Explaining Hypergraph Neural Networks: From Local Explanations to Global Concepts

## Abstract

Abstract Hypergraph neural networks are a class of powerful models that leverage the message passing paradigm to learn over hypergraphs,
a generalization of graphs well-suited to describing relational data with higher-order interactions.
However, such models are not naturally interpretable,
and their explainability has received very limited attention.
We introduce SHypX, the first model-agnostic post-hoc explainer for hypergraph neural networks that provides both local and global explanations.
At the instance-level, it performs input attribution by discretely sampling explanation subhypergraphs optimized to be faithful and concise.
At the model-level, it produces global explanation subhypergraphs using unsupervised concept extraction.
Extensive experiments across four real-world and four novel, synthetic hypergraph datasets demonstrate that our method finds high-quality explanations which can target a user-specified balance between faithfulness and concision, improving over baselines by 25 percent points in fidelity on average.

## 1 Introduction

Relational data in the form of graphs arises naturally in social networks (Fan et al., ), natural sciences (Zhang et al., ; Cranmer et al., ; Wang et al., ), traffic dynamics (Jiang & Luo, ), and knowledge databases (Schlichtkrull et al., ).
The neural approach (Kipf & Welling, ) has enjoyed exciting successes, setting new state-of-the-art and expanding the reach of machine learning to new modalities (Ektefaie et al., ; Battaglia et al., ).

However, graphs can only describe pairwise relationships.
This is insufficient to model real world systems that depend crucially on multi-way or group-wise interactions (Benson et al., ; Agarwal et al., ; Estrada & Rodríguez-Velázquez, ).
A data structure that is well-suited to capturing higher-order correlations is the hypergraph.
Whereas each edge in a graph joins two nodes, each hyperedge in a hypergraph joins an arbitrary number of nodes.
Message passing principles extended to hypergraphs give rise to hypergraph neural networks (hyperGNNs)  (Feng et al., ).

Unfortunately, graph neural networks (GNNs) and hyperGNNs share a key concern with all black-box neural models: their lack of explainability.
In response, many post-hoc explainers (Ying et al., ; Luo et al., ; Yuan et al., ; Magister et al., ; Yuan et al., ) and interpretable-by-design architectures (Zhang et al., ; Magister et al., ) have been developed for GNNs. However, the literature for hyperGNN explainability remains exceedingly sparse, with the
hypergraph modality posing new challenges as the space of possible explanations is substantially larger than the graph counterpart.

In this work, we introduce SHypX(Subhypergraph-based HyperGNN eXplainer), the first post-hoc hyperGNN explainer that produces explanations both at the instance level and global level.
Our explanations take the form of subhypergraphs. Its core idea is to approximate subhypergraph sampling with a collection of independent Gumbel-Softmax samplers,
and use gradient feedback from a loss function to obtain good explanation as per user specifications.
This instance-level optimization is combined with concept extraction to produce global explanations, where concepts represent significant, recurring subhypergraphs.
The design choices of our explainer are guided by several considerations:
ensuring explanations are faithful to the hyperGNN under study,
keeping explanations concise and legible,
and avoiding the introduction of another black-box model in the explanation method.

To the best of our knowledge, this is the first global explainer designed for hypergraphs. For instance-level explanations, the only existing hypergraph explainer (Maleki et al., ) relies on learning an attention map to attribute the importance of each node-hyperedge link and induce the explanation subhypergraph. However, it remains contentious whether attention provides a valid explanation (Jain & Wallace, ; Wiegreffe & Pinter, ; Bibal et al., ). In contrast, SHypX is simple, effective, and
doesn’t rely on additional black-box networks to explain the hyperGNN.

In addition to introducing an effective hypergraph explainer, we also propose a set of synthetic datasets, designed to better assess the quality of hypergraph explanations along with suitable metrics for evaluation. As our experiments show, the current real-world datasets used in the previous work (Maleki et al., ) barely take into account the hypergraph structure, making it difficult to properly evaluate explainers. We believe that our datasets, which entirely depend on the higher-order structures, have the potential to speed up the advancements in the field of hypergraph explainability.

Our main contributions are summarized as follows:

- 1.
We develop a model-agnostic post-hoc explainer for hyperGNNs that finds salient subhypergraphs for both instance-level and global-level explanations.
- 2.
The instance-level explainer alleviates the need for black-box attention mechanisms used in the previous work. We integrate our instance-level explainer with unsupervised concept extraction to produce a global-level explanation – a novelty in the field of hypergraph explainability.
- 3.
We introduce the first hypergraph explainiability benchmark containing four synthetic datasets
which are highly structure-dependent and thus offer a challenging testbed for explainability. Moreover, we generalize the fidelity metric for explanation faithfulness, making it more sensitive to deviations induced by the explanation subhypergraph.
- 4.
We conduct extensive evaluations on both synthetic and real-world datasets, showing that our explainer obtains coherent explanations for each class, outperforming existing methods.

## 2 Related Work

Hypergraph neural networks.
HyperGNNs operate over hypergraphs, taking inspiration from the message-passing paradigm of GNNs.
HGNN (Feng et al., ), HyperGCN (Yadati et al., ), and HNHN (Dong et al., ) generalize GCN (Kipf & Welling, ) to hypergraphs.
HCHA (Bai et al., ), HERALD (Zhang et al., ), and HEAT (Georgiev et al., ) introduce attention mechanisms for hypergraphs to dynamically learn the incidence matrix, analogous to GAT (Veličković et al., ).
UniGNN (Huang & Yang, ) proposes leveraging GNN architectures for updating node representations.
AllSet (Chien et al., ) and EDHNN (Wang et al., ) use universal approximators to learn multiset functions for node and hyperedge updates. Our work proposes a model-agnostic explainer, producing hypergraph explanations regardless of the architectural choice.

GNN explainers.
The majority of GNN explainers are local, finding an explanation subgraph pertaining to a specific input instance.
Pope et al. () and Sanchez-Lengeling et al. () apply gradient-based attribution techniques from vision and language to graphs to produce local explanations.
GNNExplainer (Ying et al., ) learns fractional edge weights and thresholds them to produce explanation subgraphs;
this framework is extended by PGExplainer (Luo et al., ), which learns a second neural network to predict edge weights.
SubgraphX (Yuan et al., ) finds the subgraphs instead by Monte Carlo Tree Search.
GraphLIME (Huang et al., ) and PGMExplainer (Vu & Thai, ) learn explainable surrogates of the original GNN.
In contrast, global explainers like XGNN (Yuan et al., ) and GCExplainer (Magister et al., ) produce explanations representative of a class:
XGNN generates explanation graphs with policy gradients and GCExplainer with unsupervised concept extraction.

To the best of our knowledge, the only existing hyperGNN explainer is HyperEX (Maleki et al., ). It optimizes an attention-based network with InfoNCE to assign importance weights to node-hyperedge links to produce local explanations. However, there is ongoing debate about whether attention mechanisms offer valid explanations (Jain & Wallace, ; Wiegreffe & Pinter, ; Bibal et al., ). In contrast, our model eliminates the need for surrogate networks, while also providing global-level explanations, a novelty in the realm of hypergraph explainability.

Figure: Figure 1: Visualization of our hypergraph explainer providing local and global explanations. (Top) Instance-level explanations are obtained by optimizing the subhypergraph structure using a loss function that incentivizes faithfulness (the explanation is able to reproduce the original prediction well) and concision (the explanation is as minimal as possible). (Bottom) Model-level explanations are obtained by combining the instance-level explainer with unsupervised concept extraction. After clustering the latent space into concepts, the closest node to each concept’s center is picked as a representative and explained using the instance-level approach to produce concept and class-level explanations.
Refer to caption: x1.png

## 3 Preliminaries

#### Notation.

A hypergraph $G=(V,E)$ comprises a set of nodes $V$ and a set of hyperedges $E$.
Each hyperedge $e=\{v_{1},...,v_{|e|}\}\in E$ is a set of nodes, and is said to be of degree $|e|$.
In this sense, graphs are a special case of hypergraphs wherein all hyperedges have degree two.
The structural content of a hypergraph is given by the incidence matrix $\bm{H}\in\mathbb{Z}_{2}^{|V|\times|E|}$, where $H_{ve}=\mathbbm{1}(v\in e)$.
$\bm{H}$ has an equivalent sparse representation as a hyperedge index of shape $(2,L)$, where $L=\sum_{e\in E}|e|$ is the number of node-hyperedge links
and each column $[v,e]$ denotes that $v\in e$.
The hypergraph has node features $\bm{X}=[\bm{x}_{1},...,\bm{x}_{|V|}]\in\mathrm{R}^{|V|\times d}$, where $d$ is the feature dimension and each $\bm{x}_{i}$ is associated to the node $v_{i}$.

Given a hypergraph $G=(V,E)$, we define a subhypergraph $G_{\textrm{sub}}=(V_{\textrm{sub}},E_{\textrm{sub}})$ to be a subset $V_{\textrm{sub}}\subseteq V$ of the nodes, and a new set of edges $E_{\textrm{sub}}$ such that each $e_{\textrm{sub}}\in E_{\textrm{sub}}$ is a subset of precisely one hyperedge in the original hypergraph.
Furthermore, we allow neither empty edges
($e_{\textrm{sub}}\neq\emptyset\ \forall\ e_{\textrm{sub}}\in E_{\textrm{sub}}$)
nor isolated nodes
($\forall\ v\in V_{\textrm{sub}},\ \exists\ e_{\textrm{sub}}$ such that $v\in e_{\textrm{sub}}$).
Altogether, this can be thought of as taking a subset of columns of the hyperedge index.

#### Problem statement.

Consider the task of node classification over a hypergraph.
(Our explainer is more general, but we defer this discussion to Appendix A.)
Let $f$ be a hyperGNN that outputs for each node $v$ a probability distribution $f(G,\bm{X},v)$ over the classes.
Our proposed model obtains both local and global explanations that are architecture-agnostic and fully post-hoc.

The goal of a local hypergraph explainer is, for each instance, to find which parts of the input hypergraph are most important to determining $f$’s output.
Accordingly, the explanation artefact is a subhypergraph.
A good explanation subhypergraph $G_{\textrm{expl}}=(V_{\textrm{expl}},E_{\textrm{expl}})$ should be able to reproduce the original prediction well (“faithful”) and also as minimal as possible (“concise”).
Loosely speaking, we want $f(G,\bm{X},v)\approx f(G_{\textrm{expl}},\bm{X}_{\textrm{expl}},v)$, where $\bm{X}_{\textrm{expl}}$ is the restriction of $\bm{X}$ to $G_{\textrm{expl}}$, for small $G_{\textrm{expl}}$.
While local explainers produce an explanation for each example, a global hypergraph explainer produces concise explanation subhypergraphs representative of each class.

## 4 Method

### 4.1 Local explainer

Given a trained hyperGNN $f$, a hypergraph $G$, and a node instance $v$ in $G$,
our goal is to produce an explanation subhypergraph that is both faithful and concise.
To achieve this, we formulate these desiderata as a joint objective
and optimize the explanation subhypergraph against this objective by discrete sampling.
Figure 1(top) gives an overview of the local explainer.

#### Objective function.

We can quantify the faithfulness of the explanation by the Kullback-Leibler divergence between the original class probabilities predicted by $f$ over $G$, and when $f$ is restricted to the explanation subhypergraph.
We can quantify concision by the $L_{1}$ norm of the incidence matrix, which is equivalent to the number of node-hyperedge links.
We denote this size measure on a hypergraph $G$ by $|G|_{1}$.
These competing objectives suggest $G_{\textrm{expl}}=\operatorname*{arg\,min}_{G_{\textrm{sub}}}\mathcal{L}$, where the loss function is

$$ $\mathcal{L}(f,G_{\textrm{sub}},G,\bm{X},v)=\lambda_{\mathrm{pred}}D_{\textrm{ KL}}\big{(}f(G_{\textrm{sub}},\bm{X},v)\ ||\ f(G,\bm{X},v)\big{)}+\lambda_{ \mathrm{size}}\big{|}G_{\textrm{sub}}\big{|}_{1},\\ G_{\textrm{sub}}\subseteq G,$ (1) $$

and $\lambda_{\textrm{pred}}$ and $\lambda_{\textrm{size}}$ are hyperparameters governing the trade-off between faithfulness and concision.

For a message passing neural networks with $d$ layers, each node’s receptive field is restricted to its $d$-hop neighborhood.
This neighborhood defines a computation subhypergraph $G_{\textrm{comp}}=(V_{\textrm{comp}},E_{\textrm{comp}})$ which contains all information that determines the hyperGNN’s output over that node.
By simplifying the loss to

$$ $\mathcal{L}(f,G_{\textrm{sub}},G_{\textrm{comp}},\bm{X},v)=\lambda_{\mathrm{ pred}}D_{\textrm{KL}}\big{(}f(G_{\textrm{sub}},\bm{X},v)\ ||\ f(G_{\textrm{ comp}},\bm{X},v)\big{)}+\lambda_{\mathrm{size}}\big{|}G_{\textrm{sub}}\big{|}_ {1},\\ G_{\textrm{sub}}\subseteq G_{\textrm{comp}},$ (2) $$

we reduce the search space of the explanation to a subhypergraph of $G_{\textrm{comp}}$, which is typically much smaller than $G$.

#### Optimization.

Exhaustively searching all $G_{\mathrm{sub}}\subseteq G_{\mathrm{comp}}$ is intractable due to the exponentially-large dimension of the search space. For a hypergraph with $n$ nodes and $m$ hyperedges of degree $d_{1}\cdots d_{m}$, selecting a subhypergraph involves choosing from
$2^{\sum_{i=1}^{m}d_{i}}$ potential subhypergraphs.
In comparison, for a graph with $n$ nodes and $m$ edges, the number of possible subgraphs is much smaller
($2^{m}$), suggesting that finding the right explanation is particularly challenging in the hypergraph domain.

Instead, our approach is to optimize a joint probability distribution of the existence of each node-hyperedge link
– in effect, a probability distribution over subhypergraphs –
and obtain candidate subhypergraphs by discrete sampling.
The sampler should be differentiable, admitting gradient updates to these probabilities.
Note that our goal is to discretely optimize the structure of the $G_{\mathrm{sub}}$,
and not the parameters of the hyperGNN, which remain fixed.

To ensure the sampler always produces a valid subhypergraph $G_{\mathrm{sub}}$, we impose the restriction that $\forall e_{\mathrm{sub}}\in E_{\mathrm{sub}}$ and $\forall v\in V_{\mathrm{sub}}$, $\ \Pr(v\in e_{\mathrm{sub}}=0)$ if $v$ was not in the original, corresponding hyperedge of $G_{\mathrm{comp}}$.
This ensures each $e_{\mathrm{sub}}$ is truly a subset of some hyperedge $e_{\mathrm{comp}}\in E_{\mathrm{comp}}$.
Thus, our goal is to sample subhypergraphs from the joint distribution

$$ $\Pr(G_{\mathrm{sub}})=\Pr(\{\mathbbm{1}_{v\in e}\}_{\forall\ v\in V_{\mathrm{ sub}},\ e\in E_{\mathrm{sub}}}),\quad v\notin e_{\mathrm{comp}}\implies \mathbbm{1}_{v\in e}=0,$ (3) $$

where $\mathbbm{1}$ is the indicator function.

We opt for a mean field approximation that decomposes the joint probability distribution into the product of marginals.
Let $\pi_{v,e}:=\Pr(\mathbbm{1}_{v\in e}=1)$.
The approximation allows us to sample each node-hyperedge link independently:

$$ $\Pr(G_{\mathrm{sub}})\approx\prod_{\forall v\in V_{\mathrm{sub}},\ e\in E_{ \mathrm{sub}}}\pi_{v,e},\quad v\notin e_{\mathrm{comp}}\implies\pi_{v,e}=0.$ (4) $$

Now we are faced with the problem of differentiable obtaining a discrete sample $y_{v,e}$ from the probabilities $\pi_{v,e}$ over each $v,e$ pair.
We accomplish this using the Gumbel-Softmax
(Jang et al., ; Maddison et al., )
over the binary categorical distribution described by
$\pi_{v,e}$.
The set of all incident node-hyperedge pairs $(v,e)$ such that $y_{v,e}=1$ forms the explanation candidate $G_{\mathrm{sub}}$.

We pass the resultant subhypergraph through the hyperGNN to evaluate $f(G_{\textrm{sub}},\bm{X},v)$.
By ensuring this entire subhypergraph sampling is differentiable, we are able to
optimize the underlying probabilities $\{\pi_{v,e}\}$, using backpropagation on the loss $\mathcal{L}(f,G_{\textrm{sub}},G_{\textrm{comp}},\bm{X},v)$ defined in Equation 2.

#### Post-processing.

Following the approach described above, we extract the subhypergraph corresponding to the lowest loss observed during optimization.
If this subhypergraph has disconnected components, we retain only the connected component containing the node $v$ being explained,
and return it as the explanation $G_{\mathrm{expl}}$.
Disconnected components do not impact the hyperGNN output, so are typically pruned away by the size penalty in $\mathcal{L}$.
However, this is not guaranteed due to the challenging loss landscape of this discrete problem.
We discard disconnected components to produce a smaller and more legible explanation artefact,
and grant the same advantage to the baselines in our evaluations.

### 4.2 Global explainer

The local explainer returns an explanation subhypergraph for a single node instance.
How can we leverage this to obtain a global explanations at the class-level?
While global explanation for hyperGNNs is an unexplored area of research, several methods were proposed for GNNs.
However, creating class prototypes by graph alignment (Ying et al., ) is NP-hard,
and graph generation with reinforcement learning (Yuan et al., ) requires expensive policy gradients optimization.
We desire a global explainer whose computation costs do not scale with the increased combinatorial possibilities of the hypergraph space.

#### Concept extraction and visualization.

We propose to obtain global explanations using unsupervised concept extraction, inspired by Magister et al. ().
Concepts are higher-level units of information, more accessible for humans than low-level neural network constructs (Ghorbani et al., ).
Similar to the GNNs domain (Magister et al., ), we find that concepts may be identified with clusters in the hyperGNN’s activation space.
We then visualize each concept by finding the local explanation subhypergraph of its representative node.

Stated more precisely, a hyperGNN $f$ learns latent node representations $\bm{z}_{v},\ \forall v\in V$.
We train a $k$-means model with $k$ centroids on $\{\bm{z}_{v}\}_{v\in V}$,
and use it to map each node $v$ onto one of $k$ concepts,
$\texttt{KMeans}(\bm{z}_{v})=c_{v}$.
To obtain a concept-level explanation for concept $c$, we take the node closest to the cluster center,

$$ $v_{c}^{*}=\operatorname*{arg\,min}_{v:\ c_{v}=c}\big{|}\big{|}\bm{z}_{v}-(1/|c |)\sum_{u:\ c_{u}=c}\bm{z}_{u},\big{|}\big{|}$ (5) $$

where $|c|$ is the number of nodes belonging to that concept.
We then produce as the explanation for concept $c$ the instance-level explanation subhypergraph for $v_{c}^{*}$, which we denote $G_{\textrm{expl}}(v)$.
This explanation is computed using our instance-level explainer described in Section 4.1.

Figure 1(bottom) illustrates the overall pipeline. Whereas GCExplainer visualizes each concept by the $n$-hop graph neighborhood of $v_{c}^{*}$, where $n$ is a hyperparameter,
the integration with our local explainer produces more legible explanation artefacts appropriate to the user’s desired faithfulness-concision tradeoff (see Appendix D for a visual comparison between the two approaches).

#### Explanation for each class.

Users may desire explanations pertaining to each class.
These explanations answer the question:
what does a representative example of each class look like, according to the hyperGNN?
To obtain such class-level explanations from our set of concept-level explanations,
we use the majority vote function,
$\texttt{MajorityVote}:\{c\}\rightarrow\{y\}$.
That is, we take the most frequently occurring class of node instances belonging to a concept, and associate the concept with that class.
The set of concepts associated with each class is taken as the explanation for that class:

$$ $\texttt{ClassExplanation}(y)=\bigl{\{}G_{\textrm{expl}}(v_{c}^{*})\bigr{\}}_{c :\ \texttt{MajorityVote}(c)=y}.$ (6) $$

## 5 Experiments

We show that our hypergraph explainer produces high quality explanations through extensive evaluations.
We test on real hypergraphs Cora, CoauthorCora, CoauthorDBLP, and Zoo from the benchmark of Chien et al. ().(^1^11We selected the latter three hypergraphs because here the hyperGNNs outperform MLP by an appreciable margin; these are expected to be the relatively discriminating test cases for explainability, as discussed in Section 5.1 For comparison, we also selected Cora, where this is not the case.)
In Section 5.1, we discuss why existing hypergraph datasets may not provide a sufficiently challenging setting for finding subhypergraph explanations,
and design challenging synthetic hypergraph datasets to complement our evaluations.
In Section 5.2, we highlight some shortcomings of the fidelity metric used to quantitatively evaluate explanations, and propose alternatives to address them.

### 5.1 Synthetic hypergraphs

#### Motivation.

Synthetic graph datasets for GNN explainability have driven substantial progress in the field.
However, no such dataset exists for hypergraphs.
We argue that synthetic (hyper)graph datasets are valuable because they guarantee the primacy of structure for solving the task.
For many real world hypergraphs like those in the benchmarks of Chien et al. (), competitive performance is already achieved by MLPs, which do not account for the hypergraph’s structure.
Accordingly, we find that node-level explanations obtained for such datasets typically comprise a “trivial” subhypergraph containing just the node itself.
While valid explanations, they suggest the dataset fails to provide a challenging and discriminating testbed for evaluating hyperGNN explainability.
Our synthetic hypergraphs ensure that labels depend critically on the hypergraph structure by construction,
complementing evaluation on real world datasets.

Figure: ((a)) Random
Refer to caption: x2.png

#### Dataset construction.

Our synthetic hypergraphs are inspired by the synthetic graphs of Ying et al. (),
which have served as a core benchmark in graph explainability.
Each hypergraph comprises a “base” component that is either random or a deterministic “hyper-binary-tree“ (Figure 2a-b),
to which various “motifs” (Figure 2c-e) are attached using a single hyperedge.
Additionally, we randomly add degree-2 hyperedges as perturbations.
Figure 2e shows an example of a hypergraph constructed in this manner.
The task is to classify nodes based on their positions in the base or motif.
See Appendix B for details.

Different combinations of these base and motif components give rise to four synthetic hypergraphs: H-RandHouse, H-CommHouse, H-TreeCycle, and H-TreeGrid.
Table 3 shows their statistics
and Table 4 benchmarks several hyperGNN architectures on these hypergraphs. Compared to benchmarks on real hypergraphs (Chien et al., ), our proposed datasets exhibits a clear gap between hyperGNNs and models that disregard structural information, such as MLPs. This indicates that the datasets represent challenging, structure-dependent tasks well-suited for evaluating hypergraph explainability.

### 5.2 Metrics

Evaluations of graph explainers often rely on comparison against the implanted motifs in synthetic datasets (Ying et al., ; Luo et al., ; Magister et al., ).
Not only is this approach impossible for real world (hyper)graphs, due to the absence of reference motifs, we argue that it is unprincipled and potentially misleading.
The implanted motifs reflect human reasoning,
but are not necessarily faithful to the neural network, which may instead rely on a variant or correlate of the motif.
Rather, a good explanation should provide users information about the hyperGNN’s predictions, reflecting its internal mechanisms.
This requirement is satisfied by the fidelity metrics (Amara et al., ):

$$ $\mathrm{Fid}_{-}=1-\frac{1}{N}\sum_{i=1}^{N}\mathbbm{1}(\hat{y}^{G_{\mathrm{ expl}}}_{i}-\hat{y}_{i}),\quad\mathrm{Fid}_{+}=1-\frac{1}{N}\sum_{i=1}^{N} \mathbbm{1}(\hat{y}^{G_{\mathrm{comp}}\setminus G_{\mathrm{expl}}}_{i}-\hat{y} _{i}),$ (7) $$

where $N$ is the number of instance-level predictions and $\hat{y}_{i}$ is the class prediction of the (hyper)GNN on the $i$th instance. The superscripts indicate a restriction of the (hyper)GNN to predict over that sub(hyper)graph.
$G_{\mathrm{comp}}\setminus G_{\mathrm{expl}}$ is the complement sub(hyper)graph to the explanation sub(hyper)graph with respect to the computational sub(hyper)graph.(^2^22The hypergraph complement is comprised of all the node-hyperedge links that exist in $G_{\mathrm{comp}}$ but do not appear in $G_{\mathrm{expl}}$. This generalizes the graph complement, which comprises the edges (and nodes at either end of the edge) which exist in $G_{\mathrm{comp}}$ but do not appear in $G_{\mathrm{expl}}$.)
A low $\mathrm{Fid}_{-}$ suggests the explanation is sufficient, and a high $\mathrm{Fid}_{+}$ suggests the explanation is necessary.
However, fidelity is vulnerable to some shortcomings,
which we identify below and address with alternatives.

#### Measuring faithfulness with generalized fidelity.

A major drawback of fidelity is that it is easily saturated.
Because correct classification suffices to maximise each term in the sum,
this metric is insensitive to more moderate perturbations to the logits.
For example, we often care if the output class was predicted with 90% probability, or by only a narrow margin.
To this end, we introduce a generalization to fidelity parametrized by a similarity function $s(\bm{p},\bm{q})$,
where $\bm{p},\bm{q}$ are probability distributions over the classes $c\in C$:

$$ $\mathrm{Fid}^{s}_{-}=\frac{1}{N}\sum_{i=1}^{N}s(\mathbf{p}^{G_{\mathrm{expl}}} _{i},\ \bm{p}_{i}),\quad\mathrm{Fid}^{s}_{+}=\frac{1}{N}\sum_{i=1}^{N}s(\bm{p} ^{G_{\mathrm{comp}}\setminus G_{\mathrm{expl}}}_{i},\ \bm{p}_{i}).$ (8) $$

Below we suggest a few good choices of $s$.
Similar to a metric introduced by Agarwal et al. (), we can instantiate $s$ as the Kullback-Leibler divergence:

$$ $s_{\textrm{KL}}(\bm{p},\bm{q}):=D_{\textrm{KL}}(\bm{p}\ ||\ \bm{q})=\sum_{c\ \in\ C}p(c)\log\bigg{(}\frac{p(c)}{q(c)}\bigg{)}.$ (9) $$

The total variation distance is another apt statistical distance for our purpose. In a discrete probability space, it is essentially the L1 distance:

$$ $s_{\textrm{TV}}(\bm{p},\bm{q}):=\frac{1}{2}\sum_{c\ \in\ C}\big{|}p(c)-q(c) \big{|}.$ (10) $$

The negative cross-entropy is also a sensitive choice of $s$. It is equivalent to the logarithmic score, a strictly proper scoring rule in decision theory:

$$ $s_{\textrm{xent}}(\bm{p},\bm{q}):=\sum_{c\ \in\ C}p(c)\log{q(c)}.$ (11) $$

Finally, the original fidelity metrics (Amara et al., ) are subsumed under this framework by choosing

$$ $s_{\textrm{Acc}}(\bm{p},\bm{q}):=1-\mathbbm{1}(\mathrm{argmax}_{c}\ p(c)- \mathrm{argmax}_{c}\ q(c)).$ (12) $$

For regression tasks, $s$ can be replaced by MSE.

#### Measuring concision with size.

By definition, a low $\mathrm{Fid}^{s}_{-}$ shows that the explanation is faithful,
since it can reproduce the original output over the full input hypergraph.
But does a high $\mathrm{Fid}^{s}_{+}$ indeed show that the explanation is also concise and necessary?
We observe that $\mathrm{Fid}^{s}_{+}$ can be especially misleading in the hypergraph context,
since a subhypergraph’s complement may also contain important nodes in $G_{\mathrm{expl}}$ .
(Further discussion in Appendix E.)
Instead, we propose to quantify concision by the size $|G_{\mathrm{expl}}|_{1}$ and density $|G_{\textrm{expl}}|_{1}/|G_{\textrm{comp}}|_{1}$ of the explanation subhypergraph.
We desire explanations of low size and low density.
Density attains the maximum value of 1 iff $G_{\textrm{expl}}=G_{\textrm{compl}}$, in which case the explanation is perfectly (if trivially) faithful.

### 5.3 Results

We compare our method against HyperEX (Maleki et al., ), which is currently the only hypergraph explainer in the literature, as well as Random, Gradient, and Attention baselines.
(See Appendix C for further details on baselines and experimental setup.)
For each dataset, all explanation methods are applied to the same model (a trained AllSetTransformer).
Separately, we perform an ablation for our explainer’s sampling technique in Section F.

**Table 1: Quantitative evaluation of hyperGNN explainers on the synthetic benchmarks. We compare explanation faithfulness, measured by generalized fidelity metrics, and concision, measured by subhypergraph size and density. Our method consistently outputs more faithful explanations than all baselines, which are given comparable or more generous size budgets ($n=20$ for H-TreeGrid, $n=10$ for all other datasets).**
|  |  | $\textrm{Fid}_{-}^{\textrm{Acc}}$ ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{KL}}$ ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{TV}}$ ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{Xent}}$ ($\downarrow$) | Size ($\downarrow$) | Density ($\downarrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| H-RandHouse | Random | 0.81 | 1.14 | 0.60 | 1.68 | 1.2 | 0.07 |
|  | Gradient | 0.36 | 0.69 | 0.32 | 1.23 | 8.3 | 0.26 |
|  | Attention | 0.61 | 0.82 | 0.45 | 1.36 | 3.6 | 0.17 |
|  | HyperEX | 0.86 | 1.09 | 0.62 | 1.63 | 0.0 | 0.01 |
|  | SHypX | 0.01 | 0.04 | 0.06 | 0.59 | 9.2 | 0.19 |
| H-CommHouse | Random | 0.78 | 3.54 | 0.76 | 3.70 | 1.0 | 0.06 |
|  | Gradient | 0.29 | 1.17 | 0.30 | 1.33 | 9.1 | 0.24 |
|  | Attention | 0.71 | 3.03 | 0.70 | 3.19 | 1.6 | 0.09 |
|  | HyperEX | 0.79 | 3.63 | 0.77 | 3.79 | 0.1 | 0.02 |
|  | SHypX | 2e–3 | 0.02 | 0.03 | 0.18 | 9.2 | 0.20 |
| H-TreeCycle | Random | 0.52 | 1.88 | 0.53 | 1.93 | 1.4 | 0.08 |
|  | Gradient | 0.29 | 1.21 | 0.28 | 1.27 | 8.3 | 0.35 |
|  | Attention | 0.26 | 0.91 | 0.31 | 0.97 | 3.0 | 0.16 |
|  | HyperEX | 0.35 | 0.64 | 0.40 | 0.70 | 0.0 | 0.00 |
|  | SHypX | 3e–3 | 0.01 | 0.01 | 0.07 | 5.6 | 0.22 |
| H-TreeGrid | Random | 0.68 | 2.11 | 0.63 | 2.30 | 8.6 | 0.35 |
|  | Gradient | 0.40 | 1.04 | 0.36 | 1.24 | 17.9 | 0.56 |
|  | Attention | 0.42 | 1.15 | 0.38 | 1.35 | 11.3 | 0.43 |
|  | HyperEX | 0.66 | 1.63 | 0.57 | 1.82 | 13.4 | 0.46 |
|  | SHypX | 0.01 | 0.02 | 0.04 | 0.22 | 15.1 | 0.45 |

**Table 2: Quantitative evaluation on four real world datasets. Our method consistently produces explanations that are both more faithful (as measured by $\textrm{Fid}_{-}^{*}$ metrics) and more concise (as measured by Size and Density) than all baselines.**
|  |  | $\textrm{Fid}_{-}^{\textrm{Acc}}$ ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{KL}}$ ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{TV}}$ ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{Xent}}$ ($\downarrow$) | Size ($\downarrow$) | Density ($\downarrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Cora | Random | 0.01 | 0.01 | 0.01 | 0.05 | 3.7 | 0.90 |
|  | Gradient | 0.01 | 0.03 | 0.01 | 0.06 | 3.9 | 0.91 |
|  | Attention | 4e–3 | 0.02 | 0.01 | 0.05 | 3.7 | 0.91 |
|  | HyperEX | 0.01 | 0.03 | 0.01 | 0.07 | 4.1 | 0.92 |
|  | SHypX | 0.00 | 5e–4 | 1e–3 | 0.03 | 1.4 | 0.61 |
| CoauthorCora | Random | 0.10 | 0.25 | 0.09 | 0.31 | 5.4 | 0.67 |
|  | Gradient | 0.01 | 0.05 | 0.02 | 0.11 | 7.2 | 0.74 |
|  | Attention | 0.02 | 0.08 | 0.02 | 0.14 | 6.4 | 0.71 |
|  | HyperEX | 0.01 | 0.03 | 0.02 | 0.10 | 7.4 | 0.75 |
|  | SHypX | 0.00 | 1e–3 | 3e–3 | 0.07 | 2.1 | 0.28 |
| CoauthorDBLP | Random | 0.11 | 0.48 | 0.14 | 0.48 | 5.5 | 0.52 |
|  | Gradient | 0.01 | 0.03 | 0.01 | 0.03 | 8.4 | 0.60 |
|  | Attention | 0.01 | 0.07 | 0.01 | 0.07 | 6.7 | 0.55 |
|  | HyperEX | 0.01 | 0.05 | 0.01 | 0.05 | 8.8 | 0.61 |
|  | SHypX | 0.00 | 3e–4 | 3e–4 | 2e–3 | 2.3 | 0.15 |
| Zoo | Random | 0.79 | 1.74 | 0.69 | 1.92 | 0.3 | 0.00 |
|  | Gradient | 0.03 | 0.06 | 0.05 | 0.24 | 9.7 | 0.01 |
|  | Attention | 0.08 | 0.26 | 0.08 | 0.44 | 3.1 | 0.00 |
|  | HyperEX | 0.04 | 0.09 | 0.06 | 0.28 | 10.0 | 0.01 |
|  | SHypX | 0.03 | 0.01 | 0.01 | 0.19 | 6.7 | 0.01 |

Synthetic hypergraphs.
Our method, SHypX, significantly outperforms all baselines across all four synthetic datasets (Table 1).
While Gradient and Attention show substantial improvements from Random (e.g. on H-RandHouse, $\textrm{Fid}_{-}^{\textrm{Acc}}$ is 0.36 and 0.61 respectively, compared to Random’s 0.81),
they don’t consistently produce faithful explanations.
On synthetic hypergraphs, HyperEX performs on par with Random.
We hypothesize that this is because it mean-aggregates nodes to produce hyperedge representations,
which constitutes a homophily assumption that is violated in the synthetic case.
In comparison, the explanations produced by our method reliably achieves near zero fidelity metrics.

Real hypergraphs.
On the real world hypergraphs, SHypX also outperforms all baselines.
For example, in Coauthor-Cora, we achieve $\textrm{Fid}_{-}^{\textrm{KL}}$ of $310-4$, compared to $0.03,0.05,0.08,0.25$ for HyperEX, Gradient, Attention, and Random respectively. While producing more faithful explanations, our model does not sacrifice concision:
it achieves this superior fidelity with the best concision on this dataset, at average size 2.1 and density 0.28.
This relative ranking between methods is consistent across all four real hypergraphs.
We also observe that the simple baselines Random, Gradient, and Attention already attain competitive performance on several real hypergraphs.
Cora is the most extreme example of this, where even Random produces faithful explanations at $\textrm{Fid}_{-}^{\textrm{KL}}=0.01$.
Indeed, SHypX’s mean explanation size of 1.4 suggests that oftentimes, just the node’s features, without neighborhood structure, suffice to achieve perfect predictions over Cora.
This “structural degeneracy” is also observed to some extent for CoauthorCora and CoauthorDBLP.
These results support Section 5.2’s discussion about complementing evaluations on real hypergraphs with our challenging synthetic ones,
and leveraging generalized fidelity as a more discriminating metric.

Comparing explanation methods across different concision budget.
In Table 1 and Table 2, for each dataset, we fixed the same hyperparameter $n$ across all baselines (to obtain the top-$n$ node-hyperedge links) such that at least one baseline produces explanations of comparable concision to SHypX; this ensures a fair comparison between the fidelity results.
We observe that the baseline explainers often do not even select components that are connected to the node being explained.
Note that, since post-processing discards these disconnected components
(see Section 4.1
),
$|G_{\mathrm{expl}}|\leq n$ the explanation size can vary across baselines despite their identical choice of $n$.
To understand how the quality of explanation varies when allowing larger subhypergraphs as explanation, we designed an experiment in which we directly control for the size of the final explanation and compare $\textrm{Fid}_{-}^{\textrm{KL}}$ (see Figure 3).
The outperformance of our method is robust across the curve,
whereas the baseline methods “buy” limited gains in fidelity with increasing size budget.

#### Trading off faithfulness with concision.

By adjusting the relative strengths of the $\lambda_{\mathrm{pred}}$ and $\lambda_{\mathrm{size}}$ coefficients, our model allows the users to effectively trade off between explanation faithfulness and concision.
Figure 3a shows H-RandHouse explanations obtained with $\lambda_{\mathrm{pred}}/\lambda_{\mathrm{size}}\in\{0.2,0.1,0.05,0.02,0.01,0.0
05\}$.
As this ratio shrinks, the extracted explanations interpolate smoothly from concise-but-less-faithful (0.36 $\mathrm{Fid}_{-}^{\mathrm{KL.}}$, mean size 4) to verbose-and-highly-faithful ($310-3$ $\textrm{Fid}_{-}^{\mathrm{KL.}}$, mean size 22).
Similarly, for Zoo, explanations obtained with $\lambda_{\mathrm{pred}}/\lambda_{\mathrm{size}}\in\{$110-2$,$510-3$,$210-3$,$1
10-3$,$510-4$\}$ form a smooth decaying curve from higher to near-zero fidelity.
Interestingly, Figure 3 suggests that for H-RandHouse, all baselines perform similarly once adjusted for final explanation size,
and that for Zoo, no baseline method reliably improves in fidelity with increasing size budget.
Finally, we note that specifying the trade-off via $\lambda_{\mathrm{pred}}/\lambda_{\mathrm{size}}$ confers our method an additional benefit:
it can dynamically adapt the explanation size for each node, according to the relevance of a node’s neighborhood in the local hyperGNN prediction.
In contrast, the baselines explainers inflexibly apply top-$n$ thresholding across all node instances.

Figure: ((a)) H-RandHouse.
Refer to caption: x8.png

Figure: Figure 4: Global concepts on H-RandHouse dataset. Class 0 is the base hypergraph, Class 1 is top-of-the-house, Class 2 is middle-of-the-house, and Class 3 is bottom-of-the-house. Concepts were extracted with 10 clusters, which sufficed to score well on the concept completeness metric (Appendix C.3).
Refer to caption: x10.png

#### Global explanations.

Figure 4 shows the concept-level explanation subhypergraphs provided by our explainer for H-RandHouse (for more datasets see Appendix D).
We find that H-RandHouse’s concept explanations are readily interpretable:
the Class 1, 2, and 3 concepts clearly show each respective top-of-house, middle-of-house, and bottom-of-house node situated within a house-like motif.
Particularly interesting is the subdivision of Class 2 into two distinct concepts:
one for the “anchor node” that is attached to the base hypergraph (includes the attaching hyperedge),
and one for the non-anchor node.
This reveals that the hyperGNN implicitly represents and reasons about two types of Class 2 nodes.
Furthermore, the Class 3 concept is visualized as a fragment of the house motif,
suggesting that this hyperGNN does not rely on the top-of-house node to make Class 3 predictions.
This mechanism is not a priori obvious,
and such information could be leveraged to debug the hyperGNN.
The remaining concepts corresponding to Class 0 reflect an eclectic variety,
representative of the diverse neighborhoods of nodes in the random base graph.

## 6 Conclusion

Explainability for hyperGNNs is an under-explored topic, but essential for their responsible deployment in critical applications.
We introduce SHypX, a model-agnostic post-hoc explainer, and demonstrate its efficacy with extensive evaluations.
At the instance-level, our method finds explanation subhypergraphs that can target a desired tradeoff between explanation faithfulness and concision.
At the model-level, we are the first to extend our instance-level method with concept extraction to efficiently derive concise global explanation subhypergraphs.
Additionally,
we design novel synthetic hypergraph datasets and
propose more general fidelity metrics,
which together allow for a challenging and sensitive evaluation of hyperGNN explainers.

## Appendix: Explaining Hypergraph Neural Networks: From Local Explanations to Global Concepts

This appendix contains details related to our proposed hypergraph neural network explainer as detailed below.
Full code associated with the proposed method will be released soon.

- •
Section A contains a discussion about additional scenarios when our model can be applied, which where not fully explored in the main paper.
- •
Section B provides more details about the proposed synthetic benchmark.
- •
Section C contains implementation details about our model and the baselines used in our experiments.
- •
Section D highlights additional global-level visualizations and a qualitative comparison between the concepts extracted by GCExplainer and the one extracted by our model.
- •
Section E includes a detailed discussion about the limitations of $\textrm{Fid}_{+}$ metric.
- •
Section F includes an additional ablation study investigating the choice of sampling technique.

## Appendix A Extensions and discussion

#### Beyond node classification.

We focused on node classification tasks to simplify exposition.
Nonetheless, our framework and methods is general to regression, as well as tasks that operate at the edge and graph level.
In regression, we simply replace the KL divergence in $\mathcal{L}$ with MSE, since $f$ outputs regression targets instead of class probabilities.
For hyperedge- and hypergraph-level tasks, $G_{\mathrm{comp}}$ may be more complex than a $d$-hop neighborhood, depending on the architecture, or even equal to $G$.
Additionally, we allow disconnected components if they contribute to the final prediction, e.g. for hypergraph-level tasks formulated with a global pooling layer.
The overall pipeline is otherwise unchanged.

#### Feature selection.

The novelty of our hypergraph explainer lies in “structure selection”, i.e. finding the explanation subhypergraph.
We may wish to simultaneously find a subset of the features which are most important to each local instance.
Ying et al. () accomplishes feature selection by learning a $L_{1}$-regularized mask $M$ over feature vectors,
which we may directly adopt to update our objective to

$$ $\displaystyle\mathcal{L}(f,G_{\mathrm{sub}},G_{\mathrm{comp}},\bm{X},\bm{M},v) =\lambda_{\mathrm{pred}}D_{\textrm{KL}}\big{(}f(G_{\mathrm{sub}},\bm{X},v)\ || \ f(G_{\mathrm{comp}},\bm{X}\odot\bm{M},v)\big{)}$ $\displaystyle\phantom{..........}+\lambda_{\mathrm{size}}\big{|}G_{\mathrm{sub }}\big{|}_{1}+\lambda_{\mathrm{feat}}\big{|}\bm{M}\big{|}_{1},\quad G_{\mathrm {sub}}\subseteq G_{\mathrm{comp}},\quad G_{\mathrm{expl}},\bm{M}^{*}= \operatorname*{arg\,min}_{G_{\mathrm{sub}},\bm{M}}\mathcal{L}.$ (13) $$

Since this approach is equally suitable for graphs and hypergraphs (and indeed, any other modality with multidimensional features), we do not focus on it in the present work.

#### Generality across architectures.

Finally, we emphasize that SHypXis model-agnostic.
It only relies on the high level message passing abstraction, which ensures that the notion of a computational subhypergraph is well-defined, and $f$ can accept any subhypergraph as an input.
Our explainer can be applied to any hyperGNN, such as HGNN, HCHA, UniGNN, and AllSet models.

## Appendix B Synthetic dataset

Our synthetic hypergraphs are designed with a “base-and-motif” construction , inspired by Ying et al. ().
For the random base, we sample a random bipartite graph with $n,m$ nodes in each of the bipartite sets respectively, and $k$ edges between them uniformly at random.
We take the largest connected component of this bipartite graph and apply the inverse star expansion to obtain a random base hypergraph (Figure 2a).
For the tree base, we enclose each triplet of a parent node and its two child nodes in a hyperedge.
This produces a tree base hypergraph that is deterministic and 3-uniform (Figure 2b).
The house, cycle, and grid motifs from Ying et al. () are also lifted to hypergraph motifs (Figure 2c-e).
In designing these, we were motivated by preserving the natural symmetries of each motif, without rendering the classification task trivial (for example, allowing motifs to be immediately distinguishable from a tree base by hyperedge degree).
In the example visualized in Figure 2e,
the hypergraph consists of a random base of 13 nodes (blue nodes and grey hyperedges), 2 house motifs, and 3 edge perturbations (pink hyperedges).

Different combinations of these base and motif components give rise to the synthetic hypergraphs H-RandHouse, H-CommHouse, H-TreeCycle, and H-TreeGrid (Table 3).
H-CommHouse comprises two H-RandHouse graphs, i.e. “communities”, stitched together with random edges.
Each node has features drawn from a normal distribution, whose mean and variance depend on the community membership.
The other three synthetic graphs have trivial features, which we choose to be all ones.
(We observed similar performance for all zeros or standard random normal features.)
Perturbations, in the form of degree-2 hyperedges, are then added randomly to simulate structural noise, increasing the difficulty of the task.
A train-validation split at 80% train nodes is applied to each hypergraph.

The benchmark task over our synthetic hypergraphs is node classification, where the node labels depend on the node’s position in the base or motif.
Each class is denoted by a distinct color in Figure 2.
In particular, all base nodes are Class 0, and all nodes in the cycle and grid motif are Class 1.
The house motif is further sub-divided into top-of-the-house (Class 1), middle-of-the-house (Class 2), and bottom-of-the-house (Class 3).

We benchmark several hyperGNN architectures on our synthetic tasks.
As claimed, the synthetic hypergraphs are challenging.
Table 4 shows that performance improves with stronger models, and the structure-agnostic MLP does no better than random.

**Table 3: Construction of novel synthetic hypergraphs. Upper section reports fundamental properties of each hypergraph dataset, such as its base and type of attached motif. Lower section reports, for each family, the default parameters used to instantiate the hypergraph used in our evaluations.**
|  | H-RandHouse | H-CommHouse | H-TreeCycle | H-TreeGrid |
| --- | --- | --- | --- | --- |
| base | random | random | tree | tree |
| motif | house | house | cycle | grid |
| node feat. | none (ones) | bimodal normal | none (ones) | none (ones) |
| # classes | 4 | 8 | 2 | 2 |
| # base nodes | 312 | 648 | 255 | 255 |
| # motifs | 100 | 200 | 80 | 80 |
| # perturb. edges | 80 | 80 | 80 | 80 |
| # inter-community edges | - | 80 | - | - |

**Table 4: Benchmarking hypergraph neural networks on the synthetic hypergraphs. Each number denotes the mean final validation accuracy, in %, over 5 random seeds. All models are three layers deep, use sum aggregation, and no dropout. AllDeepSets and AllSetTransformer have dimension-16 message passing and classifier layers; MLP, HGNN, HCHA have dimension-80 hidden layers, which ensures all models have comparable parameter count. All models are trained with the Adam optimizer at 0.001 learning rate, for 2000 epochs (MLP, HGNN, HCHA) or 500 epochs (AllDeepSets and AllSetTransformer), which we observed sufficed to achieve convergence. Other hyperparameters are per Chien et al. ()’s defaults. Boldface indicates the best model.**
|  | H-RandHouse | H-CommHouse | H-TreeCycle | H-TreeGrid |
| --- | --- | --- | --- | --- |
| MLP | $38.65_{\pm 0.00}$ | $28.91_{\pm 2.39}$ | $65.31_{\pm 0.00}$ | $73.85_{\pm 0.00}$ |
| HGNN | $79.75_{\pm 10.34}$ | $60.30_{\pm 1.59}$ | $85.44_{\pm 2.57}$ | $92.62_{\pm 2.80}$ |
| HCHA | $56.32_{\pm 20.48}$ | $26.12_{\pm 10.47}$ | $65.31_{\pm 0.00}$ | $78.26_{\pm 9.58}$ |
| AllDeepSets | $89.20_{\pm 7.18}$ | $93.33_{\pm 9.87}$ | $\bm{86.26}_{\pm 9.09}$ | $87.49_{\pm 4.39}$ |
| AllSetTransformer | $\bm{95.09}_{\pm 6.95}$ | $\bm{97.15}_{\pm 2.29}$ | $83.95_{\pm 12.38}$ | $\bm{90.05}_{\pm 4.79}$ |

## Appendix C Further experiment details

### C.1 Baselines

We compare our explainer against four baseline methods: Random, Gradient, Attention, and HyperEX.
Each of these baselines is parametrized by $n$,
such that the top-$n$ node-hyperedge links are selected according to each method’s importance ranking.

- •
Random.
The importance score of each node-hyperedge link is randomly assigned as a random variable drawn from $U(0,1)$.
That is, we get the subhypergraph induced by $n$ uniformly random node-hyperedge links.
- •
Gradient.
We compute the gradient of the logit on the predicted class of the node being explained, with respect to the hypergraph edge index.
We then get the subhypergraph induced by the $n$ node-hyperedge links with the largest non-zero gradients by absolute value.
Note that our gradient baseline is significantly more competitive than the ostensibly similar saliency and integrated gradients baselines constructed by Maleki et al. (). Our gradient baseline computes gradients on the hyperedge index, and thus selects a set of node-hyperedge links. Their gradient baselines compute gradients over the input nodes, and thus selects a set of nodes.
- •
Attention.
This baseline is only feasible for hyperGNNs with an attention mechanism.
Since we produce all explanations with respect to the AllSetTransformer architecture, this is satisfied.
We compute the mean of the attention weights from each layer.
For each AllSetTransformer layer, this includes attention weights in both the node-to-hyperedge and hyperedge-to-node directions.
We then get the subhypergraph induced by the $n$ node-hyperedge connections with the largest non-zero attention weights by absolute value.
- •
HyperEX.
The hypergraph explainer by Maleki et al. () proposes to calculate importance weights between nodes and hyperedges with a shallow attention model surrogate parametrized as
$\alpha_{ve}=\frac{\exp(\omega_{ve})}{\sum_{\tilde{e}:v\in\tilde{e}}\exp{\omega
_{v\tilde{e}}}},\quad\omega_{ve}=(\bm{W}_{Q}\bm{z}_{v})^{T}\cdot(\bm{W}_{K}\bm
{h}_{e})\cdot s_{v},$
(14)
where $\bm{W}_{Q},\bm{W}_{K},s_{v}$ are learnable weights.
$\bm{z}_{v}$ is the latent representation of node $v$, per the trained hyperGNN,
and $\bm{h}_{e}$ is the latent representation of hyperedge $e$, which Maleki et al. () compute by mean aggregation of its neighborhood:
$\bm{h}_{e}=\frac{1}{|\mathcal{N}(e)|}\sum_{v\in\mathcal{N}(e)}\bm{z}_{v}$.
HyperEX’s code is not publicly released, but was shared with us in private communications.
To facilitate fair comparison with our method and other baselines,
we adapt their implementation into our own pre- and post-processing pipelines.
Like the original authors,
we choose the hidden dimension of the attention surrogate model to be 16 and
train on 50% of the node instances with InfoNCE loss.
We choose the learning rate 0.1 by hyperparameter search.
HyperEX requires retraining a new model for each choice of $n$.

For the baselines in each dataset, we choose $n$ such that the density of the gradient or attention explanations is comparable to, or greater than, the density of our explanations.
This ensures our method does not have an unfair advantage.
We find that $n=10$ for all datasets except $n=20$ for H-TreeGrid suffices to achieve this comparison.
Note that these size budgets are greater than the mean size of explanations produced by our method on their respective datasets.
Alternatively, Figure 3 compares all methods across the entire curve of varying explanation size budgets.

For consistency, all explanation methods operate over the same AllSetTransformer model for each dataset.
This model’s task performance is reported in Table 5.
All explanation methods benefit from identical pre-processing, which reduces the search space to the computational subhypergraph.
They are also subject to the same post-processing, which retain only the connected component containing the node being explained (as described in Section 4.1).
This means the mean explanation size obtained is generally less than $n$.

### C.2 Hyperparameters for SHypX

For the main results of Table 1 and Table 2,
we choose our explanation concision budget by setting $\lambda_{\textrm{pred}}=1$, and $\lambda_{\textrm{size}}=0.05$ for the synthetic datasets and $\lambda_{\textrm{size}}=0.005$ for the real world datasets.
Alternative choices of $\lambda$ for two select datasets are reported in Figure 3.
The explanation subhypergraph is sampled with Gumbel-Softmax at temperature 1.0, and optimized with Adam for 400 epochs at learning rate 0.01.
The probability of sampling each node-hyperedge link ($\pi_{v,e}^{(1)}$) is initialized uniformly to $\approx 95\%$ across the computational subhypergraph.

### C.3 Concept extraction and concept completeness

We extract concepts by k-means clustering,
as described in Section 4.2.
The quality of concept extraction is quantified by concept completeness, the accuracy of a decision tree classifier that optimally maps the set of concepts onto the set of class labels (Magister et al., ).
Optimal is defined such that each node instance, featurized only by its concept label, is mapped to a class label with high accuracy.
Since the concept label is a discrete class, the decision tree classifier is optimized by performing majority vote within each concept, as proposed in Section 4.2.
We consider the concept extraction successful if its concept completeness is close to the task accuracy,
since this overall procedure relies on the latent representations learned by the hyperGNN.

Table 5 shows that across all datasets, the latents are indeed such that we can successfully extract meaningful concepts that score well on concept completeness (i.e. within a few percentage points of the task accuracy).
We find that $k=10$ suffices to achieve this condition on all datasets, but that it is beneficial to increase to $k=15$ for H-CommHouse.

**Table 5: Task performance (accuracy on train, validation, and test splits) for each dataset, and the concept completeness of extracted concepts. (Note we did not use a separate test split for the synthetic datasets in our experiments.) The decision tree classifier used to compute concept completeness uses the same train/validation split as the base task.**
|  | Train Acc. | Val Acc. | Test Acc. | $k$ | Concept completeness |
| --- | --- | --- | --- | --- | --- |
| H-RandHouse | 0.98 | 0.96 | - | 10 | 0.96 |
| H-CommHouse | 1.00 | 1.00 | - | 15 | 0.96 |
| H-TreeCycle | 0.99 | 0.98 | - | 10 | 0.98 |
| H-TreeGrid | 0.96 | 0.96 | - | 10 | 0.94 |
| Cora | 1.00 | 0.79 | 0.77 | 10 | 0.72 |
| CoauthorCora | 1.00 | 0.84 | 0.82 | 10 | 0.87 |
| CoauthorDBLP | 1.00 | 0.91 | 0.91 | 10 | 0.93 |
| Zoo | 1.00 | 0.96 | 0.96 | 10 | 1.00 |

## Appendix D Further concept visualizations

### D.1 Concepts for other hypergraphs

We report concept visualizations for H-CommHouse (Figure 5), H-TreeCycle (Figure 6), and H-TreeGrid (Figure 7), analogous to Figure 4 for H-RandHouse.

Figure: Figure 5: Concepts for H-CommHouse.
Refer to caption: x11.png

Figure: Figure 6: Concepts for H-TreeCycle.
Refer to caption: x12.png

Figure: Figure 7: Concepts for H-TreeGrid.
Refer to caption: x13.png

### D.2 Improvement over GCExplainer

Visualizing concepts by the $n$-hop neighborhood of their representative nodes, as suggested by directly generalizing the GNN explainer of Magister et al. (),
can produce crowded hypergraphs that obscure the crucial neighborhood important to that node instance.
In Figure 8 and Figure 9 for H-RandHouse and CoauthorCora respectively,
we demonstrate with a few examples of concepts extracted from each hypergraph.
For H-RandHouse, we see that our method (bottom row) more clearly reveals the house motif when explaining nodes located in the motif.
For CoauthorCora, the frequent appearance of the trivial subhypergraph (i.e., comprising only the node being explained) in our explanations
reveals that class labels depend more strongly on features than local structure.
This observation is not apparent from visualising $n$-hop neighborhoods (top row).

Figure: Figure 8: Concepts visualized by the $n$-hop expansion (setting $n=3$, the depth of the hyperGNN) for CoauthorCora (top row), and their respective visualizations when simplified using our method (bottom row). First column: Class 0 node from base. Second column: Class 3 node from house motif. Third column: Class 2 node from house motif.
Refer to caption: x14.png

Figure: Figure 9: Concepts visualized by the $n$-hop expansion (setting $n=1$, the depth of the hyperGNN) for H-RandHouse (top row), and their respective visualizations when simplified using our method (bottom row).
Refer to caption: x20.png

## Appendix E Limitations of Fid+

**Table 6: Fidelity and size metrics on the explanation complement, for two select datasets. We find that this can be a misleading metric.**
|  |  | $\textrm{Fid}_{+}^{\textrm{Acc}}$ ($\uparrow$) | $\textrm{Fid}_{+}^{\textrm{KL}}$ ($\uparrow$) | $\textrm{Fid}_{+}^{\textrm{TV}}$ ($\uparrow$) | $\textrm{Fid}_{+}^{\textrm{Xent}}$ ($\uparrow$) | Size ($\uparrow$) | Density ($\uparrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| H-TreeGrid | Random | 0.59 | 1.86 | 0.55 | 2.06 | 29.3 | 0.65 |
|  | Gradient | 0.73 | 2.23 | 0.67 | 2.43 | 30.6 | 0.78 |
|  | Attention | 0.42 | 1.36 | 0.39 | 1.56 | 33.1 | 0.80 |
|  | HyperEX | 0.77 | 2.23 | 0.70 | 2.42 | 24.4 | 0.54 |
|  | SHypX | 0.77 | 2.29 | 0.70 | 2.48 | 22.8 | 0.55 |
| CoauthorDBLP | Random | 0.32 | 0.95 | 0.33 | 0.95 | 22.2 | 0.48 |
|  | Gradient | 0.56 | 1.75 | 0.61 | 1.75 | 19.3 | 0.40 |
|  | Attention | 0.47 | 1.52 | 0.49 | 1.52 | 21.0 | 0.45 |
|  | HyperEX | 0.72 | 2.25 | 0.80 | 2.25 | 18.9 | 0.39 |
|  | SHypX | 0.07 | 0.33 | 0.09 | 0.34 | 25.4 | 0.85 |

Figure: ((a))
Refer to caption: x26.png

The $\textrm{Fid}_{+}$ metric has been used to measure whether an explanation is “sufficient“, that is, whether it is free of superfluous information.
A large $\textrm{Fid}_{+}$ indicates that the explanation’s complement does not contain useful information for the hyperGNN’s prediction.
This has been thought to suggest that the explanation has successfully isolated the useful information.
However, this reasoning is flawed – a successful explanation (achieving the user-desired balance of faithfulness and concision) could nonetheless induce a complement subhypergraph that can also reproduce the hyperGNN’s prediction.
This can be seen with a simple intuition: when the explanation subhypergraph is concise – that is, all of its parts are necessary, as desired – the complement is large.
The complement is therefore likely to include a large number of hyperedges and neighbors directly incident to the node being explained.
This allows the complement to reproduce the hyperGNN’s prediction with high fidelity, producing low $\textrm{Fid}_{+}$ scores.
See Figure 10 for an illustrative example.

To concretely illustrate some of these failure modes, we expose the $\textrm{Fid}_{+}$ scores of H-TreeGrid and Coauthor-DBLP in Table 6.
For H-TreeGrid,
the similar $\textrm{Fid}_{+}$ for Gradient and our method suggest that they are comparably successful at isolating relevant information to the explanation subhypergraph.
However, this does not align with our natural understanding of which explanations are more “sufficient” – whereas Table 1 shows that our method achieves an extremely low fidelity at mean explanation size of 15.1 and mean explanation density of 0.45,
Gradient produces explanations with $\textrm{Fid}_{-}^{Acc}=0.40$ while being almost 3 links larger and 10 percentage points denser.
For CoauthorDBLP, our method yields the most faithful and most concise explanations (average size 2.3 and average density 0.15) of all baselines (Table 2).
However, the small size of these explanations induces a large complement,
contributing to its unfavorable $\textrm{Fid}_{+}$ scores.

Based on these observations,
we opt for hypergraph size $|G|_{1}$ (Section 5.2) as a cheaper and less artefact-prone measure of explanation minimality.

## Appendix F Sampler ablation

**Table 7: Ablating the choice of Gumbel-Softmax sampler to two alternatives: relax-and-thresh (Ying et al., ) and sparsemax (Martins & Astudillo, ). Here, the loss function has coefficients $\lambda_{\textrm{pred}}=1$ and $\lambda_{\textrm{size}}=0.005$. Lowest losses are in boldface.**
|  |  | Loss ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{Acc}}$ ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{KL}}$ ($\downarrow$) | $\textrm{Fid}_{-}^{\textrm{TV}}$ ($\downarrow$) | Size ($\downarrow$) | Density ($\downarrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| H-RandHouse | gumbel-softmax | 0.10 | 0.00 | 0.00 | 0.01 | 19.5 | 0.32 |
|  | relax-and-thresh | 0.15 | 0.07 | 0.06 | 0.06 | 18.2 | 0.31 |
|  | sparsemax | 0.58 | 0.28 | 0.52 | 0.25 | 12.5 | 0.21 |
| zoo | gumbel-softmax | 0.04 | 0.03 | 0.01 | 0.01 | 6.7 | 0.01 |
|  | relax-and-thresh | 0.14 | 0.07 | 0.09 | 0.06 | 10.4 | 0.01 |
|  | sparsemax | 0.08 | 0.05 | 0.04 | 0.04 | 6.5 | 0.01 |

In this section, we investigate the choice of sampling technique.
Since this choice pertains to the optimization, we are primarily interested in which sampler achieves the lowest loss given a fixed objective function (Equation 2).
Table 7 compares the loss attained by the Gumbel-Softmax sampler (our choice) against two alternatives,
as well as reporting their respective fidelity and size metrics for reference.

We compare against a “relax-and-thresh” method, which is the continuous relaxation for GNN explanations popularized by GNNExplainer (Ying et al., ).
Relax-and-thresh learns real-valued probability weights over the incidence matrix, which are optimized by gradient descent.
To encourage discrete sampling, it employs an entropy penalty to softly regularize these weights to 0s and 1s.
Discreteness is only strictly enforced during post-processing:
after optimization, the probability weights are binarized by thresholding (typically at 0.5) to produce the explanation subhypergraph.
(Upon binarization, the entropy loss becomes trivially zero.)

We also try replacing Gumbel-Softmax with a sparsemax sampler (Martins & Astudillo, ).
Whereas the familiar softmax function maps logits $\bm{z}_{i}$ onto a probability distribution by $\textrm{softmax}_{i}(\bm{z})=\exp(\rm{z}_{i})/\sum_{j}\exp(\rm{z}_{j})$,
sparsemax proposes an alternative transformation:

$$ $\textrm{sparsemax}(\bm{z})=\operatorname*{arg\,min}_{\bm{p}\in\Delta^{K-1}}|| \bm{p}-\bm{z}||^{2},$ (15) $$

where $\Delta^{K-1}=\{\bm{p}\in\mathbb{R}^{K}\ |\ \bm{1}^{T}\bm{p}=1,\bm{p}\geq\bm{0}\}$ is the $(K-1)$-dimensional simplex.
Whereas the softmax probability distribution has full support, the
sparsemax probability distribution is likely to be sparse.
This is because it is the Euclidean projection of $\bm{z}$ onto the probability simplex and is likely to hit the boundary.
For fair comparison, we also optimize with an entropy loss term during sparsemax sampling, and binarize post-optimization.

We performed this ablation for one synthetic (H-RandHouse) and one real (zoo) dataset.
Table 7 shows that Gumbel-Softmax achieves better losses than both relax-and-thresh and sparsemax:
0.10 (vs 0.15 and 0.58) on H-RandHouse, and
0.04 (vs 0.14 and 0.08) on Zoo.
Even without reference to the quantitative results, we know that relax-and-thresh and (to a lesser extent) sparsemax suffer from the so-called “introduced evidence problem” (Dabkowski & Gal, ; Yuan et al., ).
Because the weighted subhypergraph seen during optimization differs from the final explanation subhypergraph obtained upon binarization,
these samplers can lead to highly unfaithful explanations.
Though relax-and-thresh attempts to mitigate this effect with entropy loss,
we find that it is insufficient to avoid this problem, particularly for hypergraphs.
The sparsity properties of sparsemax make it less prone to this failure mode (it achieves a much higher rate of zero entropy loss), but does not eliminate the problem completely.
Note that HyperEX (Maleki et al., ) is also prone to the “introduced evidence problem”, since it also thresholds attention weights to obtain the final explanation subhypergraph.

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 3
presentation: 2
questions: - Line 212 you use a mean-field approximation, can you justify this more, add some
literature that justifies such approximations for hypergraphs? Why can you do that
(specially in the context of hypergraphs)? Are there other approximations you explored
or considered? This is a pretty strong assumption.
 
- Please define a concept line 251; it seems to be specific to a GNN, and to be a cluster
of points in the GNN embedding space, and its ”representative” is the node closest to
the geometric center in the embedding space, is that correct?

- Line 249: Can you specify what the latent embedding space is for a node? Is it the
final layer’s output before the softmax?

- The local and global explainer seem to be two completely decoupled methods; please
point this out in the paper. How are they related?
 
- Have you tried other types of global explanations (e.g., GLGex-
plainer [1]) based on this local explainer? 

- Have you tried other local explanations techniques (there are plenty of work [2])?

- Has a similar sampling method for local explanations been used in GNN explanations?

Some minor comments:

– Equation (3) line 209 is not clear, I would write it differently.

– Some typos: “explainiability” line 75, “P r(v ∈ esub = 0)” line 206

– “coherent explanations” line 80: please clarify what this means.

– What is “InfoNCE” line 106? add context about noise-contrastive estimation.

– Line 259, GCExplainer is mentioned in the method, but there has not been a
reference since the introduction. It should be added on line 244 along with the
Magister paper.

– For the results, it would be helpful to separate the local from the global explanations.

– The paragraph starting line 259 compares GCExplainer with the method of the
paper, this is not clear, as it is never explained what GCExplainer does.

[1] Azzolin, Steve, Antonio Longa, Pietro Barbiero, Pietro Liò, and Andrea Passerini. "Global explainability of gnns via logic combination of learned concepts." arXiv preprint arXiv:2210.07147 (2022).

[2] Kakkad, Jaykumar, Jaspal Jannu, Kartik Sharma, Charu Aggarwal, and Sourav Medya. "A survey on explainability of graph neural networks." arXiv preprint arXiv:2306.01958 (2023).
soundness: 2
strengths: - The topic is interesting, and still in unexplored territory.
- The paper is well written, the experiments seem to yield good results and the experimental design looks solid.
summary: The paper studies a relatively unexplored problem of hypergraphs neural networks explanation for the task of node classification, both from the local and global perspective. The authors describe two methods they use for local and global explanations. They
then benchmark the methods on synthetic and real-world datasets against current baselines, notably with an updated fidelity metric.
weaknesses: - There is not a lot of theory in this paper, and the methods used are pretty straightforward. 
- The literature could be a bit more developed (the one other baseline on the problem, global explanations of GNNs, mean field approximation in the hypergraphs context).
- The local and global explanations techniques are completely decoupled, with the global
explanation in particular being transposed from a method already studied for GNNs. This is a liitle surprising, as the titled implied a deeper relationship.
- There should be more local and global explanations derived from this work and derived
from the GNNs explanation literature: benchmark different global explainer heuristic
based on the local explanation (as opposed to just one); for the local explanation, an
approximation technique with a more refined method, or at least explain why other
methods would not work.

Overall the paper needs a major revision.

---

confidence: 4
contribution: 4
initial_rating: 5
presentation: 3
questions: **1.Insufficient Analysis of Graph-to-Hypergraph Explainability**
The paper provides some reasons why hypergraphs need specialized explainers, such as larger search spaces and structural differences. However, it lacks an in-depth analysis of what specifically would fail if traditional GNN explainers were used on constructed hypergraphs, which can be represented as graphs. Including a comparative study that illustrates concrete failure cases when applying regular GNN explainers to hypergraphs would significantly strengthen the motivation for developing specialized hypergraph explainers.

**2. Constrained Real-World Evaluation**
- The real-world datasets used in the paper are relatively small, which limits the ability to generalize the results. 
- Moreover, the paper acknowledges that the selected datasets may not sufficiently test the model's ability to understand hypergraph structure, given that even simple MLPs achieve competitive performance. While the inclusion of synthetic datasets partially addresses this limitation, it would be more compelling to include more complex real-world hypergraph datasets that present a greater challenge. Such datasets would better demonstrate the practical utility and robustness of the proposed method. - Furthermore, the constrained real-world evaluation raises the question: does the limited complexity and availability of these datasets indicate that the problem lacks substantial practical applications?

Minor Comment: In Figure 1, what does the "concept-to-class decision tree" icon represent? Providing a brief explanation would help improve clarity.
soundness: 3
strengths: The paper presents multiple novel contributions: 
1. The first to provide both local and global explanations for hyperGNNs. 
2. Introduces a new sampling-based approach for local explanations that avoids using attention mechanisms. 
3. It develops techniques specifically designed for hypergraph structures. 
4. The synthetic datasets also represent an original contribution by creating structure-dependent tasks for evaluating hypergraph explainability.
5. This work addresses an important gap in making hyperGNNs more interpretable.
summary: The paper introduces SHypX, the first model-agnostic post-hoc explainer for hyperGNNs that provides both local and global explanations. 
At the local level, it finds salient subhypergraphs to explain individual predictions using Gumbel-Softmax sampling, while balancing faithfulness and concision. 
For global explanations, it extracts concepts by clustering network representations and visualizing representative examples.

In experiments, the paper introduces four synthetic hypergraph datasets and generalized fidelity metrics for proper evaluation. 
Through experiments on both synthetic and real datasets, SHypX demonstrates superior performance over existing baselines while maintaining architecture independence, making it a significant contribution to hypergraph machine learning interpretability.
weaknesses: **Insufficient Analysis of Graph-to-Hypergraph Explainability**

**Constrained Real-World Evaluation**

---

confidence: 5
contribution: 2
initial_rating: 5
presentation: 3
questions: - For the task of node classification, what would happen if a method returns only the first immediate neighbors of a node as the most important sub-hypergraph? Specifically, how would the fidelity metrics mentioned in this paper perform in this scenario? My main point is that having only an explainability model for node classification may not be of significant importance.

- Could you present a hypergraph as a bipartite graph and apply graph-based explainability models to compare their performance with your method? It would be particularly useful to try GNNExplainer, as this method is very intuitive, and it is not immediately clear why it would not perform well in your setting.

- For the global explainer, you are introducing yet another black-box, despite mentioning that SHypX "doesn’t rely on additional black-box networks." Could you clarify this contradiction?

- It is necessary to visualize the results of the explanations for at least the synthetic datasets to visually assess which sub-hypergraphs SHypX identifies as important.

- Figure 4 is very confusing, and the results you are trying to convey are not clear. Is there an alternative way to evaluate global explanations that might be more effective?
soundness: 3
strengths: - The paper is well-written and clearly presented.
- The authors address both instance-level and global-level explainability for HyperGNNs while latter has not been addressed previously.
summary: This paper addresses the post-hoc explainability of hypergraph neural networks (HyperGNNs) for the task of node classification in hypergraphs. The authors propose a method for providing both instance-level and global-level explanations for these models. The core idea of their approach is to sample node-hyperedge pairs in the computation graph of a specific node in a way that minimizes the loss function. Additionally, they introduce a set of synthetic hypergraphs to evaluate their method. Their experimental results demonstrate that the proposed method outperforms existing baselines.
weaknesses: - My main concern is that this paper only addresses explainability for the task of node classification in hypergraphs, which is a less interesting and important problem compared to (hyper)graph classification.
- The baselines used in this paper are not sufficient. While there might not be many related works on explainability in hypergraphs, explainability methods for graph neural networks could be applied and compared with the proposed method.
- The quantitative evaluation on real-world datasets shows very low performance, and the differences between the various baselines are minimal, making it difficult to determine if the proposed method performs significantly better than the other baselines.
- While having synthetic datasets for an explainability method is crucial, it is not clear how the proposed synthetic datasets are particularly useful in this setting.

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 2
questions: 1. In Line 171, it says "our goal is to produce an explanation subhypergraph" for the local explainer. However, a single subhypergraph may not be enough to fully explain the prediction. Why do you only consider to produce a single explanation subhypergraph for each graph sample? 
2. The proposed method doesn't look to be specifically designed for hyperGNNs. For example, modules like Gumbel-Softmax samplers and global clustering. Why don't apply your method also to traditional GNNs? And why not use the explainers for those traditional GNNs, (which didn't use Gumbel-Softmax samplers, but some other methods to pick crucial subgraphs) to explain hyperGNNs? 
3. For the Fidelity evaluation, why do you only evaluate the Fidelity-, how about Fidelity+?
soundness: 2
strengths: 1. This paper is the second work on hyperGNN explainability. 
2. They introduce a new benchmark that contains four synthetic datasets to evaluate hyperGNN explanations.
summary: This paper proposes an explainer for hyperGNNs at both instance-level and global-level, which is called SHypX. At both instance-level and global-level, they extract salient subhypergraphs as the explanation. Four synthetic datasets were introduced in this paper, although none of them are open-sourced.
weaknesses: 1. Related work is not thoroughly reviewed, and more recent studies should be discussed. The limitations of existing approaches for GNN explainability in the context of hyperGNNs are not clearly addressed. Since most existing methods are model-agnostic, they should be able to provide explanations for hyperGNNs too. 
2. The newly introduced datasets are not provided. 
3. The presentation is not very good. For example, the caption of Fig 2 is confusing. 
4. The performance improvements on real world datasets seems minimal. 
5. The evaluation on the global-level explainability is too few. It doesn't seem that this explainer actually produce global-level explanations.  
6. Some concerns about motivation and metrics. See questions.

Meta-review: The paper introduces a model-agnostic post-hoc explainer for hyperGNNs, providing both local and global explanations. For local explanations, it identifies important sub-hypergraphs to explain individual predictions using Gumbel-Softmax sampling, balancing faithfulness and conciseness. On the other hand, for global explanations, it derives concepts by clustering network representations and visualizing representative examples.

However, despite the rebuttal discussion, the reviewers remain unconvinced about the comprehensiveness of the empirical benchmarking and the validity of claims regarding human interpretability. As a result, the paper is considered not ready for publication in its current form.

Decision: Reject