweighted, L coincides with the classical graph Laplacian L. If G is directed and we set q = 0, L coincides with the classical sheaf Laplacian associated with the undirected version of G.
Let Trivial Directed Cellular Sheaf be any Directed Cellular Sheaf with d = 1 where, for all directed edges e = (u, v), Fuge = 1 and Fuge T(T With the next theorem, we show that, for a given directed graph without weights, _ generalized the Magnetic Laplacian and, when choosing q = 4. also the Sign-Magnetic Laplacian. The following holds:
Theorem 4. Letting G be a directed graph with unit weights, the Directed Sheaf Laplacian L² associated with a Trivial Directed Cellular Sheaf coincides with the Magnetic Laplacian L(d), In the special case where = 4. LF also coincides with the Sign-Magnetic Laplacian L
is well-known that the classical Laplacian matrix L defined for an undirected graph can be equivalently defined L = D A or L BB", where B {-1, 0, 1]axm is the node-to-edgee incidence matrix of the graph in which either of the two entries of each column has been arbitrarily multiplied by 11 While, the best of our knowledge, no similar construction is known for the Magnetic Laplacian and the Sign-Magnetic Laplacian, with the following theorem we show that one such decomposition exists and that it can be obtained via the lens of our Directed Sheaf Laplacian (which generalizes both). Indeed, we have the following:
Theorem 5. Let G be a directed graph with unit weights. Assuming a Trivial Directed Cellular Sheaf. the conjugate transpose 8" of the Directed Coboundary Operator boils down to the complex-valued node-to-edge incidence matrix B E Cu defined for edge E E incident to a vertex u:
{
1
ife (u, v) or e = fu, 2} with < v
Buc
if e {u, v) with u v e uv,
-T(19)
(9))
It follows that L(4) = BB. Also, when setting = }. we have L(t) = --= -BB.
Incidentally, this result leads to substantially simpler proofs of the positive semidefiniteness of both Laplacian matrices than those reported in their original papers.
THE DIRECTED SHEAF NEURAL NETWORK (DSNN)
The sheaf diffusion process on a graph G. introduced in Hansen & Gebhart 2020) as a generalization of the classical heat diffusion process that governs classical spectral-based GNNs Kipf & Welling (2117), follows the differential equation
(1)= = - (t),
where (() is time-dependent graph signal X. More precisely, X,, is the stalk of each node V. and it coincides with matrix Rdxf, where d denotes the dimensionality of the vertex stalk and f is the number of feature channels. X is typically obtained starting from matrix of node features of size n x f to which one applies a linear projection to obtain an n x (df) matrix, which is then reshaped to (nd) x f.
By relying on our proposed Directed Sheaf Laplacian L² we introduce the Directed Neural Sheaf Diffusion process the following generalization of the Neural Sheaf Diffusion process proposed by Bodnar et al. (2022):
X(1) -a (IX1) & x()0))()))()).
(6)

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where W1
Rddd W2 activation function.
R/X/ are two time-dependent weight matrices and a is nonlinear
We then define the Directed Sheaf Neural Network (DSNN) the convolutional neural network whose convolution operator is obtained from the discretized version of Equation 6
x(++1== ( a ( @ x(")
where X x((1) E Codxf.
The expressiveness of Eq. 7 is further enhanced by a learned parameter E 1], by which we adjust the relative magnitude of the features in each component of a stalk. The update rule thus:
((-1)) diag(1 e)x T.N
(()) [T,, 60 w(")
x
(8)
where e E [-1, 1]nd is obtained by concatenating e times. As activation function o, we adopt complex extension of the ReLU function, defined for a given 2 E C, as
if i()) 2 0, otherwise.
( (o
This choice is consistent with previous work on complex-valued GNNs and HNNs, such as (Zhang et al. 2021b Fiorini et al. (2044).
Finally, since our model operates in the complex domain, we project the output of the final layer to the real domain using unwind operation. Given X Caxe, the projection is defined as:
(Xwind(X(r)) = (R))(+)) ((X())) E R=*22
where 7 is the last convolutional layer of the network, || denotes concatenation along the feature dimension, and e is the output dimension.
Learnable Sheaf Laplacian. A key strength of SSNs is their ability to operate over richer structures, sheaves, rather than just the underlying graph. Since multiple sheaf structures can be associated with the same graph, effectively modeling the most suitable one is critical for meaningful representation learning. our proposed models, the restriction maps are learned end-to-end as a function of the input vertex features. Specifically, for each edge E E with endpoints u, V E V. each d matrix Fase is parameterized as Fuse Φ( || v..), where 2y and x,, denote the feature vectors of the nodes incident to and + is MLP. The resulting vector is reshaped into d d matrix, thus obtaining the linear restriction map Fase.
Connection with Neural Sheaf Diffusion. The Neural Sheaf Diffusion process proposed by [Bdnnan et al. (2022) relies on the Normalized Sheaf Laplacian L instead of on our proposed Directed Sheaf Laplacian T Eq. 7 Since, as shown in Theorem 3 L I when the graph in undirected, NSD is special case of SNN when the graph is undirected.
Computational Complexity. Letting f be the number of channels, assumed constant throughout the layers, we focus on single convolutional layer. In the case of an undirected graph, where all restriction maps are real-valued, the complexity of DSNN identical to the complexity of NSD, and reads O O(n(c² + dd) + m(cdd + d))), which, with d = 1, coincides with the complexity of classical spectral-based GNNs (Kipf & Welling 2017), which is O(ne² + me). In the experiments, we use d E {2. 5), which only introduces a small, constant overhead with no asymptotic impact. For -aluud, and thus, the stalks complex-valued from layer 2 onward. This, though, only leads to an extra multiplicative cost of about 4, which is independent of the graph and size of the network and plays no role in the complexity of DSNN.
directed graph, the restriction maps
For the proof of the theorems in this section, and for additional details on DSNN's inference complexity, please refer to Appendix D Appendix P respectively. Computing times are reported in Table in in the appendix.
EXPERIMENTS
We compare DSNN against different state-of-the-art baselines on two complementary tasks, node classification and direction prediction, using both real-world and synthetic datasets. Following (Bod)
al. 2022), experiment with three types of d x d blocks in the Directed Sheaf Laplacian T.F.
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diagonal, orthogonal, and general, which lead to three variants of DSNN: Diag-DSNN, O(d)-DSNN, Fit) Gen-DSNN. From Definition 1, the normalized directed sheaf Laplacian LN depends on a parame-
ter q. As F(t) is used in the convolutional layer (see Equation 7. we treat q as hyperparameter in this work. In the tables, the best results are reported in boldface and the second-bes are underlined. The datasets and code used are available on GitHub (see Appendix B Further details our experiments are reported in Appendix
Baselines. We compare DSNN against large set of GNN and SNN baselines from five categories:
i) classical GNN models: GCN (Kipf & Welling| (017). GAT (Veličković et al. 2018200));; 8); ii) GNN
models designed for heterophilic graphs: Geom-GCNI Pei al. (2020). H2GCN (Zhu al. 2020),
GPRGNN (Chien et al. 202T), FAGCN (Bo et al. 2021). GGCN Yan et al.) 2022 ; HSGNN Lu et al.
2024 ii) GNN models that address the oversmoothing problem: GCNII (Chen al., 2020 iv)
GNN models that incorporate edge directionality: DiGCL (Tong et al.
202I), MagNet (Zhang et al.
202 SigMaNet (Fiorini et al. 2023), DirGNN (Rossi et al. (0024). HaarNet (Badea & Dumitrescu
2025).
CAGN Xu et al. 22255; ; V SNN models: NSD (Bodnar et al. 2022).
Real-world datasets.The Texas, Wisconsin, Cornell, and i 1.lm datasets are provided by Pei et al. (2020); Citeseer, PubMed, and Cora by Yan et al. (2022); Squirrel, Chame leon, Roman Empire, re, and Questions by Platonov et al. (2023): and Te Telegram by byooett & Grindrod
(2020). Since GNNs are known to struggle on heterophilic graphs (where neighbors often have different labels), we evaluate DSNN across datasets with a wide range of edge homophily coefficients.
Node classification on real- datasets. We follow the evaluation protocol of Bodnar al. (2022) using 10 splits. For Texas, Wisconsin, Film, Cornell, Citeseer, Pubmed, and Cora, we rely on the fixed splits provided by Yan al. (202)) and report results on all 10 predefined splits. For Chameleon, Squirrel, Roman- Empire, and Questions, we adopt the splitting strategy from Platonoy et al. (2023). For Telegram, we use the split introduced in Zhang al. (b)2 We report mean accuracy for all datasets, and ROC AUC for Questi i Lons, which, as noted by Platonov et al. (2023), is highly imbalanced with 97% of the users belonging to the majority class. Table shows that DSNN attains the best results on 10/12 benchmarks, spanning both heterophilic and homophilic graphs. In partic- ular, Texas, Wi cons i., F Film, Chame 1 eon, Cornell, Citeseer, PubMed, Te Telegram, Roman-Empi re, and Quest inss are won by DSNN variant, while the strongest baselines nar- rowly lead only on Squirre
and Cora and only by a small margin (0.22 and 0.79, respectively). Compared to the SNN baseline (NSD), adding directionality within the sheaf systematically helps, as DSNN surpasses NSD on the majority of datasets, with notable margins on Quest ions, Texas, Telegram, and Roman-Emp ire. Relative to irrection-awart GNNs (DirGNN, SigMaNet, Mag- Net), DSNN delivers consistently stronger performance on 10/12 datasets, suggesting that combining cellular-sheaf expressivity with an explicit oriented Laplacian yields more effective message-passing bias. Overall, these results indicate that learning direction-aware restriction maps within a sheaf framework provides robust gains across many graph regimes, especially in challenging heterophilic settings. See Appendix H for the q values of DSNN compared to the ones of MagNet, Appendix | for an analysis of d values in DSNN, and Appendix L for preliminary analysis of q when set as learnable parameter.
Node classification on synthetic datasets. We further investigate the role of directionality by comparing DSNN against NSD on set of synthetic graphs generated using the Direct Stochastic Block Model (DSBM) (He et al. 2022). For this experiment, the DBSM datasets are generated with 2500 nodes, C = 5 classes, intra-cluster density oa = 0.1, probability of an edge taking certain direction 3,y = 0.2, and with an increasing inter-cluster density ai E {0.05, 0.08, 0.13. Features are intentionally minimal (scalar in/out-degree sums). We run the experiments 10 times per dataset with a 80% rraning/validatiioneesiin Across all settings in Table g the three DSNN variants achieve near- near-perfect accuracy (typically 96-99%; best: 99.14% at ag = 0. 0.10), substantially outperforming NSD (~~ 20%, i..., 3 chance for C = 5). Direction-aware GNNs trend upward with density, but still trail DSNN: MagNet rises from 78.64% to 01.58% and SigMaNet narrows the gap at higher density (up to 98. 68.6 yet remains below DSNN on average. DirGNN shows instability with a large standard error, suggesting sensitivity to directed community structure under our minimal features. Together, these results indicate that explicitly encoding orientation in the sheaf Laplacian yields a decisive advantage when communities are directionally biased, even when node features carry little information. Consistent with Table [1] (in the appendix), setting = 0-i.e., discarding edge orientation inside the lheaf-leads to a drop in
split, and report the mean accuracy and standard deviation.
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Published as a conference paper at ICLR 2026
Table 1: Node classification results. ROC AUC curve is reported on Questions while mean and standard deviation accuracy is reported for all other datasets
Modkl
Remmnn-Fuginn
Trass
Wiscrnsi
Filr
Squiel
Chancioon
Crreell
Telggnnn
Clicacer
Pulrncd
Crea
Quc-eloons
Homuph ld
0.05
0.11
0:21
0.23
0.32
0.23
0.50
0.30
0.74
0.80
0.81
0.84
Nodes
22,022
185
251
7,600
2,223
390
1K3
245
5.227
18.717
2,708
48.921
Biges
32,027
205
466
26,752
46 16.
8.834
280
R.012
4,676
44.327
5,278
153,500 2
Classes
15
5
5
Blog DSNN
90 90.40e00331
88.654955
90 882
38.34e 01
45 21
46
$7.8455770
70
94.4263.00
00
79.801
49
90.336 64
NY N1.361441
79 79
Odd--ISN
92.18-8.22
NT. 5744 B4
89 Wo:332
37. 98
2464 26
45
Shat 1M
N667 26
24
V000000 50
79 2462666
Gen-DSNN
92.08290.36
87.5765.43
43
22:31
38.40a9.755
45:34
47.1663.55
$7.54:5.56
94.81=2.28
79.8861.21
90.1740.44
67.5660772
72
79.55
Disp-NDD NSD
53.20
85 65.6726
88.63s2775
37.79t1100 001
45 23
46
36.55
66.49
12.01 58
77.3441.65
59.4240.43
87. 17.1421.66
75.82e1 LS
D6b-NSD
6367240 14
85 9545 91
89
11.81-1
48.992233
46
7he1 11
91 5955
89 4940
77. 19+1 37
Gen NSD
$3.80t00.00
82.975.13
39.21
29:21
37.80el 22
45.3122.05
45 45
136
85/68
91.7322.44 44
76. 76.3261.6
59 59.33
87 87,30211.1
77.3621.32
Haa
85.4220.43
11.57e4.18
31.3626.69
36.35e1 101
40.52a333
42.43a3.50 58
13.91a7.57
91.12 50
16.51
58, 2960.66
82.6821.54
75,01=909
CACI
00M
75 3567-7.15 15
11-4.11
14 86-111 16
10 79-1.99
79.99 11.99-7.55
66.75-1.00
COM
162244100
DOM
DSCCI
52.71 32
57 17.66..15 15
4..
29 85-077
18 900001
41 41:21-2000 20
(6:5.12
80 01.57-2.25
76.
DOM
DurGan
91 12250.13 32
14.245997
21.5726.77
29.30el 23
45 15.5 16
61 3633
92.81-2.07
Ma. 64
M0.0021
18
76.574388
SigNtaNet
25 10000000 2M
1K 19
2021-5507
36 30.99e1333
55
1921
40 10982188 8M
M. 12113440 4M
15.11+1114 14
MagNa
88.07:0.27
79.4668:3
81:18:2.80
36.51.0.9
41.04e1.54
43.82a4.55
75.09.5.59
67.62.2.00
B3.4760.54
96.3241.39
75.6600
#28GNN
60 55:99:445
72 83
78. 78.23.5.22
36.75z1 35
37.00e1.211
41 1..1423.60
74.05:5.44
62 62.69:3.05
77.1763.36
56 86.9480.42
82.41=1.42 42
74.20:22.55
GOCN
76 112540.00 48
84 8--4 35
31 Stal 3h
407552244
1971-115
75 1.-
NN 1540.37
1A 19+1 10I
H2GCN
60:11:0:52
35.70=1.00 97.
37.77a1
42.07=4.33
59.37-335 82.70u55338
17.1
89.4PH0.35
87.5761.20
15.30a11
GPRONN
082540.27
36.3644336
82.444211
34.63e122
36.622.28
10.27
74.1566-45
77.13e1.67
57.5420.38
57.9541.18
73. 29
90-2.72
10
00.31
26.40+2.01
GCNU
83.70=0.51
40.39:3.00
37,44g1
42.3222.1
43.7k22.0
77
77.3361.4
88.3711.2
78. 78.0040084
GCN
73.0950.74
55. 14:5.16 16
27.52=1
40.9944.12
00.3825.30
16.30+
18.4240.5
GAT
69.77:0.83
52 1646.63
27.44 40.
39
35.6222.06
06
39 39.112308
11.3955.55
7261:7755
76.558 23
67. 67.30al. 10
96. 66.3340.4
76. 26.5623.99
MLP
64.9220.52
80.814475
85. 39.31
36.53e0.70
40.45 41 42.79x3.50
$1.39b6,40
46.34:2.47
74,02.1.00
57. 16.337
73.6942
71.2360.94
Table 2: Node classification results on synthetic dataset. Performance comparison with different Osi values can be seen on the tables.
Model/ /a
0.05
0.08
0.10
Diag-DSNN
Oid--DSNN
97.28:0.6 68
08.34
97. 17.22.0.55 58
98.42x0.61
99.14:0.36
98 80s0.2
Model a
0.05
0.68
0.10
Gon-DSNN
96.64-0.54
98. 65
[22GNN
20.52.0.88
21.06.2. 16
20.28::0.3
Dig
Gnn-NDD
OGD--SD
20.15e1 45
20.20-1 08
2164+1.34 34
21 21.42=1005
20 20.42-1.49 49
20.57a1:25
20 +0,99
20 105811.22
40
H2GCN
BAGCN
GGCN
GPRONN
20.14:0.31
20.3421.49
20.4620992
40.02:0.06
20.02.0.06
20.48a1.02
20.68a1:23
40.
20.2420..51
20.01 78
20.28.0.54
38.96.0.73
HandNet
08.18-0553
9% 25e0
28 52
GCNH
20.
20.71a0.33
20.73.141
CAGIN
92.71:706
93. 93.31
96 962622 36
23.3423.13
21.96.2.00
21.58s1144
DIGCL
30.13:5.12
31 84.5446.8
24.73:9.1 12
GCN
20.0620.18
20.4620:22
20.01:101
MagNer
78.64=1.29
87 57.52=1.22
59:1.00
MLP
29.28.0.69
20.3120993
20.28.0:45
SigMaNet
87,44:0999
96. 1420.4
98 0020.31
DiGGNN
83.96e7.91
83. $3.52=12.92
83, 16.17.31131
performance, whereas q >> 0 restores the gains. This confirms that DSNN's improvements stem from using directionality inside the restriction maps rather than from the increased capacity of the network.
Direction prediction on the real-world datasets. We further test the performance of DSNN on the direction prediction task: a binary classification task where the model is asked to predict whether v.. E E or (v, u) E E. Following Zhang et al. (2021b). we split edges into 15% test, 5% validation, and the rest for training. and perform 10-fold cross-validation while preserving graph connectivity. As T Wisconsin, Cornell, Citeseer, Squirrel, Pubmed, Questions). On the remaining ones, DSNN is a close runner-up: on Cora it is within 0.01 accuracy of the best method, and on Film it trails the strongest direction-aware GNN by 0.18. We observe that using nonzero g is beneficial on most benchmarks. In the appendix (Table 10). the best settings adopt g > 0 in most dataset combinations, corroborating that explicitly encoding edge orientation inside the directed sheaf Laplacian - improves disambiguation of (u, v) versus (v, u). Overall, these results mirror our mode--lassificattee findings: injecting direction into the sheaf (via complex restriction maps and 1.²) yields a robust advantage on tasks where edge orientation is key.
DSNN attains state-of-the-art performance on 6/10 datasets (Texas, s,
Training and inference time. Wall-clock time (training+testingg, averaged over 10 folds, 1k epochs) and peak GPU memory are reported in Tables and (see Appendix E for more details). DSNN comes out as consistently faster than generic direction-aware GNNs, but (as expected) incurs moderate constant-face factor overhead vs. NSD due to complex- valued arithmetic. On small-to-medium graphs, DSNN runs in x8-10 s per experiment (vs. 6.5-7.8 s for NSD), while denser larger graphs times are 19-33s s for DSNN (vs. 10-16 for NSD) and 107 s for DSNN vs 47.5 s for NSD on the largest benchmark (Questions Memory usage shows similar pattern: DSNN peaks at

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Published as a conference paper at ICLR 2026
Table 3: Results on direction prediction. Mean and standard deviation accuracy are reported.
Model
Texas
Wisconsin
Cornell
Cora
Citesoor
Film
Squirrel
Chamcloon
Pubmed
Questions
DSag-DSNN
93.36+2..4
87. 17.1844.02
566464444 41
82.54=1
10=0.79
81 20H0.62
95.
92.04=0775
95.41=0.20
90.48=0.11
Odd--DSNN
94.35=4.11
88.3+
91.06+3.6
K2 98-99-111 18
85.4420.82
81. 1840.2
95.5340.31
92.52-043
95 23.0.+0233
90. 16+0.13
Gen-DSNN
93.6213.56
87.8414:04
90.8343.7
82.71a1.35
85 01.0111.01
81 2080.61
95.27 26
92/4120.49
95.5620.19
10.25:0.15
Diag.NSD
88. 88.42e4.64
85.6643.76
85.78a3 39
82.76e1119
15.0551143
80, 00.96.0.55
91.5640.66
90.12+1.44
95.2550.18
90.29
Old)-NSD
88.5364 44
85 15.6444.13
85.3844 97
82.71=1.14
85. 15-097
81.37+0.60
92.19+1 23
88.22=0.74
95.36+0.22
12+0.11.11
Gen-NSD
89,59:4.14
85 85.8914.33
85.5814.86
82 62.58:0.73
06:0.85
80.00 80.0910.59
88.9610 45
90.98:0.92
94.9920.25
90.21:0012
HaazNet
86.15=5.44
86. 31
85.55a6.23
82.53
95.04a 37
92.11
95. 15.1320.27
90.10 18
CAGN
87.3413.88
85.7913300 022
84.8111.31
81 $1.95
81.7320.67
80 80.9710.77
95.0840 19
94,8420.00
95. 1820.322
OOM
Dir-GNN
88, 88.35=4.66
86. 13+3.99 91
85.5944 95
82, 82.9920.822
84.31
80.56 00.56+0..4 42
96.2340.26
93,54=0.63
95.22e0.21
90.03=0.17
SigMaNet
86.53+3.79 89.37+3.65
85.37+4.51
84 84.29-0.9%
94,9840 44
92.01-0.59
95.05+0.22
90.05-0.14
MagNot
88.9113.96
86 86.6513.34
85.7713.55 52
82.25:0.84
84.64:1001
81.0110.51
94,9910.41
92.0920.58
95.2880.31
90.03:0.21
H2SGNN
90.19
62433.4
86.3643.41
81.21=1106
84.69=0.88
00.32200.75
91.14=0.53
94.80=0.22
18.98s0.25 25
GGCN
89.5914408
85, 85.67A2.82
16.1663
82.73
84.93
B1 81.2990.68
93.9340.55 55
91.5540.72
94, 91.9820.20
90.17.0.18
H2GCN
87.64+6.17 17
84. 84.466+4.11 13
84.59+7.118
81 42242.000 91
84 34.22=00.98
80 82+0.56
02.188+0.2 26
89, 96
94 14.69=0.26
88, 08:0.25
GPRGNN
84.41+55.2 29
46
89.55:0.66
14-82+0:55
85+0.41
PAGCN
18-15-5
86. 26+4478
83.06+7.29
82.15=0.94
64.68-1.42
00.61
04.0840410
91,22++1.11
95.0 95.08+0.47
89. 15+0.31
GCNI
89, 89,39124.81
85 66666122 87
84.4314.57
62.5990.95
84.9820.87
81 02100.474
95.3840 40
92.0 92.06:00.6
95 05.09200.21
90. 17.177.0.11 18
GAT
85, 65.7743.62 89.15+5.52
85.0145.74
82.14s 1.27
84.54
78. 18.5911.01
95.0740 26
92.25
95.04.0.21
67.73=0.99
GCN
86.90+3.68
86. 90+3-68
76.20+6.3
81.78=1.04 04
83.63=1.05
77.61+0.47
90.70=0.56
86.95=0.56
77. 77.86+0.35
MLP
90.21
86.6124.30
87.1244.05
82.54e1.09
84.9820.82
81.3850.666
93. 93.4240.32
90.08:0.69
95.03=0.18
90.01
(04-4462.) MiB versus 384-3360 MiB for NSD (i.e., typically a 5-35% increase). Importantly, both methods share the same asymptotic complexity, as shown Appendix B
Table 4: Training and testing time (s), averaged over 10 folds and 1,000 epochs.
Model
Toxas
Wiscomsin
Film
Squirrel
Chameleon
Corsll
Citeseer
Pohmed
Com
Roman- Empire
Questions
Telegram
DSNN (p
0.25.55
8 8:23
8. 8.30s0.13
19 19.2000.08
25, 19.0..11
9.7580.18
8,27a0 9 9.9660.1 10 33, 15
9. 9.10:0.21
28, 4360.066
106 99.0.99
9.58 0.32
DSNN (g
8.23a0:24
8.2460.10 10
19 19.2540.08
08
25 22
9.7760. 12
8.3300 19
91100.1
33, 14:0.5
9 9.10a0.18 18
28 8400.00
106. 50.0.06
9 0.24
DSNN (Q-0)
8.20
8.4250177
19.20s0:09
25.7920:11
9.9120.22
8,30.0.17
9.72211.10
33 33.22:007
9.0120.19
28.22
107 107.1020..3
9.46 0.22
NSD
6.5920.15
6,6100100
10.7250.14
14.4320122
7.79)0.24
6.39.0.10
7.30g0.117
16.320008
7.29
14.1520.084
47.51a0.12
7.69 a 0.35
Table Peak GPU memory (MiB).
Model
Texas
Wisconsin
Film
Squirrel Chameleon Comell Citeseer Pubmed Cora Romun- Empiie Questions
Telegram
DSNN
404
408
1078
1362
598 554
404
830
1744
606
1100
4462
414
NSD
384
400
920
1180
384
588
1338
522
730
3360
390
6 CONCLUSIONS
We introduced the Directed Cellular Sheaf, from which we derived the Directed Sheaf Laplacian L²= By encoding the edge direction in its imaginary components, F carries directional inductive bias thanks to which we obtain a convolution operator implementing a message-passing scheme capable of handling asymmetric interactions. We embedded such an operator in the Directed Sheaf Neural Network (DSNN). Our theoretical results showed that DSNN generalizes several well-established graph-leaming models, including NSD, MagNet. and SigMaNet. Empirically. DSNN exhibits strong performance across both real-world and synthetic datasets, consistently outperforming both traditional GNNs and SNNs. This demonstrates that DSNN's explicit treatment of directionality leads to superior generalization, particularly in heterophilic graph settings. Future works include extending our approach to temporal graphs to broaden applicability to dynamic settings.
ACKNOWLEDGEMENT OF SUPPORT
Stefano Fiorini's work was carried out during research visit at the University of Cambridge, supported by the ELSA - European Lighthouse on Secure and Safe AI project, and funded by the European Union under grant agreement No. 101070617. Stefano Coniglio's work was partially supported by the European Union under Next Generation EU - the Italian National Recovery and Resilience Plan (PNRR), PRIN 2022 PNRR (project code P20227CTY3, CUP D53D2301 8800001), project title "HEXAGON: Highly specil-spee EXact Algorithms for Grid Operations at the National level".
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Published as a conference paper at ICLR 2026
ETHICS STATEMENT
We adhere to the ICLR Code of Ethics. This work studies sheaf-based learning on graphs and does not involve human subjects, personally identifiable information, sensitive attributes. The real-world benchmarks we use are standard public datasets (with licenses referenced in Appendix B); our synthetic graphs are generated procedurally as described in Section and Appendix [P We release a code repository under permissive license to facilitate verification and reuse.
As with any concerns around privacy, fairness, or surveillance. Our contribution is methodological and evaluated on public or synthetic data; nevertheless, we encourage practitioners to assess nomain-specine risks, follow applicable regulations, and adopt appropriate safeguards (e.g., data minimization, bias checks) when deploying such models.
technique, downstream applications juman-centered data could raise
REPRODUCIBILITY STATEMENT
We took several steps support reproducibility. All model components, including the Directed
Sheaf Laplacian 1,F, training objectives, and update rules, are fully specified in the main text, with
additional implementation details in Appendices E and G Dataset sources, preprocessing, and
synthetic graph generation (parameter grids for Ougj Bij. and q) are documented in Appendix F We
report splits, evaluation protocols, and hyperparameter search spaces Section 5 and Appendix G and we include hardware information and training schedules.
Code and scripts to reproduce all tables (including random seeds and configuration files) accompanies this submission (Appendix BB. Note that exact bitwise determinism can depend on backend/library settings (e.g., CUDA), but we fix seeds and document any sources of nondeterminism.
LLM USAGE STATEMENT
We did not use large language models (LLMs) for deriving, checking, or producing any proofs or theoretical results in this paper. All theorems and proof's were conceived, implemented, and validated by the authors.
LLMs were used only as general-purposs assistants for: (i) light prose edits (clarity/grammar) and (ii) minor иÓтх refactoring (e.g., formatting environments). All such edits were manually reviewed. No human-subject data, personally identifiable information, or proprietary datasets were provided to any ILM,, and all experimental code runs independently of L.LM services.
REFERENCES
Malbor Asllani, Renaud Lambiotte, and Timoteo Carletti. Structure and dynamical behavior of non-normal networks. Science advances, 4(1 4(12)eeaau9403, 2018.
Theodor-Adrian Badea and Bogdan Dumitrescu. Haar-laplacian for directed graphs. IEEE Transac- fions on Signal and Information Processing over Networks, 2025.
Federico Barbero, Cristian Bodnar, Haitz Sáez de Ocáriz Borde, and Pietro Lio. Sheaf attention networks. In NeurIPS 2022 Workshop on Symmetry and Geometry in Neural Representations, 2022.
Ginestra Bianconi, Natali Gulbahce, and Adilson E Motter. Local structure of directed networks. Physical review letters, 100(11): 18701, 2008.
Deyu Bo, Xiao Wang. Chuan Shi, and Huawei Shen. Beyond low-frequency information in graph convolutional networks. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pp. 3950-3957, 2021.
Cristian Bodnar, Francesco Di Giovanni, Benjamin Chamberlain, Pietro Lio. and Michael Bronstein. Neural sheaf diffusion: A topological perspective on heterophily and oversmoothing gnns. Advances in Neural Information Processing Systems, 35:18527-18544 2022.
11

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Published as a conference paper at ICLR 2026
Alexandre Bovet and Peter Grindrod. The activity of the far right on Telegram. ResearchGate preprint, DOI: 10.51140RRG22.677000566611111111 2020.
Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. International conference on machine learning, pp. 2020.
PMLR,
Eli Chien, Jianhao Peng, Pan Li, and Olgica Milenkovic. Adaptive universal generalized pagerank graph neural network. International Conference on Learning Representations, 2021.
Fan RK Chung. Spectral graph theory, volume 92. American Mathematical Soc., 1997.
Justin Michael Curry. Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania, 2014. URL
3255
Iulia Duta, Giulia Cassarà, Fabrizio Silvestri, and Pietro Liò. Sheaf hypergraph networks. Advances in Neural Information Processing Systemx, 36: 2087-2297,,,,, 2023.
Stefano Fiorini, Stefano Coniglio, Michele Ciavotta, and Enza Messina. Sigmanet: One laplacian to rule them all. In Proceedings of the AAAI Conference Artificial Intelligence, pp. 7568-7576, 2023.
Stefano Fiorini, Stefano Coniglio, Michele Ciavotta, and Alessio Del Bue. Let there be direction in hypergraph neural networks. Transactions Machine Learning Research, 2024. ISSN 2835-8856. URL httptttss///peeeeee///
net www.eeeet/oooe forum? 48R 6pmvi
Edouard Grave, Piotr Bojanowski, Prakhar Gupta, Armand Joulin, and Tomas Mikolov. Learning word vectors for 157 languages. arXiv preprint arXiv:1802.06893, 2018.
Jakob Hansen and Thomas Gebhart. Sheaf neural networks. arXiv preprint arXiv:20/2.06333, 2020.
Jakob Hansen and Robert Ghrist. Toward spectral theory of cellular sheaves. Journal of Applied and Computational Topology, 3(4):315-358, 2019.
Uzi Harush and Baruch Barzel. Dynamic patterns of information flow in complex networks. Nature communications, 8(1):2181, 2017.
Yixuan He, Gesine Reinert, and Mihai Cucuringu. Digrac: Digraph clustering based on flow imbalance. In Learning on Graphs Conference, pp. 21-1. PMLR, 2022.
Thomas. N. Kipf and Max Welling, Semi-supervised classification with graph convolutional networks. In 5th International Conference on Learning Representations, ICLR 2017 Conference Track Proceedings, 2017.
Yujia Li, Richard Zemel, Mare Brockschmidt, and Daniel Tarlow. Gated graph sequence neural networks. In Proceedings of ICLR' 16, April 2016.
Elliott H Lieb and Michael Loss. Fluxes, Laplacians, and Kasteleyn's theorem. In Statistical Mechanics, pp. 457-483. Springer, 1993.
Derek Lim. Felix Hohne, Xiuyu Li, Sijia Linda Huang, Vaishnavi Gupta, Omkar Bhalerao, and Ser Nam Lim. Large scale learning on non-homophilous graphs: New benchmarks and strong simple methods. Advances in neural information processing systems, 34:20887-20902 2021.
Kangkang Lu, Yanhua Yu. Zhiyong Huang, Jia Li, Yuling Wang, Meiyu Liang, Xiting Qin, Yimeng Ren, Tat-Seng Chua, and Xidian Wang. Addressing heterogeneity and heterophily in graphs: A heterogeneous heterophilic spectral graph neural network. CoRR, 2024.
Hongbin Pei, Bingzhe Wei, Kevin graph convolutional networks. arXiv preprint arXiv:2002.05222 2020.
Chang, Yu Lei, and Bo Yang. Geom-gen: Geometric
Oleg Platonov, Denis Kuznedelev, Michael Diskin, Artem Babenko, and Liudmila Prokhorenkova. A critical look at the evaluation gnns under heterophily: Are we really making progress? In The Eleventh International Conference Learning Representations, 2023.
12

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Published as a conference paper at ICLR 2026
Emanuele Rossi, Bertrand Charpentier, Francesco Di Giovanni, Fabrizio Frasca, Stephan Günnemann, and Michael M Bronstein. Edge directionality improves learning on heterophilic graphs. In Learning on graphs conference, pp. 25-1. PMLR, 2024.
Allen Dudley Shepard. A Cellular Description of the Derived Category of a Stratified Space. PhD thesis, Brown University, 1985. URL https:t///// /www. math. www.ooollssoce..ooooooeeoo ~akp/ siles/shepard.ppe
Zekun Tong, Yuxuan Liang, Changsheng Sun, Xinke Li, David Rosenblum, and Andrew Lim.
Digraph inception convolutional networks. Advances in Neural Information Processing Systems, 2000-Decembrr(Neer/Pyy---111
2020a. ISSN 10495258.
Zekun Tong, Yuxuan Liang, Changsheng Sun, David S. Rosenblum, and Andrew Lim. Directed graph convolutional network, 2020b.
Zekun Tong. Yuxuan Liang, Henghui Ding, Yongxing Dai, Xinke Li, and Changhu Wang. Directed graph contrastive learning. Advances in Neural Information Processing Systems, 34: 14:19580-19593, 2021.
Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks. In International Conference on Learning Representations, 2018.
Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay Sarma, Michael M Bronstein, and Justin M Solomon. Dynamic graph enn for learning point clouds. Acm Transactions On Graphics (1og),38(55) 3-15):1-12, 2019.
Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and S Yu Philip. A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems, 32(1):4-24 2020.
Dong Xu, Ziye Liu, Fengming Li, and Yulong Meng. Complex graph neural networks for multi-hop propagation. Neurocomputing, pp. 130364, 2025.
Yujun Yan, Milad Hashemi, Kevin Swersky, Yaoqing Yang, and Danai Koutra. Two sides of the same coin: Heterophily and oversmoothing graph convolutional neural networks. In 2022 IEEE International Conference on Data Mining (ICDM), pp. 1287-1292. IEEE, 2022.
Olga Zaghen, Antonio Longa, Steve Azzolin, Lev Telyatnikov, Andrea Passerini, et al. Sheaf diffusion goes nonlinear: Enhancing gnns with adaptive sheaf laplacians. PROCEEDINGS OF MACHINE LEARNING RESEARCH, 2024.
Jie Zhang, Bo Hui, Po-Wei Harn, Min-Te Sun, and Wei-Shinn Ku. smge: A complex-valued graph convolutional network via magnetic laplacian for directed graphs, 2021a.
Xitong Zhang, Yixuan He, Nathan Brugnone, Michael Perlmutter, and Matthew Hirn. Magnet: A neural network for directed graphs, 2021b.
Jiong Zhu, Yujun Yan, Lingxiao Zhao, Mark Heimann, Leman Akoglu, and Danai Koutra. Beyond homophily in graph neural networks: Current limitations and effective designs. Advances in neural information processing systems, 33:7793-7804, 2020.
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A
NOTATION SUMMARY
Table lists the notation used in the main paper.
Table 6: Notation
Symbol
Meaning
G = (V,
Directed graph with vertex set V, edge set E
n VV. |E]
Number of vertices and edges
(0, 1] j
Adjacency matrix of G Symmetrized adjacency matrix Imaginary unit (i² -1)
A, (A+ + AT)
i
Th) -
- -A- ))
matrix (Magnetic term) Directionality parameter
QER
F(u) E Cd
Stalk (vector space) at vertex w. Stalk at edge e
F(e) E Cd
Fuge
Restriction map from node u to edge
Fige
Real base map (before applying T())
CG Dvev Flu)
Space of O-cochains
(GGFF) DrEF F(e)
Space of 1-cochains
-
Directed coboundary operator
((z), Fudetu Fugeety
Action on cochain æ
L =
Directed Sheaf Laplacian (DSL)
L D 112 L D-1122
Normalized DSL Degree block for node Block degree matrix
D. = Ee E(E Fi TigeFuse Fuge
D diag( Di..... Dn)
T(u)
Set of edges incident to (ignoring direction) Sets of undirected and directed edges
EL
X C
Node feature cochain at layer t Learnable weight matrices at layer Learned scaling of stalk components Complex ReLU activation
w w((t)
1-
o.) unwind(2)
Concatenation real and imaginary parts of
Complex incidence matrix (for trivial sheaf)
((a)
Magnetic Laplacian
L
Sign-Magnetic Laplacian
B CODE REPOSITORY AND LICENSING
The code written for this research work is available at htttppp:///ttthbb..m////aaaaeet DSNN and freely distributed under the Apache 2.0 license!"
The Cora, Citeseer, and PubMed datasets are available at https::/liins.oor datasets//citeseee PubMed, and Cora are sourced from (Yan al. 2022). The Squi rre Chame Roman Empire and Quest i ons datasets come from Platonov et al. (023).
Telegram provided by Bovet Grindrod 2020))
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Regarding the models used in this paper, we rely publicly available implementations from the following sources:
MLP, GCN, GAT, GGCN, GCNII, GPRGNN: bttps:/github.com
with MIT license.
H2GCN: [att:ppiileecc...ooooea/eeeaaaaa Hatp:///tteeemaaaaa//2C
/ http:://gitibb.cmmmoooffffhoonn
FAGCN:
M | with Apache License
2.0.
http://ffttss.///tt...cc///sssss with Apache License
2.0.
DirGNN: di - -network with Apache
License 2.0.
httpss///tttuuuccc////tteeeeeeer peuuzal-Sieeaa-daafaaaaaaiinnnn Apache License 2.0.
NSD:
HaarNet: http:////ttth/oo/hhhhhaaaaaaaaaa main
Cktpatttee///thhao.oo/////////a2 with MIT license.
H2SGNN: http:///sss/a/aaaooo/////aaaaa/a
DiGCL: httpt//t/tssaaaaaaococ/////////a with MIT license.
C
DERIVATION OF THE EQUATION OF L
Since, by construction, _ 5, the following equation holds:
1²(x) = Σ
uge i Fa
(F
(F Tigyy (F Fughtt
eTT(u)
(M,MEE
self-lrop
inflow
FT
50
T(9)
Σ
FP F
u usee
c=
auillow
undirected
For a directed graph G, the uu component of L
can be rewritten as follows:
1 L
Σ Fi
Fuge
Σ Fr
Fuge
uns
rET(w)
e (M,))E
Σ
(
Σ
Fr use.
uge
(W,)EE
Equation by
is obtained by combining this equation with the previous one and factoring each summation
(F See
D PROOFS OF OUR THEOREMS
Theorem 1. L is Hermitian and 3 > O, and the same holds for L
Proof. By definition, we have I = 5. Therefore, for any pair indices v E V, I.F
T0 = 35 T.
and = 52,5am Since this implies = (I,²a.) for all of its entries u, U, conclude that the
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matrix is Hermitian. It follows that the spectrum of L. is real. By following essentially the same arguments using ŌD 2, one can show that the spectrum of Ly is real as well.
Since, again by definition, L 8 8, its associated quadratic form 8 ox (with E C)) can be rewritten as x* 00 (6x)² (ōx) |0x: Since | is norm, || 2 0 holds for all C, thus implying L = 0 for all x= E C. Thus, L = 0. By following the same arguments using 8' =
= 8D- &, one show that T N well.
Theorem 2. FN 2I.
Proof. Let QN :- -- for the case where Ō has not been given an arbitrary orientation (this is in line with the classical construction of the Signless Laplacian for undirected unweighted graphs). Since Q is the product of a matrix and its conjugate, we have Q ž 0. It is easy to show that Q 2T T. From this, we deduce:
211
This shows that not only L has nonnegative spectrum, but also that its spectrum upper-bounded by 2.
Theorem 3. If is undirected, T coincides with the classical sheaf Laplacian T.F for any choice of E R. Also, if the sheaf trivial and is undirected and unweighted, Li coincides with the classical graph Laplacian L. If G is directed and we set q = O, L coincides with the classical sheaf Laplacian associated with the undirected version of G.
Proof. Part 1. If G is undirected, all restriction maps of the Directed Cellular Sheaf are real for every choice of RR- because, for all u, E V. A implies R(T()) and ((((2)) 0 for any choice g. This implies (FFwe) = 0 for all e E E' where u is one of its endpoints; therefore, L is real valued and _ LT.
Part 2. Under the same assumptions on G, the Directed Cellular Sheaf is trivial d and E with being one if its endpoints. Thus, 1 -1 if (u, o) E E and
Fage = for all edges 0 otherwise, while - = lfee E E = fu, w} }/; by definition, it follows that L coincides with the classical Laplacian matrix L = D - A with A E {0, 10,
Part 3. Setting g = leads to, for all u, u V, T = cos(0) + isin(0) = 1. Thus, L coincides with the Directed Sheaf Laplacian = associated with the undirected version of G which is obtained from it by preserving each of its edges and making all of them undirected-this coincides with discarding 3(F = 0 for all e E where is one of its endpoints.
Theorem 4. Letting G be a directed graph with unit weights, the Directed Sheaf Laplacian I associated with Trivial Directed Cellular Sheaf coincides with the Magnetic Laplacian L(9), In the special case where q 1. L also coincides with the Sign-Magnetic Laplacian L".
Proof. Part 1. First, we show that, when adopting a Trivial Directed Cellular Sheaf for a directed graph G with unit weights, we have:
_ (((g)
u, u. V #0
- [[u)]
V.
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Eq.
and read:
Ft uuga DSc - FO T())
use ugge
if e (u, 0)
L
Ft
ude Fuee -FFo U Tilii FFFeee
if c = (t u)
Fi
ugle use us Uuee
--FT
if {u, v) otherwise
0
uu
Σ
F
use
F
uee.
(ET(u)
When considering a Trivial Directed Cellular Sheaf, we have
Fusle = -Fage = 1 fu, v) E E and, thus, L = - -T(9)
(th latter is because
Auv Avu implies T:l cos (0) isin(0) 1).
Fusee = and Fuge =(1) e = (u, v) and, thus, L -T
Fuge -T(1) and Foge = 1 if = (v, u) E E and, thus, L - -TTI = =-T(9))
Each diagonal term
Lig of L reads
Σ
Σ
F uage uge use
usee use
eET(u)
em (E.
Σ
(TD Fugee 2 (TTla)) Fuge T(g)
Σ
ft Fugerus use
e (O.))E
[u.
=(T(f') (T!?)
=[](u)),
where ((()))((TT)))))))))))) = holds since T( With this, Part shown.
Part 2.
The Magnetic Laplacian reads
(1) D. (1),, with ((0) = A, @ exp (12mg (A A))),
with A, A+A and D. diag(1,,A)..
2
By definition we gave of (g(g) for component with u, V, we have:
(4) = D Drav H Dane
Asue T(1) uv
Part 2a. Let's assume G undirected. such a case, we have we have As.. and A As.. otherwise. This implies D [m]).. Thus, have:
whenever {u, E E
L A Tri)) _T(9) 1
u, U.EV V +
I uuu Da... A T(9) Da [T(1)
T
F
V.
where the last equation holds since T 0 for any q. Thus, I I(w).
Part 2b. Let's assume G directed without digons. In such a case, we have A whenever either
(u, v) E or (v,w) E and As... 0 otherwise. This implies D (u)) Thus, we have:
I (g) As,o T(g)
1 1(())
F
u. V
2
L (4) = -Dans A Tig)
D
1
T(u)
Luu
3
E V,
2
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where the last equation holds since Tid 0 for any q. Thus, L =2 Notice that the scaling factor is immaterial when the Laplacian matrix is embedded a GCN/SNN, as it is directly subsumed by either W1 or W2 in equation (only by the latter in a GCN, where Wi is not present).
Part 3. Since, shown in Fiorini al. 2003), L(a) and L coincide with q 4. the last part of the claim follows directly from Parts 2a and 2b.
Theorem Let G be a directed graph with unit weights. Assuming Trivial Directed Cellular Sheaf. the conjugate transpose 8* of the Directed Coboundary Operator boils down to the complex-valued node-to-edge incidence matrix B E Cu defined for an edge E E incident to a vertex u.
Buc -
1
{u, 4 with if = {u,v] with u > v
if (u, v)
-T(9)
ife (v,
follows that L(1) BB* With q 1 L(D --= BB
Proof. (First, notice the arbitrary orientation that was given to the undirected edges).
From the proof the previous theorem, we know that, if G has unit weights and the Directed Cellular Sheaf is trivial we have:
_ -T(g)
u. u. V. u
_ [(u)l
ueV.
Let's consider (BB*)uc )uo share single edge. Calling it e, we have (BB*) Bux (Bwe) if E E or 0 if they share no edge at all. Let's assume they do, and considering three cases:
EV'E Buer (Bye)) B
Since we are considering graph, u, can only
If = (u, with u < U, Bue and (Bue)* -1 with an arbitrary orientation and, thus, (Bee)
-T(1) (this is correct since, shown before, 7 is always equal to Auv = Avu).
-1 =1
Ife (u, with u >> v, Buc - and (Bre)* with arbitrary orientation and, thus, Bue (Bue)* = -1 =T -T((g) (as shown before, the latter is always equal to if Anv Agu).
If v), Buc 1 and (Boe)* = (-T(2) )r and, thus, Bue (Bee)" = ( (-T11))
-T(9) since ((t)) is Hermitian by construction.
If (v, u). Buc -T()) and (Bee)" and, thus, Bud = -
This shows that, if has unit weights and assuming a Trivial Directed Cellular Sheaf, we have 1, = B The fact that (with a scaling factor of 2, when needed) I = BB* and L B when g = follow from the previous theorem.
E COMPLEXITY OF DSNN
As mentioned in the paper, the complexity of DSNN coincides, asymptotically, with that of NSD. This is because the adoption of complex-valued restriction maps-which are specific to DSNN and not present NSD does not affect the asymptotic inference complexity of DSNN. This because complex-valued synaptic weights, pre-activations, and activations only incur a constant multiplicative overhead (approximately a factor of 4) in the forward pass and, thus, not alter the asymptotic complexity from the real-valued case analysis. To better see this, consider three complex-valued matrices:
A = Ar ++++A iiA,
= -R+iXX +ix,, = =YYR+ +iY,
with
A A E Nn. EXNER'
Yn. E
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satisfying the complex linear equation Y AX. This equation be rewritten purely in the real domain using the lifting transformation:
X X1
YR
Ar Ar
-A AR
X
E
R2nX,,
YR
E
2mxp,
As
E
2m22n
so t = XR holds. Hence, complex-valued operations can be reduced to real-valued opera- tions with constant factor overhead, which is immaterial in the asymptotic complexity.
Experimental Scalability We evaluate the scalability of DSNN relative to NSD by reporting computation time and memory usage in the node classification task. For DSNN, we report results using the most essource-demanding configuration (General), with d 16 hidden channels, and 2 layers. We obtained the results included in the Tables 4 and using single Nvidia RTX A6000 GPU.
F FURTHER DETAILS ON THE DATASETS
Real-world dataset. The Texas, Misconsin, and Cornell datasets are part of the WebKB collection, modeling links between websites from different universities. In these datasets, nodes are labeled as student, project, course, staff, or faculty.
The iill 1m dataset is derived from -il--irector--- director-actor-oee actor- writer network. Each node represents actor, and edges indicate co-occurrence on the same Wikipedia page. Node features correspond to keywords extracted from these Wikipedia pages. The nodes are classified into five categories based on the content of the actors' Wikipedia entries.
The Ci teseer dataset contains 3,312 scientific publications classified into six categories. The citation network includes 4,732 links. Each publication represented by a binary word vector indicating the presence or absence of words from dictionary of 3,703 unique terms.
The PubMed dataset consists of 19,717 scientific publications related to diabetes, categorized into three classes. The citation network contains 44,338 links. Each publication described by a TF-IDF weighted word vector derived from dictionary of 500 unique words.
The Cora a dataset includes 2,708 scientific publications classified into seven classes, with citation network comprising 5,429 links. Each publication is represented by a binary word vector indicating the presence absence of words from dictionary of 1,433 unique terms.
The Squirrel and Chame leon datasets consist of articles from the English Wikipedia (December 2018). Nodes represent articles, and edges represent mutual links between them. Node features indicate the presence of specific nouns in the articles. Nodes are grouped into five categories based on the original regression targets.
Telegram is an influence network that analyses the interactions and influences between distinet groups and actors who associate and propagate political ideologies. This pairwise-influence network between 245 Telegram channels with 8912 links. The labels are generated following the method discussed in (Bovet & Grindrod 2020). with a total of four classes
The Quest ionesiions dataset is derived from the quusstion-answert users, and an edge connects two nodes one user answered the other's 's question within one-year interval (September 2021-Augus 2022). To limit dataset size, Platonov et al. (023) focus users interested in the topic "medicine." " The task is to predict which users remained active (i.e., were not deleted or blocked) by the end of the period. Node features embeddings (Grave et al. (2018) of words in the user description, with an additional binary feature indicating the 15% of users lacking description. The final dataset contains 48.9K nodes with an average degree of 6.28.
platform Yandex Q. Nodes represent
computed as the mean of fast
Table 7 reports the statistics of the datasets used in this paper.
Synthetic dataset.
Following Zhang et al. (0211). generate synthetic graphs using the directed stochastic block model (DSBM) as follows. Let n be the number of nodes and the number of equal-sized communities (Ct,..,, Cc). First, we sample undirected graph by connecting each
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Table 7: Summary of datasets: number of nodes, edges, density (%), and percentage of directed edges.
Dataset Nodes
# Edges
Density (%)
% Directed Edges
Roman-Empire
22,662
32,927
0,006
65.24
Texas
183
295
0.009
89.25
Wisconsin
251
466
0,007
89.11
Film
7,600
26,752
0.046
87.74
Squirrel
2,223
46,998
0.009
90.60
Chameleon
890
8,854
1.120
85.01
Cornell
183
280
0,008
93.50
Telegram
245
8,912
14,900
90.33
Citeseer
3,327
4,676
0.042
98.78
PubMed
18,717
44,327
0,013
99.97
Cora
2,708
5,278
0.072
97.14
Questions
48,921
153,540
0.064
98.97
pair of nodes E C and v E Cj independently with probability 0zj [0, Ogy Oji. where Oi controls intra-community edge density and ony for i i controls inter-community connectivity. To obtain directed graph, we introduce a rule to transform the graph from undirected to directed:
we define a collection of probabilities {8k J where Big E [0. 1]. such that Big + By, = 1. with probability Bij. and v with probability 3yt.
C2 and Cj, orient the edge
The synthetic datasets we used do not exhibit strong homophily, On the contrary, they lack significant homophilic structure, shown by the following table, where their homophily is measured according to both (Pe et al. 2020) and (Lim et al. 2021):
Dataset
Homophily (Pei al. 2020)
Homophily (Lim et al. 2021
Synthetic with ai 0.05
0.33
0.36
Synthetic with ai 0.08
0.23
0.25
Synthetic with aiy = 0.10
0.20
0.20
Table 8: Homophily values for synthetic datasets under different ai settings.
G FURTHER DETAILS ON THE EXPERIMENTS
Hardware. The experiments were conducted on 2 different machines:
1. An Intel(R) Xeon(R) Gold 6326 CPU @ 2.90GHz with 380 GB RAM, equipped with an NVIDIA Ampere A100 40GB.
2. A 12th Gen Intel(R) Core(TM) 19-12900KF CPU 3.20GHz CPU with 64 GB RAM, equipped with NVIDIA RTX 4090 GPU.
3. A Intel(R) Xeon(R) Silver 4210R CPU 2.40GHz with 64 GB RAM, equipped with an NVIDIA A100 80GB.
4. A AMD Ryzen Threadripper PRO 5975WX 32-Cores with 256 GB RAM, equipped with an NVIDIA RTX A6000 GB.
Model Settings. We trained every learning model considered this paper for to 1000 epochs with early stops of 200. We adopted learning rate of {1 10 2,2 10 2 10 and employed the optimization algorithm Adam.
We adopted hyperparameter optimization procedure identify the best set of parameters for each model. For every model, we searched for the optimal combination of the following hyperparameters for the link prediction:
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Dropout: (0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9)
Number of layers: (2, 3, 4, 6}
Hidden channels: {8, 16, 32, 64).
For some specific models, we also included additional hyperparameters in the search space:
NSD-comp and NSD: shea __act (elu, tanh, relu); d (2,3, 4, add_ _ True, False); hpd_ E (True, False)
DirGNN: oDIONN E (0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0); jk (cat, max)
MagNet: o E (0.0, 0.05, 0.1, 0.15, 0.2, 0.25)
GCNII: ĀGCNI E [0.0, 0.1, 0.2); À (0.0, 1.0, 1.5]
FAGCN: (0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0)
GGCN: denay
(0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2)
GPRGNN: AGPRONN E (0.0, 0.1. 0.2, 0.3, 0.4. 0.5, 0.6, 0.7, 0.8, 0.9, 1.0); G______PGGNNNNN E [0.0, 0.1, 0.3, 0.4, 0.6, 0.7,0.000.. 0.9.
DSNN: E
E (0.0, 0.05, 0.1, 0.15, 0.2, 0.25), sheaf_ach act felu, tanh, relu), d {2,3,4, 4, 5); 1pd_1p E (True, False]: apd_ E (True, False)
H SENSITIVITY ANALYSIS ON
The values of y for DSNN, alongside those reported by MagNet (Zhagg al. 2021b), summarized in Table The results indicate that both approaches tend to favor either small (close to 0) or large (close 0.25) values for q. This analysis suggests that there is no clear correlation between heterophilyhhomophily and the use of directionality in either DSNN or MagNet. Instead, both models exploit edge directionality adaptively, using when enhances information propagation for the learning task at hand. In particular, the table shows that, in 30 out of 36 computations involving DSNN, setting a strictly positive g leads to a better performance.
Table 9: Best values used for DSNN variants compared to MagNet (Zhang et al. 2021b across different datasets
Roman Empire
Texas
Wisconsin Film Squirrel Chameleon Comell Telegram Citeseer Pabmed Cora Questions
Ding-DSNN
0.1
0
0.1
0.1
0.25
0.1
0.25
0.1
0.25
0.2
0.25
0.2
O(d)-DSNN
0
0.2
0.2
0.25
0
0.1
0
0.2
0.25
0.15
Gen-DSNN
0.15
0.1
0.2
0.2
0.15
0.1
0
0.2
0.25
0.25
0.1
0.05
MagNet
0.20
0.15
0
0.15
0.05
0.15
0.15
0.15
0.2
0.25
Table 10 reports the best values for the direction prediction task. In 27 out of 30 cases, incorporating directionality (i.e., setting q F 0) improves performance, further confirming the benefits explicitly modeling edge orientation.
Table 10: Best q values for the direction prediction task. Incorporating directionality improves performance in most cases
4
Texas
Wisconsin Film Squirrel Chameleon Coreell Citeseer Pubmed
Cora
Questions
Diag-DSNN
0.25
0.25
0.05
0.25
0.05
0.05
0.25
0.15
0.15
O(d)-DSNN
0.25
0
0.1
0.1
0.2
0.10
0.20
0.25
0.05
0.05
Gen-DSNN
0
0.25
0.15
0.1
0.2
0.10
0.10
0.2
0.05
0.25
MagNet 0.20
0.10
0.10
0.15
0.10
0.25
0.15
0.10
0.10
0
Table I reports the results of a sensitivity analysis on q carried out on the synthetic datasets. The table confirms that the best performance is obtained with q 전 0 also this dataset.
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Published as a conference paper at ICLR 2026
Table 11: Sensitivity analysis on q for synthetic datasets. Each entry reports mean and standard deviation accuracy
a/9
0.00
0.10
0.20
0.25
0.05
20,40#1.05
08.00=0.93
98.16+0.77
97.28+0.78
0.08
19.64+1.03
97.10+0.70
96.7420.77
88.78+1.21
0.10
20.1840.76
98.60+0.46
99.14+0.36
98.66+0.35
Avg
20.07:0.95
97.90+0.70
98.01+0.64
94.91=0.78
Table 12: Best d values for each dataset.
Model
Texas
Wisconsin Film Squirel Chameleon Comell Citeseer Pubmed Cora Telegram Questions
Roman- Roman-Empire
Diag
4
5
5 5
5
3
2
5
4
Bundle
Generul
3
5
4
3
3
I SENSITIVITY ANALYSIS ON d
Table 12 reports the best d values selected for each dataset and model variant. While the optimal varies across datasets and models, some clear patterns emerge. The Diag model tends to favor smaller values on smaller datasets (e.g., Wisconsin, Cora), whereas Bundle and General models often require slightly higher d larger or more complex graphs (e.g.. Squirrel, Chameleon). Overall, the most common best d value across all datasets and models is 4, the minimum value of 2 never occurs, and the selected d generally ranges from 3 to This demonstrates that moderate slightly higher values consistently preferred, and that d should be adapted based on both the model variant and dataset characteristics to achieve optimal performance.
We also perform the scalability of the method by selecting different d and calculate the peak GPU memory (MiB). We adopt the same setup as reported Appendix E, corresponding to the most resource-demanding configuration (General) with 16 hidden channels, and 2 layers.
Table 13: Peak GPU usage (MiB) for different values of d.
d
Texas
Comell
Wisconsin
Cora
Citeseer
Puhmed
Film
Squirrel
Chameleon Questions
Roman-Fmpire
2
402
402
406
598
823
1453
926
1103
548
3514
1414
403
403
407
600
826
1573
981
1207
565
3904
1516
4
404
404
408
606
830
1744
1078
1362
598
4462
1646
404
404
409
630
834
1954
1204
1567
636
5173
1812
10
417
417
428
842
952
3667
2203
3253
960
10966
3135
15
437
437
458
1183
1253
6469
3850
6058
1497
20500
5295
Table 13 reports GPU memory usage (in MiB) across all datasets for different values of d. Even for the largest configuration considered (d =d= 15), our method runs on a 24GB GPU for small and medium datasets (e.g., Texas, Cornell, Wisconsin), demonstrating its efficiency and scalability. For larger datasets such as Pubmed, Questions, and Squirrel, memory usage increases with d as expected, yet remains fully manageable, showing that our approach can handle both small and large-scale graphs without exceeding typical hardware limits.
J EXPLICIT CONSTRUCTION AND NUMERIC EXAMPLE OF L FOR = 0.25
J.1 SETUP AND DEFINITIONS
Consider the graph G = (V, E) with node set V = (A, B, C}. We equip G with a directed cellular sheaf F of stalk dimension d = 2. The directionality parameter is q E \ {0) and we define 2nq.
Edges and Adjacency.
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Published as a conference paper at ICLR 2026
C1: directed are A + B. Adjacency: AAB ABA 0.
E2, eg: an undirected connection B - C. modeled as two antiparallel directed arcs B -
C' and eg C B. Adjacency: ABC ACB 1.
B
C3
Figure 2: Visualization of the graph = (V, E) with nodes V A, C. Edge e represents the directed are A B, while edges 62 and 63 form two antiparallel ares between B and C, modeling an undirected connection.
Magnetic Restriction Maps. modulated by magnetic phase:
For edge (u, v), the restriction map at the target node is
Fugg F Tée
with T(s) exp(i0(Auv> Ava)) la,
and we assume the source restriction maps Fuge are the identity la.
Phase Factor Calculation.
Edge A B: AAB ABΛ 1, phase exp(ii) cos(O) + isin(0).
Edge B - C: ABC
Aсв = 0, phase 1.
Matrix Assignments. We define the base maps Fuge using and matrices M1, M2 E x2x2
Edge e1: FAse, 12, FBee, = T12.
Edge e2: Fngm MI, FC = 12.
Edge ex: Foses M2, FBge, I2-
J.2 MATRIX CONSTRUCTION
Directed Coboundary Operator J.
The operator ò El acts on a cochain as (ōx).
Fageta Fosetyp Explicitly:
B
61
T
- T12
02
T2
-
TT
02
8-
62
02
M1
- -1
02
Mi
- T2
02
-Ig
-1
M2
00
- I
M₂
Directed Sheaf Laplacian
F.F.
The Laplacian is defined as I
525, with 8" the conjugate
transpose:
In
02
0g
Ig
-12
02
3
- -
M
J
02
M1
- 12
0g
T
M2
02
12
M2
Block Computation.
The block matrices are:
Lī 12, Lin -T22, Lia = -712
Lig = 212 + MX Mi, 1 -MM] + M2), 16 = - + MT).
. T2 My Mz.
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Explicit Matrix Decomposition.
With cos(O) + si) and M1
(b M2 (1
have
1
0
(os
0
0
0
sin
0
cos (0)
0
0
0
0
sin (
L
cos
())
0
-2
0
sim ())
0
0
0
COS
(
-2
-2
0
sin ())
0
3
-2
1
2
0
Real
Imaginory
This decomposition clearly shows that the imaginary part encodes edge directionality, while the real part encodes the undirected topology.
Setting 9 0.25, the magnetic phase factor becomes T cos(2 + i sin (2 Substituting
this value into the Laplacian yields the explicit numeric form of the Directed Sheaf Laplacian:
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
-1
0
L
0
0
3
1
-2
0
+i
1
0
0
0
0
0
0
0
1
4
-2
-2
0
1
0
0
0
0
0
2
2
3
0
0
0
0
0
O
0
2
0
0
0
Imaginary
The real component encodes the undirected topology, wile the imaginary component captures all directional information introduced by the magnetic phase. This numeric example provides explicit demonstration of how the Directed Sheaf Laplacian L separates directional and undirectional contributions in simple 3-node graph.
K
SPECTRAL COMPARISON
Figure 3 depicts the spectra of the DSNN's Directed Sheaf Laplacian (with d = 1) and of the Magnetic Laplacian on Cornell, Texas, and Cora, for different values the charge parameter g. The figure shows that there substantial differences between the two spectra when the stalk has dimension 1.
L
LEARNABLE
While the parameter was treated as a fixed hyperparameter the main paper, our architecture is compatible with learning q jointly with the model. We trained Gen-DSNN with q learnable scalar using three different initialization values (0.25, 0. 0.1 0). Table reports the mean and standard deviation of the learned g over 10 folds for five datasets, together with the fixed q value that produced the best performance in the paper.
Table 14: Learned values (mean + std)
Dataset
init = 0.25
init g 0.125
init = 0
for best results
Cora
+ 0.008
0 + 0.005
0 0.004
0.10
Wisconsin
0.15 + 0.06
0.11 0.03
0.05 0.06
0.20
Corelll
0.18 + 0.08
0.10 0.09
0.01 + 0.02
0.00
Texas
0.15 = 0.04
0.14 0.06
0.09 0.08
0.10
Telegram
0.19 = 0.07
0.10 + 0.09
0.04 0.05
0.20
We observe that the learned q converges stably across different initializations: for some datasets (e.g.. Cora) it consistently converges close zero, suggesting that directional information is less relevant, while for others (e.g.. Wisconsin, Texas, Telegram) it converges to moderate values that are close to the best fixed choice used in the paper. This indicates that learning is both feasible and meaningful.
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Published as a conference paper at ICLR 2026
372
C010 9e03
CORA
506
12:00
2.00
2000
200
2000
12:0
Figure 3: Spectra of the DSNN's Directed Sheaf Laplacian and of the Magnetic Laplacian on Comell,, Texas, and Cora, for different values of the charge parameter g and d = 1.
25


Reviews:
[{"review_id": "RgXYlHh0Cs", "reviewer": "Reviewer_Xiwg", "summary": "This paper introduces the Directed Cellular Sheaf and corresponding Directed Sheaf Laplacian (DSL) and  Directed Sheaf Neural Network (DSNN), extending Sheaf Neural Networks (SNNs) to directed graphs through complex-valued restriction maps. The authors prove that the DSL is Hermitian, positive-semidefinite, and upper-bounded by 2, and that it generalizes both the classical Sheaf Laplacian and the Magnetic/Sign-Magnetic Laplacians. Empirically, DSNN consistently outperforms GNN and SNN baselines on node-classification and direction-prediction tasks across 12 real-world and several synthetic datasets.", "questions": "1. Could q be made learnable per edge or per graph?\n2. Have you examined the spectral properties or phase distributions of learned complex restriction maps?\n3. Could you comment on DSNN’s scalability?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* Novelty and rigor: The formulation is original and mathematically principled. The theoretical results are sound and clearly proven.\n* Unifying framework: DSNN subsumes NSD, MagNet, and SigMaNet as special cases.\n* Empirical performance: DSNN variants achieve top performance on 10/12 node-classification and 8/10 direction-prediction benchmarks, with large margins on heterophilic and directed graphs.\n* Writing quality: The paper is nice to read.", "weaknesses": "* Empirical scope: Experiments focus on small-to-medium graphs; scalability to large or temporal directed graphs remains untested.\n* Ablations: Only sensitivity to q is analyzed. Additional studies on stalk dimension d or learned restriction-map architectures would strengthen the empirical section.\n* Baselines: Comparison omits some recent direction-aware or heterophilic GNNs (e.g., DiGCL, DPGNN).\n* Minor clarity issues: Dense notation and a few typos (“SNNs approaches”).\n* Limited intuition: The geometric meaning of complex restriction maps and the global parameter q could be discussed more intuitively; why complex numbers versus real skew-symmetric forms?", "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": "fOcvN1yF8V", "reviewer": "Reviewer_47m4", "summary": "This paper introduces Directed Cellular Sheaves and a corresponding Directed Sheaf Laplacian, enabling Sheaf Neural Networks to incorporate edge directionality, a missing capability in current SNNs. The authors prove key spectral properties and show that this formulation recovers classical sheaf Laplacians and magnetic Laplacians as special cases. They further propose DSNN, demonstrating consistent gains on both real-world graphs and synthetic directional SBM settings.", "questions": "1. Is $q$ learned or tuned per dataset? If tuned, how stable is performance across $q$ values?\n\n2. Does DSNN maintain benefits in settings where directional edges are sparse or only weakly informative?\n\n3. How does performance degrade if a portion of edge directions are flipped or randomized?\n\n4. Why the random split is adopted for the node classification instead of the widely-used public splits? Performance on small homophilic and heterophilic benchmarks can vary noticeably with random seeds, so it would be useful to justify this choice and clarify whether public splits are also tested.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "**1. Principled directional sheaf formulation**\n\nThe paper introduces directed cellular sheaves and a corresponding directed sheaf Laplacian, providing the first rigorous sheaf-theoretic framework for directed graphs and addressing a clear limitation of existing SNNs.\n\n**2. Solid theoretical foundation**\n\nThe authors prove Hermiticity, PSD spectrum bounds, and show that the proposed operator recovers classical sheaf Laplacians and magnetic Laplacians as special cases, demonstrating a sound and unifying mathematical design.\n\n**3. Comprehensive experimental results**\n\nAcross both real-world and synthetic benchmarks, the model outperforms existing SNNs and competitive direction-aware GNNs, with especially strong results in heterophilic and direction-dominated settings, validating the benefits of directional sheaf modeling.", "weaknesses": "**1. Limited intuition for the directional mechanism**\n\nWhile the mathematical construction is provided, the paper provides limited high-level insight into **how and why** the complex restriction maps enhance directional information flow in practice. The introduction of the complex phase feels algebraically motivated rather than guided by an intuitive model of directional propagation.\n\n**2. Scope of experimental evaluation**\n\nThe evaluation focuses primarily on small-to-medium-scale datasets. There is no demonstration on larger real-world directed benchmarks (e.g., OGB-ArXiv, and arxiv-year). \n\n**3. Ablations could be deeper**\n\n3.1. The effect of stalk dimension $d$\n\n3.2. Sensitivity to direction sparsity or unreliable edge orientation (i.e., direction noise)\n\n3.3. Effect of learning vs. fixing the phase $q$\n\n**4. Writing clarity**\n\nThe definition and construction of the directed cellular sheaf are mathematically sound but presented in a dense, notation-heavy manner. Adding intuitive explanations, intermediate steps, and conceptual guidance (e.g., how complex phases encode directional flow at a high level) would make the framework more accessible and easier to follow for a broader audience beyond sheaf specialists.", "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": "YmUmK5KLgn", "reviewer": "Reviewer_YHDx", "summary": "The paper introduces the Directed Cellular Sheaf, which incorporates edge orientation into cellular sheafs. Based on the novel introduction, the paper proposes the Directed Sheaf Laplacian that serves as the backbone of the Directed Sheaf Neural Network. Experimental results validate the efficacy of the proposed GNN, with theoretical analysis on the properties of the Directed Cellular Sheaf.", "questions": "What does $\\tilde{F}^0$ mean?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The topics of Sheaf Neural Networks and directed graphs are significant.\n- The proposed Directed Cellular Sheaf admits satisfactory properties.\n- The proposed Directed Sheaf Neural Network seems to work well in experiments.\n- Complexity analysis is provided together with implementation code.", "weaknesses": "- It is not clear what $\\tilde{F}^0$ means from Definition 1 without a clear reference or prior definition.\n- The description of DSBM is not clear and without a reference in the main text. Should refer to [1].\n- Some citation styles are not correct. For example, \\citep should be used for the reference at the end of line 291.\n- Some more works can be considered for comparison, e.g., [2] and [3].\n-  The concept can benefit from some motivating examples of the Laplacian. Also, there is a figure sheaf.png in the anonymous github link that may help with illustration.\n\nReference:\n\n[1] He, Y., Reinert, G., & Cucuringu, M. (2022, December). Digrac: Digraph clustering based on flow imbalance. In Learning on Graphs Conference (pp. 21-1). PMLR.\n[2] Badea, T. A., & Dumitrescu, B. (2025). Haar-Laplacian for directed graphs. IEEE Transactions on Signal and Information Processing over Networks.\n[3] Lin, L., & Gao, J. (2023, June). A magnetic framelet-based convolutional neural network for directed graphs. In ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (pp. 1-5). IEEE.", "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": "OicNXhNKXa", "reviewer": "Reviewer_uTjh", "summary": "This paper extends Sheaf Neural Networks (SNNs) to directed graphs by introducing the Directed Cellular Sheaf and the corresponding Directed Sheaf Laplacian (DSL), which explicitly encode edge orientation through complex-valued, direction-aware restriction maps. Building on this framework, the authors propose the Directed Sheaf Neural Network (DSNN), enabling principled learning on directed and heterophilic graphs. Experiments on synthetic and real-world datasets demonstrate that DSNN effectively captures directional dependencies and outperforms existing SNN and GNN models where edge directionality is crucial.", "questions": "1. The introduction should better articulate why extending SNNs remains valuable in 2025. Specifically, it should discuss the advantages of combining SNNs with GNNs for handling directed graphs and heterophily, and explain why these structural refinements are still meaningful in the era of more unified and general graph learning paradigms.\n\n2. The comparison set is mostly limited to models from 2020–2022, with only one 2024 method included. More recent GNNs addressing heterophily and oversmoothing (from 2024–2025) should be added to strengthen the empirical evaluation and demonstrate DSNN’s relevance against state-of-the-art models.\n\n3. The paper does not include baselines from graph prompting, graph foundation, or pre-trained graph models, which have recently become dominant in node classification and general graph learning. Including such comparisons would position the work more clearly within the modern landscape of graph representation learning.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper introduces a mathematically principled extension of SNNs to directed graphs through the Directed Cellular Sheaf and Directed Sheaf Laplacian.\n\n2. It effectively captures asymmetric and directional relationships while maintaining robustness to heterophily.\n\n3. The experiments demonstrate consistent performance gains on both synthetic and real-world directed graph datasets.", "weaknesses": "1.  Most compared models are from 2020–2022, with only one from 2024. The evaluation lacks more recent direction-aware or topology-based GNNs, which weakens the empirical evidence for DSNN’s claimed advantages.\n\n2. The paper does not clearly justify why Sheaf Neural Networks are the right framework for addressing heterophily or extending to directed graphs. Given the recent shift toward graph foundation models and unified architectures, it remains unclear whether adapting SNNs is the most effective or timely direction, rather than developing more generalizable approaches.", "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)