Cooperative Sheaf Neural Networks
Authors: André Ribeiro
Ana Luiza Tenorio
Juan Belieni
Amauri H Souza
Diego Mesquita
Venue: ICLR 2026 Poster (2026)
Keywords: sheaves
message-passing
graphs
neural networks
Primary area: learning on graphs and other geometries & topologies

Abstract: Sheaf neural networks (SNNs) leverage cellular sheaves to induce flexible diffusion processes on graphs, generalizing the diffusion mechanism of classical graph neural networks. While SNNs have been shown to cope well with heterophilic tasks and alleviate oversmoothing, we show that there is further room for improving sheaf diffusion. More specifically, we argue that SNNs do not allow nodes to independently choose how they cooperate with their neighbors, i.e., whether they convey and/or gather information to/from their neighbors. To address this issue, we first introduce the notion of cellular sheaves over directed graphs and characterize their in- and out-degree Laplacians. We then leverage our construction to propose Cooperative Sheaf Neural Network (CSNN). Additionally, we formally characterize its receptive field and prove that it allows nodes to selectively attend (listen) to arbitrarily far nodes while ignoring all others in their path, which is key to alleviating oversquashing. Our results on synthetic data empirically substantiate our claims, showing that CSNN can handle long-range interactions while avoiding oversquashing. We also show that CSNN performs strongly in heterophilic node classification and long-range graph classification benchmarks.

Published as a conference paper at ICLR 2026
COOPERATIVE SHEAF NEURAL NETWORKS
André Ribeiro
Ana Luiza Tenório Getulio Vargas Foundation ana teg.riofgof... br
Juan Belieni
Getulio Vargas Foundation andre. gu www.guiaaasss..... Egv.br
Getulio Vargas Foundation
Juanbelies lenni 8gma com
Amauri H. Souza
Diego Mesquita
Federal Institute of Ceará, 26 Al
Getulio Vargas Foundation, 26 AI diego. www.mmiuisssgg.....em.....
amauriholandee iholanda@laaee fce. edu. br
ABSTRACT
Sheaf neural networks (SNNs) leverage cellular sheaves to induce flexible diffusion processes on graphs, generalizing the diffusion mechanism classical graph neural networks. While SNNs have been shown to cope well with heterophilic tasks and alleviate oversmoothing, we show that there is further room for improving sheaf diffusion. More specifically, we argue that SNNs do not allow nodes to independently choose how they cooperate with their neighbors, i.e., whether they convey and/or gather information to/from their neighbors. To address this issue, we first introduce the notion of cellular sheaves over directed graphs and characterize their in- and out-degree Laplacians. We then leverage our construction to propose Cooperative Sheaf Neural Network (CSNN). Additionally, we formally characterize its receptive field and prove that it allows nodes to selectively attend (listen) to arbitrarily far nodes while ignoring all others in their path, which is key to alleviating oversquashing. Our results on synthetic data empirically substantiate our claims, showing that CSNN can handle long-rang' interactions while avoiding oversquashing. We also show that CSNN performs strongly in heterophilic node classification and long-range graph classification benchmarks.
INTRODUCTION
Graph neural networks (GNNs) have become the standard models for an array of predictive tasks over networked data, with far-reaching applications in, e.g., physics simulation "Sanchez-Gonzalee
et al. 2020), recommender systems (Ying et al., 2018), and molecular modeling (Duvenaud et al.
Gilmer al.) 2017). Nonetheless, classical GNNs have well-known pitfalls. For instance, they typically struggle on heterophilic tasks (Zhu et al. 20220) - i.e. cases where connected nodes often belong to different classes have dissimilar features. Furthermore, GNNs may also be susceptible oversmoothing (Oono and Suzuki| 2020) and oversquashing (Alon and Yahav) (022I) Oversmoothing occurs when stacking multiple GNN layers yields increasingly similar node representations, whereas oversquashing refers to the loss of information when carrying information through increasingly long paths due to the compression of exponentially growing information into fixed-size vectors.
2015
A recent line of works (e.g., Hansen and Gebhart 2020 Bodnar et al. 2022 Bamberger et al. 22255) which we henceforth refer as Sheaf Neural Networks (SNNs), proposes modeling node interactions using cellular sheaves to achieve principled solution to deal with oversmoothing and heterophilic tasks. cellular sheaf over an undirected graph associates (i) vector spaces F(i) and F(e), known as stalks, each vertex i and each edge e, and (ii) linear map Fiae, known restriction map, to each incident vertex-ed eddee pair 4l e. These mathematical constructs induce a sheaf Laplacian which is governed by the restriction maps and generalizes the conventional Graph Laplacian.
In parallel line of investigation, Finkelshtein et al. 2024) have recently shown that GNNs generally lack the flexibility to allow for nodes to individually select how they cooperate with their neighbors, ie., choose whether they convey and/or gather information to/from their neighbors. This selective communication (also called cooperative behavior) is especially desirable trait tackle oversquash-

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ing, it allows controlling the amount of information flowing between nodes. A natural question ensues: Can sheaf neural networks achieve cooperative behavior?
In this paper, provide negative answer to this question. More precisely, for SNNs to zero out all the incoming information at a node i, they must set Fi = 0 for all incident edges e, which also implies the information flowing from is suppressed (see Figure 2). To circumvent this limitation, introduce the notion of directed cellular sheaves (Definition 3.2) and define their in- and out-degree Laplacians (Definition 3.3). Leveraging these notions, we propose Cooperative Sheaf Neural Networks (CSNNs). Importantly, we show that cooperative behavior can be achieved using only a pair of restriction maps per node, which considerably increases computational efficiency compared to full sheaves - in which the amount of restriction maps increases linearly with the number of edges.
Our theoretical results show that CSNN allows for nodes to selectively listen to other arbitrarily distant nodes, which is desirable trait to alleviate weer-squshinn. Our results on synthetic data specifically designed induce over-squashing (Alon and Yahav (202T) substantiate our claims, showcasing CSNNs' superior potential to handle long-range dependencies. In addition, extensive real-world experiments on 11 node-classification benchmarks and 2 long-range graph-classificttion tasks demonstrate that CSNN typically outperforms existing SNNs and cooperative GNNs.
In summary, our contributions are:
1. We introduce the notions of in- and out-degree Laplacians for cellular sheaves over directed graphs, which can be used to model asymmetric relationships between nodes. We treat undirected edges pair of directed ones and leverage these constructions to propose CSNN - provably extending the flexibility of sheaf diffusion to accommodate cooperative behavior;
2. We provide a theoretical analysis of CSNN, showing that: (a) for each layer t. in CSNN, nodes may be affected by information from nodes at distance up to 2t-hop neighbors (Proposition 4.2). instead of up to t-hop neighbors in usual GNNs; and (b) there exist restriction maps which make the embedding of a node i layer & highly sensitive the initial feature of a node j. where & is also the distance between i and j (Proposition 4.3).
3. We carry an extensive experimental campaign to validate the effectiveness of CSNNs, encom- passing both synthetic and real-world tasks. Our experiments on synthetic data show that CSNNs is remarkably capable of mitigating over-squashing and modeling long-range dependen- cies. Meanwhile, results on over 13 real-world tasks show CSNNs typically outperform prior sheaf-based models and cooperative GNNs.
2 BACKGROUND
For the sake of completeness, here we provide a summary of core concepts concerning cellular sheaves undirected graphs. We also briefly discuss neural sheaf diffusion and cooperative GNNs.
In this work, we denote an undirected graph by a tuple G = (V, E) where V is a set of vertices (or nodes) and E is set of unordered pairs of (distinct) vertices, called edges, with = [V] and m = E We denote the neighbors of a node i in G by N(i) = {j [i,j) E F.J. Next, Definition 2.1 introduces the notion of cellular sheaves over undirected graphs.
Definition 2.1. cellular sheaf (G,F) over a (undirected) graph G (V, E) associates:
1. Vector spaces F(1) to each vertex i E V and F(e) to each edge e e E,, called stalks.
2. Linear maps Fige: F() F(e) to each incident vertex-edge pair i ite, called restriction maps.
Hereafter, we assume all vertex and edge stalks are isomorphic to Rdd. If all restriction maps are equal to the identity map, we say the cellular sheaf is constant. Moreover, d= 1, the sheaf is said trivial.
Given the importance of the Laplacian operator for graph representation learning, it is instrumental to define the Laplacian for undirected cellular sheaves a key concept in the design of SNNs. Towards this end, we introduce in Definition the the spaces of 0- and 1-cochains.
Definition 2.2. The space of O-cochains, denoted by CO, F), and the space of 1-cochains, C' FG, of a cellular sheaf (G,F) are given by
(G, FF ① F(i) and (G, = ! F(e). ieV
(1)
cEE
where + denotes the (external) direct sum.
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Now, for each E E choose orientation i=i- j and consider the coboundary operator 5: (G, - C (G. defined by (6X), = Fge - Figex, Then, the sheaf Laplacian is defined by LF = d I is the trivial sheaf, 8 can be seen as the incidence matrix, recovering the n x n graph Laplacian. A more explicit way to describe the Laplacian is the following:
Definition 2.3. The sheaf Laplacian of cellular sheaf FG,F) is the linear operator L (G, F) that, for a O-cochain (G, outputs
Ly(X);
Σ Fise (FigeX; F)g(x,) LJ
VieV.
The Laplacian LE can also be seen as a positive semidefinite matrix with diagonal blocks Lia Lise Fige and non-diagonal blocks Taj 1.l FFls jge.
To build intuition around Definition 2.1 we may interpret the node stalks F(i) as the space of private opinions held by an individual i, following the perspective of Hansen and Ghrist (202T). For an edge e connecting nodes i and the stalk F(e) corresponds to the public opinions exchanged between them.
That being said, note that ker LF = (X E FF, Figex; = FjseXj Ve = {i.j} E E(G)). This can be understood as the space of public agreement between all pairs of neighboring nodes i and j. Note that i and j can have distinct opinions about the same topic on their respective private opinion spaces F(i) and F(j); however, when they publicly discuss this topic, they may prefer to not manifest their true opinion. Alternatively, since the edge stalks may be different from the node stalks, some topies of the private opinion spaces do not need be discussed all. In both cases, the apparent consensus lies in Ker Ly.
Vector bundles. When restriction maps are orthogonal, we call the sheaf vector bundle. In this case, L=(X);; := jite (x; - (x Figee for any E V. Flat vector bundles are special cases of vector bundles in which we assign an orthogonal map O to each node i and set Fige = O, for all e incident to i. This entails (X), :- Li (x; O, O,x;), for any E V. Note that flat vector bundles only comprise n restriction maps as opposed to 2m maps in general cellular sheaves. Prior works (Bodnar et al., 2022; Bamberger al.
2025) have leveraged these simpler constructions to
propose computationally efficient based neural networks.
Neural Sheaf Diffusion (NSD). Bodnar et al. (2022) introduce NSD building on Euler iterations of X. First, we project the initial node features X into l channels using an MLP n), i.e. Xo = (x) E Rudxh, Then, NSD recursively computes
the heat equation induced by A,, i..., X A
X = (1 + (InxA @ E)) Ox - (Ar WI.)X.We.).
(3)
where 1,xh is an h matrix of ones. AF(e) is the sheaf Laplacian at layer t, e [-1, is (learned) vector scaling the features along each stalk dimension, a is an element-wise non-lincan non-lincarity, and W1. Rdxd W2, hR are weight matrices. Importantly, the restriction maps which govern A AF(1) are learned in an end-to-end fashion, alongside WT, and W2.--
Cooperative GNNs. Finkelshtein al. (2024) recently proposed flexibilizing message-passinn passsng GNNs by treating nodes as players that can choose how they cooperate with their neighbors. More specifically, cooperative GNNs employ an auxiliary GNN, called the action network, that decides individually how each node partakes in the message passing of the base GNN, or environment network. The action network decides whether each node only propagates information (PROPAGATE), only gathers information from neighbors (LI does none (ISOLATE) or does both (STANDARD). Cooperative GNNs learn the action and environment GNNs simultaneously. using the straight-through Gumbel-Soffman estimator to propagate gradients through the discrete actions of the action network.
3 CELLULAR SHEAVES FOR DIRECTED GRAPHS
We kick off this section addressing our initial research question: Can SNNs achieve cooperative behav- ior? Recall that communication between nodes in an SNN is governed by its sheaf Laplacian, which is induced by the restriction maps. Thus, in SNNs, picking state among PROPAGATE, LISTEN, ISCLATE, and STANDARD node i translates to choosing a suitable configuration for its respective restriction maps. The following result says that SNNs cannot fully alternate between these action.
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F(jt)
F(i)
4
F(i)
F(1)
F(e)
F(1)
<
Fltj)
Figure 1: On the left, a cellular sheaf shown for a single edge of an undirected graph with stalks isomorphic to R²a The restriction maps Fide, Fjze move the vector features between these spaces. On the right, the analogous situation for sheaf on a single pair of directed edges. Then there are four, possibly distinct, restriction maps Fighy. Figji, Fog Fjgjs.
Proposition 3.1. Lerie V. Lx(); does not depend on x, for any E neighbor of then L=(X), oor (X), = J
Put plainly, Proposition 3.1 states that the sheaf diffusion provides framework where node that does not I.T STEN (since it does depend on j) must not PROP AGATE (since the update of j does depend on i), independently of the action that j takes. In other words, PROPAGATE implies LISTEN, which means it collapses to ISOLATE (see Figure 2).
We can circumvent this limitation by treating undirected edges as pair of directed ones, creating an additional channel of communication between nodes. To accommodate directed edges (i.e., E S x V), we propose using cellular sheaves over directed graphs.
Cellular sheaves over directed graphs must distinguish the restriction map where i is the source node of an edge from the restriction map where i the target node of an edge. Therefore, we change the edge notation from to ij and jł to make this distinction explicit. See Figure for an illustration.
Definition 3.2. cellular sheaf (G,F) over (directed) graph G (V, F)) associates:
1. Vector spaces F(1) to each vertex i E V and F(ij) to each edge ij e E, called stalks.
2. A linear map Figij F(i) F(ij) for each incident vertex-edge pair 1 ij and linear map
Figh F(i) F(i) for each incident vertex-edge pair i 4 jt. called restriction maps.
For simplicity, again, henceforth assume all node and edge stalks d-dimensional.
We are now left with the task of defining sheaf Laplacians that can be used for information diffusion in directed graphs. For directed graphs, it common to define both in- and out-degree Laplacians (Agaev and Chebotarev 2005). Given a directed graph with possibly asymmetric adjacency matrix A, the out-degree Laplacian is T Dont A and the in-degree Laplacian is I h A, with Di and Dour denoting the diagonal matrices containing in- and out-degree of nodes in V(G). Definition 3.3 below generalizes these notions to sheaves over directed graphs.
Definition 3.3. The out-degree sheaf Laplacian of a cellular sheaf (G,F) is the linear operator L C (G, O (G, F) that, for a O-cochain
(G, outputs
Σ JEN())
(FT Fiai Figiyx FlagFFsjex,)
VieV.
(4)
The in-degree sheaf Laplacian of cellular sheaf (G,F) is the linear operator T. (G, C (G,F) that, for a O-cochain X (G, outputs
(X); = Σ (Fajoiigi Fla
VieV.
(5)
JEN(()
We note that if (G, is the trivial sheaf, then L = (L T and L i
Flat vector bundles over directed graphs. We can also improve the parameter efficiency of cellular sheaves over directed graphs using flat vector bundles. Since the graphs are directed, we need to distinguish between edges with identical endpoints but with different orientations. Thus, for each node i, we assign a source conformal map S, and a target conformal map T,, and set Fagis = S, and Figji T; for all neighbors j of
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4 COOPERATIVE SHEAF NEURAL NETWORKS
In this section, we leverage the sheaf Laplacians in Definition 3.3 to propose Cooperative Sheaf Neural Networks (CSNNs), an SNN which allows nodes to independently to decide how they participate in message diffusion, choosing whether to broadcast their information and/or to listen from the neighbors. To exploit the asymmetric communication induced by sheaves over directed graphs, we convert our input undirected graph into directed one by replacing undirected edges with pair of directed ones.
We design CSNN's diffusion mechanism by composing the out-degree and the transposed in-degree sheaf Laplacians. In practice, we use their normalized versions
/ D L out
and (A = Dm (1)) D
where Din Dow are the block-diagonals of the in and out Laplacians, reseectvelly
define a CSNN layer by augmenting the Euler discretization of our novel heat equation X (Aj)) T Arr X with linear transformations and nonlinear activation function a:
X = (1+ (Iaxk & z)) @ X, - <((4)))) A out F()) (I, & W W,
(6)
where Iuch is an n-by-h matrix of ones, & E [-1,1] dx1 and Wi, E Rdxd and W2, Raxh are learned matrices responsible for mixing node features and channels, respectively.
Efficient implementation. For computational efficiency, we use flat vector bundles to define both the in- and out- degree sheaf Laplacians. More precisely, for each node i, we define source conformal map S; and a target conformal map T, for all neighbor of i. Thus, out-degree sheaf Laplacian simplifies
out (X); EjeN(() (S TTS,
while the transpose of the de-reer sheaf Laplacian is
((L (x));: == Ejen()) (T - STTs,x)))
(8)
Note these matrices have block structure, with diagonals (L = ΣT T. (L ESSS. and remaining blocks (, = (L = T Sj.
We point out that, since conformal maps are of the form S, = Q, and T; Cr. for some orthogonal matrices Qs, R, and scalars Cs., Cr., computing their inverses and normal- izing the Laplacian becomes trivial. this case, block scaling simplifies to a block matrix of scalars time identity and the normalization both numer- ically stable and computationally efficient.
4 o,
S. 1
The first step of each layer consists of comput- ing the conformal maps T, and S.,. We do this through learnable functions St,, = G,, i) and Tie dGG, "(,,X,,),, i), Vi E V. As in prior works on SNNs Bodnar et al. (2022) y [Bamberger et al.)
Figure O creates the effect of isolating the node i. Directed edges allow for LISTEN and PROPAGATE separately.
(0025). we use neural networks learn the restriction maps. In addition, we use Householder reflections (Mhammedi et al. 2017 Obukhov 2021) to compute orthogonal maps and multiply them by learned positive constant for each node.
4.1 ANALYSIS
We now show that CSNN can achieve cooperative behavior, characterize its receptive field, and prove that appropriate conformal maps help CSNNs handle long-range interactions.
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We note that CSNN allows each node i drift from the STANDARD behavior (S,, T, # 0) by zeroing- out its conformal maps. Setting T; * 0 drives to LISTEN. Setting S; * 0 corresponds to PROPAGATE. Finally, S, T. 0 implies ISOLATE.
We highlight the importance of considering the directions: for undirected graphs, there is a single map O for each node where O, 0 could only mean that does not communicate all. In other words, the possible actions are only STANDARD and ISOLATE. We illustrate this in Figure 2
Thus, to show model achieves cooperative behavior, we must ensure the following: (a) if i does not listen, then its update cannot depend on Xj. Vj E j of i; and (b) if has a non-propagating neighbor k; then its update cannot depend on X;- PProposition 4.1 shows that CSNN satisfies these conditions, while Figure 3 iillustrates them and delineates limitations of the NSD model.
Proposition 4.1. If the target map T; zero, then ((L)t) 1 (X)), 0. If the source map S for some neighbor k i then ((L)) T L F (X)); does not depend on Xk-
Moreover, our model has the ability to reach longer distances. In most GNNs, if is the distance between two nodes i and j. then they can only communicate after layers. CSNN enables communication between these nodes after [t/2] layers.
(a)
(b)
(c)
Proposition 4.2. In each layer I, the features of node can be affected by the features of nodes up to 21-hops.
Figure 3: Given graph (a) we illustrate
the consequences of preventing node 3 from
listening. For NSD (b), this means L (X)s must not depend on Xj, for j = 1. 4 implying Fige not depending on X3. preventing node 3 from propagating information. In CSNN
We also show in Proposition 4.3) that CSNNs with layers are capable of making i and communicate while ignoring all the other nodes on a path from i to j such that | - il & I. This feature is an asset to handle over-squashing in long range tasks, allowing CSNN to selectively tend to information from distant nodes. Example 4.4 illustrates this result in a four-node graph.
and leading to (X)(X)
(c), we can set T3 = O. Provided S3 # O,
outbound communication is possible.
Proposition 4.3. Let i and j be nodes distance t. In CSNN, i can learn ignore all the 1 nodes in the shortest from i joj while receiving the information from j in the l-layer. Moreover, if we choose a path with the (n + 1)-layer.
nodes between and j. then receives the information from j in
Example 4.4. Consider a directed graph with vertex set V = (1,2,3444 and edge set E' = (((12),
(2, 22, (3, (3.4),(C (4,3), (3, 2), (2, 1)}. We follow the proof Proposition 4.3 show we can propagate a message from node 4 to node 1, while the latter ignores all remaining nodes. We denote the target/source map of node at layer t by Ti, and S.,, respectively. To achieve our desired result:
a. In the first layer, must have zero source and target maps except for T3.1 and S4,1- A simple verification gives that ((L')) T (X)) = for all k # 3. So ((L)) " (X)) 0, for all le + If 3, then ((I4)) T (X))s = S4,xx
(0) with x denoting the feature
vector of k at layer t, thus x will be the only feature vector to influence x,. x.,
b. In the second layer, we must have that all source and target maps are zero except for T2,2 and Sa.2. Then ((D))) (X)) 0, for all k # 2 and ((L))) Lo (X))2 -TT S
Thus (1) will be the only feature vector to influence x2
c. In the third layer, all source and target maps must be zero except for Ti,3 and Sa.3. Then
((L)) ( (X))x 0, for all k + and ((L)) T L out (X)), = -T... (3) Thus x is influenced only by x
(3)
(2) which was influenced only by x3
(1) which is influeceed only by x
Consequently, x(3) is affected by (0) while ignoring the features of all other nodes in all other layers.
This configuration shows that although CSNN can achieve 2-hop neighbors, it can also refrain from this behavior to only access 1-hop neighbors per layer. Moreover, this flexibility indicates there are multiple forms to stablish communication between two distant nodes while ignoring others.
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S
T3
S
T3
S;
T2
2
3
4
T:
S2 T2
S3
T3
S1
Figure 4: Illustration of Example 4.4 At layer t, we consider that all maps but Ta and S4 (t-1),
are O, enabling the flow of information from right to left following the bottom edges.
Observe that the derivative of xi (3) in relation to (0) can be high as the values of the non-zero Ti and S; permit. This shows our model can mitigate over-squashing. which refers to the failure of an information propagating (2022), studied over-squashing in message passing neural networks through a bound on the Jacobian (x(") /ox!") c' Aij, where is the layer, e a constant that depends on the architecture of the model,
distance nodes. Di Giovanni al.. (2023) and [Topping al.
and Á the normalized adjacency matrix. Moreover, over over- squashing occurs when we have small derivative ex /xx"/xx(") /ax("), since it means that after layers, the feature at is mostly insensitive to the information initially contained at j. i.e., the information was not propagated properly. [Proposition 4.3) states that the feature at node can be sensitive to the information initially contained at node independently of the distance, given enough layers and an appropriate configuration of restriction maps. Consequently, Proposition 4.3) suggests that the restriction maps regulating the sensibility between distant nodes can provide higher upper bounds to ox(") ox(*/ax fax, while decreasing the value of ax!") lax for other nodes k.
RELATED WORKS
Cooperative GNNs. Finkelshtein et al.] (2024) were the first to propose flexibilizing message passing by allowing nodes to choose how they cooperate with each other. Each layer of their model, CO-GNN, employs an additional GNN that chooses an action for each node. While CO-GNNs can be employed with arbitrary base and action networks, their main caveat is that training can become increasingly difficult as these networks become more complex - the grid of hyper-parameter grows considerably and the stochastic nature of the action network may affect model selection. Different from CO-GNN, our CSNN does not rely on discrete actions and can smoothly modulate between cooperative behavior patterns.
Sheaf Neural Networks. Besides works on SNNs for graph data with real-valued node features, recent works have expanded the literature accommodate heterogeneous edge types (Braithwaitel et al. 2024). hypergraphs (Duta et al.) (2023), nonlinear Sheaf Laplacians (Zaghen et al. (2024), and node features living on Riemann manifolds (Battiloro al.) 2024p. While recent works on SNNs learn restriction maps in an end-to-end fashion, there are also prior works in which they are manually constructed (Hansen and Ghrist 2019) or computed as a pre-processing step (Barbero al.) (022).
Quiver Laplacians, Sumray et al. (2024) propose sheaf Laplacians over quivers (directed graphs w/ self-loops) to improve feature selection on tabular data, with no learning component involved. The in- and out-I Lut-Laplacians we defined here are not particular cases of the Laplacians over quivers. The former are positive semi-definite matrices, while our Laplacians may have complex eigenvalues with negative real parts.
6 EXPERIMENTS
We provide both synthetic and real-world experiments to evaluate the performance of CSNN, including node- and graph-level prediction tasks. [Section 6.1 assess CSNN's capacity to cir- cumvent over-squashing using the MeighborsMatch benchmark proposed by Alon and Yahav (2021) Section 6.2 presents experiments on eleven node classification tasks, showcasing the effectiveness of CSNN for heterophilic graphs. Finally, [Section 6.3 consider the Peptides datasets from the Long Range Graph Benchmark (Dwivedi et al., 2022) to substantiate the capability of model mitigate under-reaching and over-squashing real-world graph

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classification. We also provide additional experiments in Appendix B Our code can be found at http:s//DEhhaccccc//aaarrrr222op Sheaf-NN
6.1 OVER-SQUASHING
In order to verify our theoretical results the capacity of CSNN to alleviate over-squashing. we reproduce the NeighborsMatch problem proposed by Alon and Yahav (202T), using the same framework. The datasets consist of binary trees of depth r, with the root node the larget, the leaves containing its possible labels, and the leaf with the same number of neighbors as the target node containing its true label. We provide the parameters used for this task in Appendix C
2019) fail to fit the datasets Figure 5 shows GCN (Kipf and Welling 2017) and GIN (Xu et al.
starting from 4 and GAT (Veličković et al., 2018) and GGNN (Li et al. 2016) fail fit the datasets starting from r = 5. These models suffer from over-squashing and are not able to distinguish between different training examples, while the CSNN model reaches perfect accuracy for all tested r.
Alon and Yahav (2021) argue that the difference in performance for the GNNs are related to how node features are updated: on one hand, GCN and GIN aggregate all neighbor information before combining it with the representation of the target node, forcing them to compress all incoming information into single vector. On the other hand, GAT uses attention to selectively weigh messages based the representation of the target node, allowing it (to some degree) filter out irrelevant edges and only compress information from a subset of the neighbors. So models like GAT (and GGNN) that compress less information per step can handle higher better than GCN and GIN.
This experiment shows that CSNN is more efficient in ignoring irrelevant nodes and can avoid loosing relevant information. Moreover, Proposition 4.3 provides theoretical support for this resuli, as it states that there are choices of parameters for which the model can listen only to the nodes along path between distant nodes i and j. enabling selective communication to diminish noise impact.
GIN
Comparison against other sheaf models. No- tably, CSNN outperforms other sheaf methods this task. BuNN reports 100% accuracy until 6. Then it drops to 71% and 42% for
(Dhe prodken nalinl
Figure 5: Accuracy for increasing tree depths in the NeighborsMatch task. CSNN consistently achieves 100% accuracy for all values of r.
and r = 8, respectively, as reported in Bambeeggrr et al. (2025). For NSD with orthogonal maps, we obtained 100% accuracy when 2, 91% for 3, and then sharp drop to 5% when 4.
6.2 NODE CLASSIFICATION
Datasets. We evaluate our model on the five recently proposed heterophilic graphs from Platonov et al. (2023), and also on six classical ones for which benchmarking results can be found in Pei et al. (2020); Rozemberczki et al. (2021): Tang et al. (2009). As pointed out in Platonov et al. (202)). the datasets Squirrel and Chameleon have many duplicate nodes, which may lead to data leakage. Following their guidelines, we use their cleaned version of these datasets to ensure meaningful evaluation. For binary classification datasets, we report AUROC, while for multiclass datasets we report accuracy.
Experimental setting, As baselines for benchmarks in Platonov et al. (2023). we use GCN (Kipf and Welling 2017), GraphSAGE (Hamilton al., (017). GAT (Veličković et al. 2018) and GT (Shi et a (021). together with the variations GAT-sep and GT-sep, which concatenate the representation of a node to the mean its neighbors instead of summing them (Zhu et al., 2020). These are all classical baselines used in Platonov et al. (2023) to compare against GNN architectures specifically developed for heterophilic settings, and that achieve the best performance in most cases. We also compare CSNN against recent models such as CO-GNN, NSD, and BuNN (Bamberger et al., al.] 2025). Results for BuNN on Squirrel, Chameleon, and the datasets in Table 2 are not available, since we do not have access their code.
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Table 1: Performance comparison on datasets from Platonov al. (2023). AUROC is reported for minesweeper, tolokers and questions, accuracy is reported for the remaining datasets. CSNN is the best-performing method in out of datasets.
Model
roman-empire
umazon-ratingg 0.38
minesweeper 0.68
tolokers 0.59
questions 0.84
squireel 0.20
chameleon 0.23
Edge Homophily
0.05
GCN
73.69 + 0.74
48.70 0.63
89.75 = 0.52
83.64 = 0.67
76.09 1.27
39.47 = 1.47
40.89 = 4.12
SAGE
0.67
53.63
93.51 0.57
82.43
0.62
36.09
1.99
37.77 4.14
GAT
80.87
49.09 = 0.63
92.01 0.68
8347
35.62
= 2.06
39.21 = 3.08
GAT-sep
88.75
0.35
83.78
76.79
35.46
3.10
39.26
GT
8.75
51.17 = 0,66
91.85
83.23
77.95
36,30 1.98
38.87 3366
ST-sep
87.32 1 0.39
52.18 0.80
92.29 a 0.47
82.52 a 0.92
78.05 0.93
1.63 36.66
40.31 2 3.01
CO-GNN
0.50
0.34
97.35 0.63
84.84 0.96
75.97
0.89
39.39
41.14
Ord)-NSD
80.41 0.72
42.76 0,54
92.15 0.84
7
69.69
1.46
3..7
37.93 = 2.24
BuNN
91.75 0.39
$3.74 +: 0.51
0.16 98.99
84.78 = 0.80
78.75
1.09
CSNN
0.50
1.00
0.25
85.45 0.53
1.22
2.23
43.09 3.17
Table 2: Accuracy for node classification datasets on the fixed splits of Pei et al. achieves the best results 2 out of 4 datasets.
2020 CSNN
Model
Texas 0.11
Wisconsin 0.21
Film 0.22
Comell 0.30
Edge Homophily
GGCN
84,86 4.55
86.86
3.29
37.54
1.56
85.68
6.63
H2GCN
84.86 1 7.23
87.65 + 4.98
35.70
1.00
82.70
5.28
GPRGNN
78.38
82.94
4421
34.63
= 1.22
80.27
8.11
FAGCN
82.43 + 6.89
82.94
7.95
34.87
= 1.25
79.19
9.79
MixHop
77.84 + 7.73
75.88
4,90
32.22
2.34
73.51
6.34
GCNII
77.57 3.83
80.39
3.40
37.44
77.86
3.79
Geom-GCN
66.76 + 2.72
64.51
3.66
31.59
= 1.15
60.54
+ 3.67
PaicNorm
60.27 + 4.33
48.43
6.14
27.40
= 1.24
58.92
3.15
GruphSAGE
82.43 6.114
81,18
5.56
34.23
0.99
75.95
5.01
GCN
55.14 + 5.16
51.76
+ 3.06
27.32
+ 1.10
60.54
4 5.30
GAT
52.16
49,41
4.09
27.44
0.89
61.89
MLP
80.81 4.75
85.29
3.31
36.53
& 0.70
81.89
1 6.40
FSGNN
87.57 4.86
87.65 + 3.51
35.62
0.87
87.30 4.53
GloGNN
84.32
87.06 3.53
37.35
1.30
83.51
ACMGCN
87.84 4,440
88.43 +: 3.22
1.09 36,28
85.14 + 6.07
CO-GNN
77.57 5,44
83.73 a: 4.03
36.26 a: 3.74
72.70 a: 5.47
Diag-NSD
85.67 + 6.95
88.63 2.75
37.79 = 1.01
86.49 7.35
O(d)-NSD
85,95
89.41 4.74
37.81 1.15
84,86 4.71
Gen-NSD
82,97 5.13
89.21 + 3.84
37.80
1.22
85.68 6.51
CSNN
87.30
90.00 # 2.83
38.03 # 1.12
81.62 = 4.32
For the remaining datasets, we compare against the classical GCN, GAT and SAGE; the models specifically tailored for heterophilic data GGCN (Yan et al. 2022), Geom-GCN (Pei al.. (020).
H2GCN (Zhu et al. 2020), GPRGNN (Chien et al.
2020). FAGCN (Bo et al. 2021I. FSGNN (Maurya
al. 2022), GloGNN et al. 2019;; and the models GCNII (Chen et al. 2020) and Pa Norm (Zhao and Akoglu 2020)
(2022), ACMGCN (Luan et al. 2022), and MixHop (Abu-El-Hail
designed to alleviate oversmoothing. We use the 10 fixed splits proposed by [Platonov et al.
2023
and Pei et al. (2020). We refer to Appendix C further implementation details.
Results. Table I andd Table shshow that CSNN the berf-pero method in 9 out of 11 datasets.
These results highlight our model's capacity to deal with heterophilic graphs of different sizes and heterophily levels. We note CSNN often outperforms both NSD and CO-GNN in Table T While we report the results in Table for completeness, to the findings Platonov et al. (2023), which highlight that the small scale of these datasets may incur unstable and statistically insignificant results.
note they exhibit high variance in accordance
6.3 GRAPH CLASSIFICATION
To assess the effectiveness of CSNNs on long-range tasks, we evaluate it on the peptides dataset from the Long Range Graph Benchmark Dwivedi et (0.22). is a dataset containing 15k graphs and two different tasks: peptides-fun ffnnc is graph classification task, while peptides-struct is a regression one. We report average precision (AP) for peptides-fune and mean absolute error (MAB) for peptides-struct.
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Setup. We follow the experimental setup of Tönshoff et al. (2024), and tune the network hyperparameters keeping the ~500k parameter budget proposed by Dwivedi et al. (2022) for fair comparison. We run the model using four different seeds and report mean and standard deviation of the evaluation metrics. The base- lines are taken from Tönshoff et al. (2024) and we also compare against the results reported for BuNN by Bambbeggrr et al. (2025).
Table 3: Performance comparison of models on the peptides datasets.
Model GCN
peptides-
peptides-struct
68.60 = 0.50
24.60 0.07
GINE
66.21 = 0.67
2 0.17
GatedGCN
67.65 = 0.47
24.77 = 0.09 0.09
DReW
71.50
25.36
SAN
0.75
25.45
GPS
6,91
2
G- G-VIT
24,49
Results. Our model achieves the best perfor- mance in the peptides-struct dataset, and second- best in the peptides-func, as shown in (Table 3] These results further strengthen our claims on the capacity of CSNNs to mitigate over-squashing and
Exphormer
65.27 = 0.43
24.81 0.07
BuNN
72.76 0.65
24.63 0.12
CSNN
71,58 = 0.80
24.32 # 0.04
perform better in scenarios where long-range and under-reaching are known issues.
7 CONCLUSION
This work proposed Cooperative Sheaf Neural Networks, a novel SNN architecture that incorporates directionality in order to increase its efficiency by learning sheaves with conformal maps, allowing nodes to choose the optimal behavior in terms of information propagation with respect to its neighbors. We provided thcoretical insights on how CSNN can alleviate over-squashing due to its capacity to smoothly modulate node behavior in information diffusion. We also validated its effectiveness node and graph classification experiments on heterophilic graphs and long-range tasks.
Limitations and Future Works. While CSNN is not computationally more taxing than other SNNs, is worth pointing that developing strategies to scale sheaf-basee networks is a major research challenge. While have used conformal maps reduce the parameter complexity of restriction maps, leave open the possibility that there are further ways to improve the scalability of CSNN. We also believe that efficient message-passing implementations could represent step towards large-scale SNNs.
Another promising direction for future works is extending SNNs to cope with high-order structures like cell- and simplicial complexes, possibly allowing for more expressive models and promoting long-range communication with fewer layers.
ETHICS AND REPRODUCIBILITY STATEMENTS
Ethics Statement. We do not foresec immediate negative societal ethical impacts of work.
Reproducibility Statement. Aiming to secure reproducibility of our work, we provide proofs of theoretical results and in experiment detail in Appendix and Appendix C) Moreover, will provide a public code once the review process complete.
ACKNOWLEDGEMENTS
We acknowledge the support by the Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) SEL-260003300034422225, SEL-226003/020094422555 and the Conselho Nacional de Desenvolvimento Cientifico Tecnológico (CNPq) (404336/2023-0,, 305692/2025-9, 408974/2025-7, 31 1006//2025-)) We also thank The Adjoint School 2022 that introduced cellular sheaves Ana Luiza Tenório, which contributed towards making this work possible.
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A PROOFS
A.I PROOF OF PROPOSITION 3.1
Proof. Suppose Ly(X); does not depend of X; for any j neighbor of i. Since Ly Ei Flge (FigeX; -F F)ycXy), this means FT Fggec 0. Therefore, FygeX ker( (en(FTe)) for any j neighhor of i. Thus, Figex, = or Xj 0, for every
Note that, Figex; = 0 implies that (X( Eijise F
If FjgeXy = 0 for every j. then (X), O.
L
A.2 REMARK REGARDING PROPOSITION 3.1 AND NONLINEAR SHEAF LAPLACIAN
If the sheaf Laplacian is nonlinear as Zaghen et al. Ensge Fla.de (FigeX, F)ye,y), where de F(e) F(e) continuous function for each edge e, then saying that Lx(X); does not depend of x, means Fig. Figex; = O. Then the proof holds similarly to the above.
2024). i.e. Lx(X),
A.3 PROOF OF PROPOSITION
Proof. We have that (L)j²) out valued at given vertex is:
)) (9)
Σ
T.T. T:
Σ (S. - TTS,),) jEN(()
T.S,
Σ
(S] TTS,x.)
(x)
WENG)
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So T, 0 (i.e. does not listen) implies ((L L 1 does not broadcast, i.c., Tς= 0, then ((L)) Lm [));
11""(X (X)); O. If listens, but certain neighbor
Σ
T
Σ (S T)S,X,) T, JEN(
Σ
(S - TJ r.JS.x.))
JeN()))
uEN())
Since the sum does not go through the index k; is not component in ((L7) ( (X));,
A.4 PROOF OF PROPOSITION
Proof. Let 1, and fix node i. Then are essentially just using the composition described in Equation (9) (up to normalization and learnable weights). In the equation we have a sum running over all neighbors j of and another sum running over all neighbors u of each j. So u can be 2-hop neighbor of and we have that was updated with information from to 2-hop neighbors. Similarly, the node u is updated by up to 2-hop neighbors. Therefore, in the second layer with information from up to 4-hop neighbors.
i was updated
If t n, assume by induction that each node receives information from 2n-hop neighbors. In the next layer + 1, will be updated by its n-update of its 2-hop neighbors. Let be a node in the 2-hop neighborhood of By the inductive hypothesis, j receives information from 2n-hop neighbors, whose distance to i is up 2n + 2 2(n + 1), concluding the proof by induction. D
A.5 PROOF OF PROPOSITION
Proof. Choose path from i So there are - vertices between an say 01; -.. - In the first layer, let Sy and T be different of zero and all other source and target maps equal zero
This results in ((4)) )T TArh 0 for every h # 1z-1 and Arrt)) ).._ depends only of zz- So, except for ze, -e,- the values 2g are not updated.
In the second layer, let S,.., and Te... different of zero, and all other maps equal zero. This results ((Aje) AT
for every h * Ut-1, where ((Atm) T Ag v._,, depends only of the x that was updated in the previous layer and depends only of znj.
We continue this reasoning until the t-layer, in which we make Se, and T, different of zero, and all other source maps equal zero. This results in ((A)) A = 0 for every k i, where
(1-1) which going backwards depends only of the original zj up transformations given by the target and source maps.
((Aja)T Agut), )e, depend only of the øez
B ADDITIONAL EXPERIMENTS
In this section, we provide other two experiments. The first is real-world large dataset and the second noisy implementation of Example 4.4 to illustrate how Proposition 4.3 may work in practice.
Real-world. We run CSNN on the Penn94, dataset with 41.554 nodes, 1.362.222 229 edges, and whose feature dimension is 5, against multiple baselines reported in Li et al. (2022) to illustrate CSNN's performance larger ambiguous heterophilic (Luan al. 2024) dataset. We highlight that CSNN also outperform baselines on Squirrel, another dataset classified as ambiguous heterophilic.
Synthetic. In the following, we consider a path graph with four nodes as in Example 4.4 where the features are initialized in similar fashion of the TreeNeighborsMatch task, but consider that all nodes have a number "blue neighbors" to add noisy information along the path, while the source and target (nodes 3 and 0) have the same number of these neighbors. The goal is transfer information from node to node
Figure 6 shows an actual run of CSNN in such graph: if the norm of T, is close zero, then the arrow pointing to i is not draw. Analogously to S, both are close to zero, the node is isolated, but if none of them are consider the following:
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If S (in the in in-Laplacian) and T,S (in the out-Laplacian) are both close to 0, then the edge (j, i) does not exist
If SJ (in the out-Laplacian) and T. T, both close then edge (j, i) does exist.
Table 4: Results on the Penn94 dataset.
Models
Penn94 0.47
Edge Homophily
MLP
73.61 + 0.40
GCN
82.47 + 0.27
GAT
81.53 + 0.55
MixHop
83.47 + 0.71
GCNII
82.92 + 0.59
H2GCN
81.31 + 0.60
WRGAT
74.32 + 0.53
GPR-GNN
81.38 0.16
ACM-GCN
82.52 0.96
LINKX
84.71 + 0.52
GloGNN
85.57 + 0.35
CSNN
86.00 + 0.39
We observe restriction maps going to in different configuration than the one exhibited in Example 4.4 but still sending z3 to node in the second layer, while suppressing other node Teatures.
Technically, the significance of a node i over in given layer / is measured by the norm of the
Layer 0
the term that multiplies (2 in the expression of (()))))) A)))))s For instance, in Figure 6 the restriction maps provided that listens to O and no one else, so it can be affected only
x
Layer
(0)
xg
by (.) and by =0 A calculation shows that (0)
Tj is multiplied by quantity about 0.7 while (0)
Layer 2
To is multiplied by 132, approximately. This guarantees that in layer 2, when the message (0) goes from node 1 to O, the feature x1 is not relevant and, in practice, is ignored.
(9)
Figure 6: Illustration of running CSNN in a path graph with four nodes along three layers.
C EXPERIMENT DETAILS
In this section, we provide the grid of hyperparameters used in the experiments. If the number of GNN layers is set to 0, we use an MLP with two layers to learn the restriction maps. Otherwise, we adopt a GraphSAGE architecture with the specified number of layers.
We also trained CO-GNN using the hyperparameter table from Finkelshtein et al. (202)), considering I and as explicit hyperparameters instead of treating CO-GNN(J,, p) and CO-GNN(E, M) as separate model variants.
All datasets except for roman-empii eaarree were treated as undirected graphs. For the roman-empin dataset, we found that using the stored list of edges preferable to doubling the edges, since the graphs from (Patoonov et al. (2023) are stored as "directed" lists where elements as (0,2) and (2,0) are regarded equivalent, for example.
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All experiments were conducted cluster equipped with NVIDIA V100, A100, and H100 GPUs. The choice of GPU depended on the availability at the time of the experiments. Each machine was provisioned with at least 80 GB of RAM.
We also present some statistics of the heterophilic benchmarks in Table
Table 5: Hyperparameter configurations used across heterophilic benchmarks.
Parameter
roman-cmpire, amazon-ratings
minesweeper, tolokers, questions
sheaf dimcnsion
3.4,5
# layers
2-5
2-5
hidden channels
32,
# of GNN layers
0-5
01-5
GNN dimension
32,64
dropout
0.2
0.2
input dropout
0.2
0.2
epochs
2000
activation
GELU
GELU
left weights
trse,
trie. true. false
right weights
trac, false
learning rate
0.02
0.002, 0.02
weight decay
10-8
107 10-*
Table 6: Hyperparameter configuration used for NeighborsMatch.
Parameter
Neighbors.Match
sheaf dimension
2
# layers
r +
# hidden channels
32
# of GNN layers
r' +
GNN dimension
32
dropout
0.0
input dropout
0.0
activation
Id
left weights
true
right weights
true
layer norm
true
Table 7: Statistics of the heterophilous datasets
roman-empire
amazon-ratings
minesweeper
tolokers
questions
nodes
22662
24492
10000
11758
48921
edges
32927
93050
39402
519000
153540
avg degree
2.91
7.60
7.88
88.28
6.28
node features
300
300
7
10
301
classes
18
5
edge homophily
0.05
0.38
0.68
0.59
0.84
adjusted homophily
-0.05
0.14
0.01
0.09
0.02
metric
acc
acc
TOC auc
FOC auc
FOC auc
D COMPLEXITY AND RUNTIME OF CSNN
Using d for dimension the stalks, h as the number of channels and c dh, the complexity of our model is follows:
(d2 for the embedding of graph features into the sheaf stalks;
O(V]dP) O(V]cd) when applying W1;
O(V[d)? O(V]ch) when applying Wa;
16

---

Published as a conference paper at ICLR 2026
h(2 O((Elcd) for the two sparse Laplacian- vector multiplication;
. (IV + [E)) = (VV + El)) for constructing the blocks of the Laplacians.
This gives total of O [ (c((((dd)))))))) Since we use 1 & d < 5, the stalk dimension contribution is small. We highlight that code also contains a completely message-passing based implementation, that does not need constructing the Laplacian. This cheaper implementation yields
complexity of V(c h) Ede)..
In the following we report the runtime of CSNN and the non-sheaf models on datasets of Table as well as the improvement compared the best baseline method.
Table Runtime comparison on datasets from Platonov seconds per epoch, averaged over 10 epochs. CSNN has a similar runtime compared to these simple baselines, and presents positive improvement on accuracy in general. The baselines achieving the best accuracy are highlighted bold.
(2023). We report the mean time in
Model
roman-empire
amazon-ratings
minesweeper
tolokers
questions
squirel
chameleon
GCN
0.05s
0.04s
0.03s
0.04s
0.07s
0.01s
0.02s
SAGE
0.07s
0.03s
0.06s
0.16s
0.01s
0.02s
GAT-sep
0.09s
0.05s
0.05s
0.12s
0.21s
0.01s
0.02s
GT-sep 0.14s
0.16s
0.07s
0.14s
0.32s
0.02s
0.01s
CSNN
0.05s
0.10s
0.06s
0.14s
0.16s
0.05s
0.03s
Improvement
4.37%
2.90%
5.50%
2.00%
1.61%
1 4.33%
5.38%
We can that sometimes CSNN is quicker than SAGE, and sometimes it equal GCN in terms of runtime. This might look counter-intuitive, but CSNN achieves its best performance with fewer parameters. For instance, for the roman-empire dataset, GCN has 2,269, 714 parameters, while CSNN has 339,90 ,0000 i.e. GCN has about 668% more parameters.
17


Reviews:
[{"review_id": "qLFGWFK0DG", "reviewer": "Reviewer_4F1X", "summary": "The authors propose an approach for learning cooperative behavour between nodes, by treating undirected edges between nodes as pairs of directed edges and introducing directed sheaf neural networks to effectively deal with them.", "questions": "Is it possible to have a high-level illustration clarifying the problem with plain SNN and the advantage of CSNN? This would help interested readers not familiar with the math behind SNN to gather the intuition behind the approach.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The work is well motivated, addressing a clear limitation of sheaf neural networks in modeling cooperation patterns between nodes.\n\nThe proposed solution has solid theretical grounds.\n\nSynthetic results clearly confirm the ability to mitigate oversquashing, and an extensive evaluation on real-world datasets shows competitive performance wrt the state of the art.", "weaknesses": "Experimental results on the classical datasets for heterophilic analysis (Table 2) show very marginal improvements (considering the high variance, most likely none of these is significant). This is not a novelty, and questions the appropriateness of these datasets as benchmarks (as Platonov et al already pointed out). I encourage the authors to briefly discuss this aspect, so as to direct further research towards more appropriate evaluation benchmarks.", "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": "jlLcArpU1c", "reviewer": "Reviewer_rWHL", "summary": "The work proposes a new Sheaf Neural Network model, called Cooperative Sheaf Neural Network (CSNN), with the goal of combining the benefits of Sheaf Neural Networks in tackling oversmoothing in GNNs and handling heterophilic data, with the property of performing selective communication between nodes, which is expected to lead to cooperative behaviour and avoid oversquashing problems. The model and its benefits are validated through synthetic and real-world experiments.", "questions": "**General concerns** \n- In your introduction, you mention that selective communication is a desirable property for SNN in order to tackle oversquashing. Recently, Nonlinear Sheaf Neural Networks have shown a similar behavior, selectively exploiting information of neighbors in complex node interactions [1]. Do you think there may be a connection between your method and the employment of a nonlinear Laplacian? \n- After definition 2.3: \"...however, when they publicly discuss this topic, they may prefer to not manifest their true opinion. \". In my understanding, since the edge stalks may be different from the node stalks, the individuals don't necessarily need to discuss the topics of the private opinion spaces. The topics in the public agreement space may also be different.\n- It is not clear how your result in Proposition 4.3 relates to your discussion on the oversquashing behavior relying on the definitions in [2] and [3], which you reference after Example 4.4. These works formalize oversquashing as a bound on the Jacobian, as you also state, and this bound is strongly influenced by the presence of $A_{i,j}^{(t)}$. In Example 4.4, although you show that at each layer one node is influenced by only a single other node, when computing the Jacobian the update matrices $T$ and $S$ will still accumulate in the product across multiple layers. So, there is still a product of $O(t)$ matrices in the bound for the Jacobian. Wouldn’t this have the same effect as including the term  $A_{i,j}^{(t)}$? Could you please elaborate on this point, perhaps by explicitly showing how your method improves this bound compared to a sheaf model that does not perform selective communication?\n\n**Experiments**\n- It would be useful to have in Table 1 and 2 an homophily measure for each dataset, to intuitively understand the setting.\n- I would recommend to rephrase the claims regarding the results. For example, in the \"Results\" of section 6.2, you state \"We note CSNN often outperforms both NSD and CO-GNN by a significant margin\", although looking at the results of Table 1 and 2, and considering the +/- confidence bound, the improvements are not as significant as stated. \n- As mentioned in the weaknesses, I believe the experiments would benefit from a direct comparison between the proposed method and its non-sheaf (CO-GNN) and non-cooperative (NSD/BuNN) counterparts - for example, by including these models in the synthetic experiments of Section 6.1 and adding CO-GNN in Table 2 to provide a more complete comparison and a clearer demonstration of the claims.\n\n**Writing/Typos**\n- In related works, in the paragraph related to Cooperative GNNs: \"... that chooses the cooperation the action each node takes\". There seems to be an error in the structure of this sentence.\n- In Table 2, last column, the second-best result is not highlighted with grey color.\n\n\n\n[1] Zaghen et al., Sheaf Diffusion Goes Nonlinear: Enhancing GNNs with Adaptive Sheaf Laplacians (2024)\n\n[2] Di Giovanni et al., On over-squashing in message passing neural networks: The impact of width, depth, and topology. (2023)\n\n[3] Topping et al., Understanding over-squashing and bottlenecks on graphs via curvature (2022)", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The motivation, goal, and methodology are well formulated.\n- The paper reads well, it has a nice structure, and the necessary background knowledge is well presented.", "weaknesses": "There are strong claims regarding (1) consistently outperforming SNNs and cooperative GNNs (both in introduction and in the experiments, see Results in Section 6.2), and (2) about not soccumbing to oversquashing (abstract). Looking at the results, it doesn't appear to improve substantially with respect to the competitors, so I would advise to reconsider the strength of the claims, which may lead to high expectations in the experiments.\n\nAdditionally, the introduction of a sheaf structure generally adds computational and runtime overhead compared to a GNN model, so its inclusion should be well justified in two respects: (1) whether the additional cost is outweighed by the gain in prediction accuracy, and (2) whether there is a real practical need for a sheaf-based model. Regarding (1), a discussion ideally comparing the model with non-sheaf baselines, is missing in the main part of the paper. Regarding (2), I would expect a convincing discussion of the advantages over other SNN methods in preventing oversquashing (due to the additional cooperative component), and over cooperative GNNs in mitigating oversmoothing and handling heterophilic datasets (thanks to the additional sheaf structure). These advantages are not evident from the experiments, as the relevant comparisons are either missing or the results/discussions are not sufficiently convincing.", "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": "vbD4CP73wR", "reviewer": "Reviewer_o6mM", "summary": "This paper extends Sheaf Neural Networks, a class of GNNs that use cellular sheaves to generalize message passing, to support cooperative communication between nodes. Classical SNNs effectively address heterophily and oversmoothing, but they lack the flexibility for nodes to independently choose how to exchange information (i.e., whether to propagate, listen, or isolate).\nTo overcome this, the authors propose the Cooperative Sheaf Neural Network (CSNN), which introduces directed cellular sheaves with separate in- and out-degree Laplacians, enabling asymmetric and selective communication.", "questions": "1. This paper seems to be an A+B work, with a combination of sheaf GNN + cooperative GNN, and some extension on directed graphs. And it is not well motivated or does not have some special designs for heterophily and over-squashing.\n\n2. Why does the sheaf neural network have to achieve cooperative behavior?\n\n3. \"our model has the ability to reach longer distances.\" It is no surprise when you multiple your in-degree and out-degree sheaf Laplacian together. However, it is found that long-range information is harmful in many heterophilic datasets [1].\n\n4. Missing comparison with some baseline models, e.g. ACMGCN [2], FSGNN [3], GloGNN [4]. More tests on malignant and ambiguous heterophilic datasets listed in [5]. Experiments on large scale datasets used in [4].\n\n\n\n\n[1] Less is More: on the Over-Globalizing Problem in Graph Transformers. In Forty-first International Conference on Machine Learning 2024.\n\n[2] Revisiting heterophily for graph neural networks. Advances in neural information processing systems. 2022 Dec 6;35:1362-75.\n\n[3] Simplifying approach to node classification in graph neural networks. Journal of Computational Science, 62, 101695.\n\n[4] Finding global homophily in graph neural networks when meeting heterophily. In International conference on machine learning (pp. 13242-13256). PMLR.\n\n[5] The heterophilic graph learning handbook: Benchmarks, models, theoretical analysis, applications and challenges. arXiv preprint arXiv:2407.09618. 2024 Jul 12.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Clear theoretical guarantees linked to long-range neighbors and over-squashing.\n\nEmpirical performance is compelling", "weaknesses": "Scalability of sheaf-based models on large-scale graphs remains to be tested.", "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": "Y7kZsU1uTP", "reviewer": "Reviewer_ZAJv", "summary": "The paper argues that standard Sheaf Neural Networks (SNNs) on undirected graphs cannot realize node-level cooperative behavior (separating LISTEN vs PROPAGATE), because silencing incoming messages at a node also suppresses its outgoing influence.", "questions": "Please see the above weaknesses", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "### Clear motivation\n* The limitation of undirected-sheaf diffusion for cooperative behavior is crisply identified and formalized. Proposition 3.1 explicitly shows PROPAGATE -> LISTEN coupling in classical SNNs; the directed-sheaf construction is a natural remedy\n\n### Empirical breadth on heterophily\n* Across 11 node-classification benchmarks (including the cleaned Squirrel/Chameleon), CSNN often outperforms both cooperative GNNs and prior SNNs\n\n### Implementation details\n* Parametrization for orthogonal maps, per-node conformal scaling, and a complexity discussion point to a careful engineering effort", "weaknesses": "### Novelty and Positioning\n* Treating undirected edges as two directed edges plus per-node source/target maps is a natural extension but arguably an incremental one within the sheaf literature. The paper acknowledges related notions (e.g., quiver Laplacians), but the novelty boundary versus prior directed/sheaf constructions (and vector bundle variants on directed graphs) is not fully pinned down\n\n### Theoretical Analysis\n* Proposition 4.3 is an existence proof: there exist restriction maps for clean long-range, path-selective propagation. The paper does not analyze whether gradient-based training reliably finds such configurations in realistic, noisy data\n* The paper notes their directed-sheaf Laplacians can have complex eigenvalues with negative real parts (unlike PSD Laplacians). Yet the stability of the discrete diffusionis not analyzed\n* The model still performs local diffusion; the 2-hop per layer can accelerate reach but may also accelerate oversmoothing. There’s no theorem bounding oversmoothing for CSNN or showing improved expressivity beyond MPNN limitations\n\n### Writing Clarifications\n* A derivation/intuition for why this specific normalization (scalar multiples of identity under conformal maps) is preferred (vs. other block scalings) would help\n* Regarding Proposition 4.3 \n  * The constructive path-only propagation is compelling but assumes the ability to zero out many maps across layers. In practice, with shared parameters and noisy optimization, how often does training approximate this regime?", "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)