Title: CAT-Walk: Inductive Hypergraph Learning via Set Walks
Venue: NeurIPS 2023
Accepted: yes
Keywords: Hypergraph Learning

Temporal Networks

Higher-order Temporal Motifs

Inductive Representation Learning
Abstract: Temporal hypergraphs provide a powerful paradigm for modeling time-dependent, higher-order interactions in complex systems. Representation learning for hypergraphs is essential for extracting patterns of the higher-order interactions that are critically important in real-world problems in social network analysis, neuroscience, finance, etc. However, existing methods are typically designed only for specific tasks or static hypergraphs. We present  CAT-Walk, an inductive method that learns the underlying dynamic laws that govern the temporal and structural processes underlying a temporal hypergraph. CAT-Walk introduces a temporal, higher-order walk on hypergraphs, SetWalk, that extracts higher-order causal patterns. CAT-Walk uses a novel adaptive and permutation invariant pooling strategy, SetMixer, along with a set-based anonymization process that hides the identity of hyperedges. Finally, we present a simple yet effective neural network model to encode hyperedges. Our evaluation on 10 hypergraph benchmark datasets shows that CAT-Walk attains outstanding performance on temporal hyperedge prediction benchmarks in both inductive and transductive settings. It also shows competitive performance with state-of-the-art methods for node classification. (https://github.com/ubc-systopia/CATWalk)

CA t -W alk : Inductive Hypergraph Learning via Set Walks

Abstract

Temporal hypergraphs provide a powerful paradigm for modeling time-dependent, higher-order interactions in complex systems. Representation learning for hypergraphs is essential for extracting patterns of the higher-order interactions that are critically important in real-world problems in social network analysis, neuroscience, finance, etc. However, existing methods are typically designed only for specific tasks or static hypergraphs. We present CA t -W alk , an inductive method that learns the underlying dynamic laws that govern the temporal and structural processes underlying a temporal hypergraph. CA t -W alk introduces a temporal, higher-order walk on hypergraphs, S et W alk , that extracts higher-order causal patterns. CA t -W alk uses a novel adaptive and permutation invariant pooling strategy, S et M ixer , along with a set-based anonymization process that hides the identity of hyperedges. Finally, we present a simple yet e ff ective neural network model to encode hyperedges. Our evaluation on 10 hypergraph benchmark datasets shows that CA t -W alk attains outstanding performance on temporal hyperedge prediction benchmarks in both inductive and transductive settings. It also shows competitive performance with state-of-the-art methods for node classification. (Code)

1 Introduction

Temporal networks have become increasingly popular for modeling interactions among entities in dynamic systems [1-5]. While most existing work focuses on pairwise interactions between entities, many real-world complex systems exhibit natural relationships among multiple entities [6-8]. Hypergraphs provide a natural extension to graphs by allowing an edge to connect any number of vertices, making them capable of representing higher-order structures in data. Representation learning on (temporal) hypergraphs has been recognized as an important machine learning problem and has become the cornerstone behind a wealth of high-impact applications in computer vision [9, 10], biology [11, 12], social networks [13, 14], and neuroscience [15, 16].

Many recent attempts to design representation learning methods for hypergraphs are equivalent to applying Graph Neural Networks (G nn s) to the clique-expansion (CE) of a hypergraph [17-21]. CE is a straightforward way to generalize graph algorithms to hypergraphs by replacing hyperedges with (weighted) cliques [18-20]. However, this decomposition of hyperedges limits expressiveness,

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ExtractedSETWALKs:

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connected totwoothernodesthatarethemselves

connected by a three-way connection,it tends to

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SETWALKcandistinguish

hthesetwounderlyingdynamiclawsbuthypergraphwalk

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simplewalkontheCEof thehypergraphfailtodifferentiatebetweenthese^cases.

leading to suboptimal performance [6, 22-24] (see Theorem 1 and Theorem 4). New methods that encode hypergraphs directly partially address this issue [11, 25-28]. However, these methods su ff er from some combination of the following three limitations: they are designed for 1 learning the structural properties of static hypergraphs and do not consider temporal properties, 2 the transductive setting, limiting their performance on unseen patterns and data, and 3 a specific downstream task (e.g., node classification [25], hyperedge prediction [26], or subgraph classification [27]) and cannot easily be extended to other downstream tasks, limiting their application.

Temporal motif-aware and neighborhood-aware methods have been developed to capture complex patterns in data [29-31]. However, counting temporal motifs in large networks is time-consuming and non-parallelizable, limiting the scalability of these methods. To this end, several recent studies suggest using temporal random walks to automatically retrieve such motifs [32-36]. One possible solution to capturing underlying temporal and higher-order structure is to extend the concept of a hypergraph random walk [37-43] to its temporal counterpart by letting the walker walk over time. However, existing definitions of random walks on hypergraphs o ff er limited expressivity and sometimes degenerate to simple walks on the CE of the hypergraph [40] (see Appendix C). There are two reasons for this: 1 Random walks are composed of a sequence of pair-wise interconnected vertices, even though edges in a hypergraph connect sets of vertices. Decomposing them into sequences of simple pair-wise interactions loses the semantic meaning of the hyperedges (see Theorem 4). 2 A sampling probability of a walk on a hypergraph must be di ff erent from its sampling probability on the CE of the hypergraph [37-43]. However, Chitra and Raphael [40] shows that each definition of the random walk with edge-independent sampling probability of nodes is equivalent to random walks on a weighted CE of the hypergraph. Existing studies on random walks on hypergraphs ignore 1 and focus on 2 to distinguish the walks on simple graphs and hypergraphs. However, as we show in Table 2, 1 is equally important, if not more so.

For example, Figure 1 shows the procedure of existing walk-based machine learning methods on a temporal hypergraph. The neural networks in the model take as input only sampled walks. However, the output of the hypergraph walk [37, 38] and simple walk on the CE graph are the same. This means that the neural network cannot distinguish between pair-wise and higher-order interactions.

We present C ausal A nonymous Se t Walk s (CA t -W alk ), an inductive hyperedge learning method. We introduce a hyperedge-centric random walk on hypergraphs, called S et W alk , that automatically extracts temporal, higher-order motifs. The hyperedge-centric approach enables S et W alk s to distinguish multi-way connections from their corresponding CEs (see Figure 1, Figure 2, and Theorem 1). We use temporal hypergraph motifs that reflect network dynamics (Figure 2) to enable CA t -W alk to work well in the inductive setting. To make the model agnostic to the hyperedge identities of these motifs, we use two-step, set-based anonymization: 1 Hide node identities by assigning them new positional encodings based on the number of times that they appear at a specific position in a set of sampled S et W alk s, and 2 Hide hyperedge identities by combining the positional encodings of the nodes comprising the hyperedge using a novel permutation invariant pooling strategy, called S et M ixer . We incorporate a neural encoding method that samples a few S et W alk s starting from nodes of interest. It encodes and aggregates them via MLP-M ixer [44] and our new pooling strategy

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S et M ixer , respectively, to predict temporal, higher-order interactions. Finally, we discuss how to extend CA t -W alk for node classification. Figure 3 shows the schematic of the CA t -W alk .

Wetheoretically and experimentally discuss the e ff ectiveness of CA t -W alk and each of its components. More specifically, we prove that S et W alk s are more expressive than existing random walk algorithms on hypergraphs. We demonstrate S et M ixer 's e ffi cacy as a permutation invariant pooling strategy for hypergraphs and prove that using it in our anonymization process makes that process more expressive than existing anonymization processes [33, 45, 46] when applied to the CE of the hypergraphs. To the best of our knowledge, we report the most extensive experiments in the hypergraph learning literature pertaining to unsupervised hyperedge prediction with 10 datasets and eight baselines. Results show that our method produces 9% and 17% average improvement in transductive and inductive settings, outperforming all state-of-the-art baselines in the hyperedge prediction task. Also, CA t -W alk achieves the best or on-par performance on dynamic node classification tasks. All proofs appear in the Appendix.

2 Related Work

Temporal graph learning is an active research area [5, 47]. A major group of methods uses G nn s to learn node encodings and Recurrent Neural Networks (R nn s) to update these encodings over time [48-55]. More sophisticated methods based on anonymous temporal random walks [33, 34], line graphs [56], GraphMixer [57], neighborhood representation [58], and subgraph sketching [59] are designed to capture complex structures in vertex neighborhoods. Although these methods show promising results in a variety of tasks, they are fundamentally limited in that they are designed for pair-wise interaction among vertices and not the higher-order interactions in hypergraphs.

Representation learning on hypergraphs addresses this problem [17, 60]. We group work in this area into three overlapping categories:

1 Clique and Star Expansion : CE-based methods replace hyperedges with (weighted) cliques and apply G nn s, sometimes with sophisticated propagation rules [21, 25], degree normalization, and nonlinear hyperedge weights [17-21, 39, 61, 62]. Although these methods are simple, it is well-known that CE causes undesired losses in learning performance, specifically when relationships within an incomplete subset of a hyperedge do not exist [6, 22-24]. Star expansion (SE) methods first use hypergraph star expansion and model the hypergraph as a bipartite graph, where one set of vertices represents nodes and the other represents hyperedges [25, 28, 63, 64]. Next, they apply modified heterogeneous GNNs, possibly with dual attention mechanisms from nodes to hyperedges and vice versa [25, 27]. Although this group does not cause as large a distortion as CE, they are neither memory nor computationally e ffi cient. 2 Message Passing : Most existing hypergraph learning methods, use message passing over hypergraphs [17-21, 25-27, 39, 61, 62, 65, 66]. Recently, Chien et al. [25] and Huang and Yang [28] designed universal message-passing frameworks that include propagation rules of most previous methods (e.g., [17, 19]). The main drawback of these two frameworks is that they are limited to node classification tasks and do not easily generalize to other tasks. 3 Walkbased : random walks are a common approach to extracting graph information for machine learning algorithms [32-34, 67, 68]. Several walk-based hypergraph learning methods are designed for a wide array of applications [43, 69-78]. However, most existing methods use simple random walks on the CE of the hypergraph (e.g., [26, 43, 78]). More complicated random walks on hypergraphs address this limitation [40-42, 79]. Although some of these studies show that their walk's transition matrix

di ff ers from the simple walk on the CE [40, 79], their extracted walks can still be the same, limiting their expressivity (see Figure 1, Figure 2, and Theorem 1).

Our method di ff ers from all prior (temporal) (hyper)graph learning methods in five ways: CA t -W alk : I Captures temporal higher-order properties in a streaming manner: In contrast to existing methods in hyperedge prediction, our method captures temporal properties in a streaming manner, avoiding the drawbacks of snapshot-based methods. II Works in the inductive setting by extracting underlying dynamic laws of the hypergraph, making it generalizable to unseen patterns and nodes. III Introduces a hyperedge-centric, temporal, higher-order walk with a new perspective on the walk sampling procedure. IV Presents a new two-step anonymization process: Anonymization of higher-order patterns (i.e., S et W alk s), requires hiding the identity of both nodes and hyperedges. We present a new permutation-invariant pooling strategy to hide hyperedges' identity according to their vertices, making the process provably more expressive than the existing anonymization processes [33, 34, 78]. V Removes self-attention and R nn s from the walk encoding procedure, avoiding their limitations. Appendices A and B provide a more comprehensive discussion of related work.

3 Method: CA t -W alk Network

3.1 Preliminaries

Definition 1 (Temporal Hypergraphs) . A temporal hypergraph G = ( V , E , X ) , can be represented as a sequence of hyperedges that arrive over time, i.e., E = { ( e 1 , t 1 ) , ( e 2 , t 2 ) , . . . } , where V is the set of nodes, ei ∈ 2 V are hyperedges, ti is the timestamp showing when ei arrives, and X ∈ R |V|× f is a matrix that encodes node attribute information for nodes in V . Note that we treat each hyperedge ei as the set of all vertices connected by ei.

Example 1. The input of Figure 1 shows an example of a temporal hypergraph. Each hyperedge is a set of nodes that are connected by a higher-order connection. Each higher-order connection is specified by a color (e.g., green, blue, and gray), and also is associated with a timestamp (e.g., ti).

Given a hypergraph G = ( V , E , X ), we represent the set of hyperedges attached to a node u before time t as E t ( u ) = { ( e , t ′ ) | t ′ < t , u ∈ e } . We say two hyperedges e and e ′ are adjacent if e ∩ e ′ , ∅ and use E t ( e ) = { ( e ′ , t ′ ) | t ′ < t , e ′ ∩ e , ∅} to represent the set of hyperedges adjacent to e before time t . Next, we focus on the hyperedge prediction task: Given a subset of vertices U = { u 1 , . . . , uk } , we want to predict whether a hyperedge among all the ui s will appear in the next timestamp or not. In AppendixG.2 we discuss how this approach can be extended to node classification.

The (hyper)graph isomorphism problem is a decision problem that decides whether a pair of (hyper)graphs are isomorphic or not. Based on this concept, next, we discuss the measure we use to compare the expressive power of di ff erent methods.

Definition 2 (Expressive Power Measure) . Given two methods M 1 and M 2 , we say method M 1 is more expressive than method M 2 if:

1. for any pair of (hyper)graphs ( G 1 , G 2 ) such that G 1 , G 2 , if method M 1 can distinguish G 1 and G 2 then method M 2 can also distinguish them,

there is a pair of hypergraphs G ′ 1 , G ′ 2 such that M 1 can distinguish them but M 2 cannot.

3.2 Causality Extraction via S et W alk

The collaboration network in Figure 1 shows how prior work that models hypergraph walks as sequences of vertices fails to capture complex connections in hypergraphs. Consider the two walks: A → C → E and H → E → C . These two walks can be obtained either from a hypergraph random walk or from a simple random walk on the CE graph. Due to the symmetry of these walks with respect to ( A , C , E ) and ( H , E , C ), they cannot distinguish A and H , although the neighborhoods of these two nodes exhibit di ff erent patterns: A , B , and C have published a paper together (connected by a hyperedge), but each pair of E , G , and H has published a paper (connected by a pairwise link). We address this limitation by defining a temporal walk on hypergraphs as a sequence of hyperedges:

Definition 3 (S et W alk ) . Given a temporal hypergraph G = ( V , E , X ) , a S et W alk with length ℓ on temporal hypergraph G is a randomly generated sequence of hyperedges (sets):

where ei ∈ E , t ei + 1 < t ei , and the intersection of ei and ei + 1 is not empty, ei ∩ ei + 1 , ∅ . In other words, for each 1 ≤ i ≤ ℓ -1 : ei + 1 ∈ E t i ( ei ) . We use S w [ i ] to denote the i-th hyperedge-time pair in the S et W alk . That is, S w [ i ][0] = ei and S w [ i ][1] = t ei .

Example 2. Figure 1 illustrates a temporal hypergraph with two sampled S et W alk s, hypergraph random walks, and simple walks. As an example, ( { A , B , C } , t 6 ) → ( { C , D , E , F } , t 5 ) is a S et W alk that starts from hyperedge e including { A , B , C } in time t 6 , backtracks overtime and then samples hyperedge e ′ including { C , D , E , F } in time t 5 < t 6 . While this higher-order random walk with length two provides information about all { A , B , C , D , E , F } , its simple hypergraph walk counterpart, i.e. A → C → E, provides information about only three nodes.

Next, we formally discuss the power of S et W alk s. The proof of the theorem can be found in Appendix E.1.

Theorem 1. A random S et W alk is equivalent to neither the hypergraph random walk, the random walk on the CE graph, nor the random walk on the SE graph. Also, for a finite number of samples of each, S et W alk is more expressive than existing walks.

In Figure 1, S et W alk s capture higher-order interactions and distinguish the two nodes A and H , which are indistinguishable via hypergraph random walks and graph random walks in the CE graph. We present a more detailed discussion and comparison with previous definitions of random walks on hypergraphs in Appendix C.

Causality Extraction. We introduce a sampling method to allow S et W alk s to extract temporal higher-order motifs that capture causal relationships by backtracking over time and sampling adjacent hyperedges. As discussed in previous studies [33, 34], more recent connections are usually more informative than older connections. Inspired by Wang et al. [33], we use a hyperparameter α ≥ 0 to sample a hyperedge e with probability proportional to exp GLYPH<16> α ( t -t p ) GLYPH<17> , where t and t p are the timestamps of e and the previously sampled hyperedge in the S et W alk , respectively. Additionally, we want to bias sampling towards pairs of adjacent hyperedges that have a greater number of common nodes to capture higher-order motifs. However, as discussed in previous studies, the importance of each node for each hyperedge can be di ff erent [25, 27, 40, 65]. Accordingly, the transferring probability from hyperedge ei to its adjacent hyperedge ej depends on the importance of the nodes that they share. We address this via a temporal S et W alk sampling process with hyperedge-dependent node weights . Given a temporal hypergraph G = ( V , E , X ), a hyperedge-dependent node-weight function Γ : V×E→ R ≥ 0 , and a previously sampled hyperedge ( ep , t p ), we sample a hyperedge ( e , t ) with probability:

where φ ( e , e ′ ) = P u ∈ e ∩ e ′ Γ ( u , e ) Γ ( u , e ′ ), representing the assigned weight to e ∩ e ′ . We refer to the first and second terms as temporal bias and structural bias , respectively.

The pseudocode of our S et W alk sampling algorithm and its complexity analysis are in Appendix D. We also discuss this hyperedge-dependent sampling procedure and how it is provably more expressive than existing hypergraph random walks in Appendix C.

Given a (potential) hyperedge e 0 = { u 1 , u 2 , . . . , uk } and a time t 0, we say a S et W alk , S w , starts from ui if ui ∈ S w [1][0]. We use the above procedure to generate M S et W alk s with length m starting from each ui ∈ e 0. We use S ( ui ) to store S et W alk s that start from ui .

3.3 Set-based Anonymization of Hyperedges

In the anonymization process, we replace hyperedge identities with position encodings, capturing structural information while maintaining the inductiveness of the method. Micali and Zhu [45] studied Anonymous Walks (AWs), which replace a node's identity by the order of its appearance in each walk. The main limitation of AWs is that the position encoding of each node depends only on its specific walk, missing the dependency and correlation of di ff erent sampled walks [33]. To mitigate this drawback, Wang et al. [33] suggest replacing node identities with the hitting counts of the nodes based on a set of sampled walks. In addition to the fact that this method is designed for walks on simple graphs, there are two main challenges to adopting it for S et W alk s: 1 S et W alk s

are a sequence of hyperedges, so we need an encoding for the position of hyperedges. Natural attempts to replace hyperedges' identity with the hitting counts of the hyperedges based on a set of sampled S et W alk s, misses the similarity of hyperedges with many of the same nodes. 2 Each hyperedge is a set of vertices and natural attempts to encode its nodes' positions and aggregate them to obtain a position encoding of the hyperedge requires a permutation invariant pooling strategy. This pooling strategy also requires consideration of the higher-order dependencies between obtained nodes' position encodings to take advantage of higher-order interactions (see Theorem 2). To address these challenges we present a set-based anonymization process for S et W alk s. Given a hyperedge e 0 = { u 1 , . . . , uk } , let w 0 be any node in e 0. For each node w that appears on at least one S et W alk in S k i = 1 S ( ui ), we assign a relative, node-dependent node identity, R ( w , S ( w 0 )) ∈ Z m , as follows:

For each node w we further define I d ( w , e 0 ) = {R ( w , S ( ui )) } k i = 1 . Let Ψ ( . ) : R d × d 1 → R d 2 be a pooling function that gets a set of d 1-dimensional vectors and aggregates them to a d 2-dimensional vector. Given two instances of this pooling function, Ψ 1 ( . ) and Ψ 2 ( . ), for each hyperedge e = { w 1 , w 2 , . . . , wk ′ } that appears on at least one S et W alk in S k i = 1 S ( ui ), we assign a relative hyperedge identity as:

That is, for each node wi ∈ e we first aggregate its relative node-dependent identities (i.e., R ( wi , S ( uj ))) to obtain the relative hyperedge-dependent identity. Then we aggregate these hyperedge-dependent identities for all nodes in e . Since the size of hyperedges can vary, we zero-pad to a fixed length. Note that this zero padding is important to capture the size of the hyperedge. The hyperedge with more zero-padded dimensions has fewer nodes.

This process addresses the first challenge and encodes the position of hyperedges. Unfortunately, many simple and known pooling strategies (e.g., S um (.), A ttn -S um (.), M ean (.), etc.) can cause missing information when applied to hypergraphs. We formalize this in the following theorem:

Theorem 2. Given an arbitrary positive integer k ∈ Z + , let Ψ ( . ) be a pooling function such that for any set S = { w 1 , . . . , wd } :

where f is some function. Then the pooling function can cause missing information, meaning that it limits the expressiveness of the method to applying to the projected graph of the hypergraph.

While simple concatenation does not su ff er from this undesirable property, it is not permutation invariant. To overcome these challenges, we design an all-MLP permutation invariant pooling function, S et M ixer , that not only captures higher-order dependencies of set elements but also captures dependencies across the number of times that a node appears at a certain position in S et W alk s.

S et M ixer . MLP-M ixer [44] is a family of models based on multi-layer perceptrons, widely used in the computer vision community, that are simple, amenable to e ffi cient implementation, and robust to oversquashing and long-term dependencies (unlike R nn s and attention mechanisms) [44, 57]. However, the token-mixer phase of these methods is sensitive to the order of the input (see Appendix A). To address this limitation, inspired by MLP-M ixer [44], we design S et M ixer as follows: Let S = { v 1 , . . . , v d } , where v i ∈ R d 1 , be the input set and V = [ v 1 , . . . , v d ] ∈ R d × d 1 be its matrix representation:

where

Here, W (1) s and W (2) s are learnable parameters, σ ( . ) is an activation function (we use GeLU [80] in our experiments), and LayerNorm is layer normalization [81]. Equation 5 is the channel mixer and Equation 6 is the token mixer. The main intuition of S et M ixer is to use the Softmax (.) function to bind token-wise information in a non-parametric manner, avoiding permutation variant operations in the token mixer. We formally prove the following theorem in Appendix E.3.

Theorem 3. S et M ixer is permutation invariant and is a universal approximator of invariant multi-set functions. That is, S et M ixer can approximate any invariant multi-set function.

Based on the above theorem, S et M ixer can overcome the challenges we discussed earlier as it is permutation invariant. Also, it is a universal approximator of multi-set functions, which shows its power to learn any arbitrary function. Accordingly, in our anonymization process, we use Ψ ( . ) = S et M ixer ( . ) in Equation 3 to hide hyperedge identities. Next, we guarantee that our anonymization process does not depend on hyperedges or nodes identities, which justifies the claim of inductiveness of our model:

Proposition 1. Given two (potential) hyperedges e 0 = { u 1 , . . . , uk } and e ′ 0 = { u ′ 1 , . . . , u ′ k } , if there exists a bijective mapping π between node identities such that for each S et W alk like S w ∈ S k i = 1 S ( ui ) can be mapped to one S et W alk like S w ′ ∈ S k i = 1 S ( u ′ i ) , then for each hyperedge e = { w 1 , . . . , wk ′ } that appears in at least one S et W alk in S k i = 1 S ( ui ) , we have I d ( e , e 0 ) = I d ( π ( e ) , e ′ 0 ) , where π ( e ) = { π ( w 1 ) , , . . . , π ( wk ′ ) } .

Finally, we guarantee that our anonymization approach is more expressive than existing anonymization process [33, 45] when applied to the CE of the hypergraphs:

Theorem 4. The set-based anonymization method is more expressive than any existing anonymization strategies on the CE of the hypergraph. More precisely, there exists a pair of hypergraphs G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) with di ff erent structures (i.e., G 1 GLYPH<29> G 2 ) that are distinguishable by our anonymization process and are not distinguishable by the CE-based methods.

3.4 S et W alk Encoding

Previous walk-based methods [33, 34, 78] view a walk as a sequence of nodes. Accordingly, they plug nodes' positional encodings in a R nn [82] or Transformer [83] to obtain the encoding of each walk. However, in addition to the computational cost of R nn and Transformers, they su ff er from over-squashing and fail to capture long-term dependencies. To this end, we design a simple and low-cost S et W alk encoding procedure that uses two steps: 1 A time encoding module to distinguish di ff erent timestamps, and 2 A mixer module to summarize temporal and structural information extracted by S et W alk s.

Time Encoding. We follow previous studies [33, 84] and adopt random Fourier features [85, 86] to encode time. However, these features are periodic, so they capture only periodicity in the data. We add a learnable linear term to the feature representation of the time encoding. We encode a given time t as follows:

where ω l , b l ∈ R and ω p ∈ R d 2 -1 are learnable parameters and || shows concatenation.

M ixer Module. To summarize the information in each S et W alk , we use a MLP-M ixer [44] on the sequence of hyperedges in a S et W alk as well as their corresponding encoded timestamps. Contrary to the anonymization process, where we need a permutation invariant procedure, here, we need a permutation variant procedure since the order of hyperedges in a S et W alk is important. Given a (potential) hyperedge e 0 = { u 1 , . . . , uk } , we first assign I d ( e , e 0) to each hyperedge e that appears on at least one sampled S et W alk starting from e 0 (Equation 3). Given a S et W alk , S w : ( e 1 , t e 1 ) →··· → ( em , t em ), we let E be a matrix that E i = I d ( ei , e 0 ) || T( t ei ):

where

3.5 Training

In the training phase, for each hyperedge in the training set, we adopt the commonly used negative sample generation method [60] to generate a negative sample. Next, for each hyperedge in the training set such as e 0 = { u 1 , u 2 , . . . , uk } , including both positive and negative samples, we sample M S et W alk s with length m starting from each ui ∈ e 0 to construct S ( ui ). Next, we anonymize each hyperedge that appears in at least one S et W alk in S k i = 1 S ( ui ) by Equation 3 and then use the M ixer module to encode each S w ∈ S k i = 1 S ( ui ). To encode each node ui ∈ e 0, we use M ean ( . ) pooling over S et W alk s in S ( ui ). Finally, to encode e 0 we use S et M ixer to mix obtained node encodings. For

hyperedge prediction, we use a 2-layer perceptron over the hyperedge encodings to make the final prediction. We discuss node classification in Appendix G.2.

4 Experiments

We evaluate the performance of our model on two important tasks: hyperedge prediction and node classification (see Appendix G.2) in both inductive and transductive settings. We then discuss the importance of our model design and the significance of each component in CA t -W alk .

4.1 Experimental Setup

Baselines. We compare our method to eight previous state-of-the-art baselines on the hyperedge prediction task. These methods can be grouped into three categories: 1 Deep hypergraph learning methods including H yper SAGCN [26], NHP [87], and CHESHIRE [11]. 2 Shallow methods including HPRA [88] and HPLSF [89]. 3 CE methods: CE-CAW [33], CE-E volve GCN [90] and CE-GCN [91]. Details on these models and hyperparameters used appear in Appendix F.2.

Datasets. We use 10 available benchmark datasets [6] from the existing hypergraph neural networks literature. These datasets' domains include drug networks (i.e., NDC [6]), contact networks (i.e., High School [92] and Primary School [93]), the US. Congress bills network [94, 95], email networks (i.e., Email Enron [6] and Email Eu [96]), and online social networks (i.e., Question Tags and Users-Threads [6]). Detailed descriptions of these datasets appear in Appendix F.1.

Evaluation Tasks. We focus on Hyperedge Prediction: In the transductive setting, we train on the temporal hyperedges with timestamps less than or equal to T train and test on those with timestamps greater than T train. Inspired by Wang et al. [33], we consider two inductive settings. In the Strongly Inductive setting, we predict hyperedges consisting of some unseen nodes. In the Weakly Inductive setting,we predict hyperedges with at least one seen and some unseen nodes. We first follow the procedure used in the transductive setting, and then we randomly select 10% of the nodes and remove all hyperedges that include them from the training set. We then remove all hyperedges with seen nodes from the validation and testing sets. For dynamic node classification, see Appendix G.2. For all datasets, we fix T train = 0 . 7 T , where T is the last timestamp. To evaluate the models' performance we follow the literature and use Area Under the ROC curve (AUC) and Average Precision (AP).

4.2 Results and Discussion

Hyperedge Prediction. We report the results of CA t -W alk and baselines in Table 1 and Appendix G. The results show that CA t -W alk achieves the best overall performance compared to the baselines in both transductive and inductive settings. In the transductive setting, not only does our method outperform baselines in all but one dataset, but it achieves near perfect results (i.e., ≈ 98 . 0) on the NDC and Primary School datasets. In the Weakly Inductive setting, our model achieves high scores (i.e., > 91 . 5) in all but one dataset, while most baselines perform poorly as they are not designed for the inductive setting and do not generalize well to unseen nodes or patterns. In the Strongly Inductive setting, CA t -W alk still achieves high AUC (i.e., > 90 . 0) on most datasets and outperforms baselines on all datasets. There are three main reasons for CA t -W alk 's superior performance: 1 Our S et W alk s capture higher-order patterns. 2 CA t -W alk incorporates temporal properties (both from S et W alk s and our time encoding module), thus learning underlying dynamic laws of the network. The other temporal methods (CE-CAW and CE-E volve GCN) are CE-based methods, limiting their ability to capture higher-order patterns. 3 CA t -W alk 's set-based anonymization process that avoids using node and hyperedge identities allows it to generalize to unseen patterns and nodes.

Ablation Studies. We next conduct ablation studies on the High School, Primary School, and Users-Threads datasets to validate the e ff ectiveness of CA t -W alk 's critical components. Table 2 shows AUC results for inductive hyperedge prediction. The first row reports the performance of the complete CA t -W alk implementation. Each subsequent row shows results for CA t -W alk with one module modification: row 2 replace S et W alk by edge-dependent hypergraph walk [40], row 3 removes the time encoding module, row 4 replaces S et M ixer with M ean (.) pooling, row 5 replaces the S et M ixer with sum-based universal approximator for sets [97], row 6 replaces the MLP-M ixer

module with a R nn (see Appendix G for more experiments on the significance of using MLP-M ixer in walk encoding), row 7 replaces the MLP-M ixer module with a Transformer [83], and row 8 replaces the hyperparameter α with uniform sampling of hyperedges over all time periods. These results show that each component is critical for achieving CA t -W alk 's superior performance. The greatest contribution comes from S et W alk , MLP-M ixer in walk encoding, α in temporal hyperedge sampling, and S et M ixer pooling, respectively.

Hyperparameter Sensitivity. We analyze the e ff ect of hyperparameters used in CA t -W alk , including temporal bias coe ffi cient α , S et W alk length m , and sampling number M . The mean AUC performance on all inductive test hyperedges is reported in Figure 4. As expected, the left figure shows that

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increasing the number of sampled S et W alk s produces better performance. The main reason is that the model has more extracted structural and temporal information from the network. Also, notably, we observe that only a small number of sampled S et W alk s are needed to achieve competitive performance: in the best case 1 and in the worst case 16 sampled S et W alk s per each hyperedge are needed. The middle figure reports the e ff ect of the length of S et W alk s on performance. These results show that performance peaks at certain S et W alk lengths and the exact value varies with the dataset. That is, longer S et W alk s are required for the networks that evolve according to more complicated laws encoded in temporal higher-order motifs. The right figure shows the e ff ect of the temporal bias coe ffi cient α . Results suggest that α has a dataset-dependent optimal interval. That is, a small α suggests an almost uniform sampling of interaction history, which results in poor performance when the short-term dependencies (interactions) are more important in the dataset. Also, very large α might damage performance as it makes the model focus on the most recent few interactions, missing long-term patterns.

Scalability Analysis. In this part, we investigate the scalability of CA t -W alk . To this end, we use di ff erent versions of the High School dataset with di ff erent numbers of hyperedges from 10 4 to 1 . 6 × 10 5 . Figure 5 (left) reports the runtimes of S et W alk sampling and Figure 5 (right) reports the runtimes of CA t -W alk for training one epoch using M = 8, m = 3 with batch-size = 64. Interestingly, our method scales linearly with the number of hyperedges, which enables it to be used on long hyperedge streams and large hypergraphs.

5 Conclusion, Limitation, and Future Work

We present CA t -W alk , an inductive hypergraph representation learning that learns both higher-order patterns and the underlying dynamic laws of temporal hypergraphs. CA t -W alk uses S et W alk s, a new temporal, higher-order random walk on hypergraphs that are provably more expressive than existing walks on hypergraphs, to extract temporal higher-order motifs from hypergraphs. CA t -W alk then uses a two-step, set-based anonymization process to establish the correlation between the extracted motifs. We further design a permutation invariant pooling strategy, S et M ixer , for aggregating nodes' positional encodings in a hyperedge to obtain hyperedge level positional encodings. Consequently, the experimental results show that CA t -W alk i produces superior performance compared to the state-of-the-art in temporal hyperedge prediction tasks, and ii competitive performance in temporal node classification tasks. These results suggest many interesting directions for future studies: Using CA t -W alk as a positional encoder in existing anomaly detection frameworks to design an inductive anomaly detection method on hypergraphs. There are, however, a few limitations: Currently, CA t -W alk uses fixed-length S et W alk s, which might cause suboptimal performance. Developing a procedure to learn from SetWalks of varying lengths might produce better results.

Acknowledgments and Disclosure of Funding

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).

Nous remercions le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG) de son soutien.

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Appendix

Table of Contents

A Preliminaries, Backgrounds, and Motivations

We begin by reviewing the preliminaries and background concepts that we refer to in the main paper. Next, we discuss the fundamental di ff erences between our method and techniques from prior work.

A.1 Anonymous Random Walks

Micali and Zhu [45] studied Anonymous Walks (AWs), which replace a node's identity by the order of its appearance in each walk. Given a simple network, an AW starts from a node, performs random walks over the graph to collect a sequence of nodes, W : ( u 1 , u 2 , . . . , uk ), and then replaces the node identities by their order of appearance in each walk. That is:

While this method is a simple anonymization process, it misses the correlation between di ff erent walks and assigns new node identities based on only one single walk. The correlation between di ff erent walks is more important in temporal networks to assign new node identities, as a single walk cannot capture the frequency of a pattern over time [33]. To this end, Wang et al. [33] design a set-based anonymization process that assigns new node identities based on a set of sampled walks. Given a vertex u , they sample M walks with length m starting from u and store them in S u . Next, for each node w 0 that appears on at least one walk in S u , they assign a vector to each node as its hidden identity [33]:

where W [ i ] shows the i -th node in the walk W . This anonymization process not only hides the identity of vertices but it also can establish such hidden identity based on di ff erent sampled walks, capturing the correlation between several walks starting from a vertex.

Both of these anonymization processes are designed for graphs with pair-wise interactions, and there are three main challenges in adopting them for hypergraphs: 1 To capture higher-order patterns, we use S et W alk s, which are a sequence of hyperedges. Accordingly, we need an encoding for the position of hyperedges. A natural attempt to encode the position of hyperedges is to count the position of hyperedges across sampled S et W alk s, as CAW [33] does for nodes. However, this approach misses the similarity of hyperedges with many nodes in common. That is, given two hyperedges e 1 = { u 1 , u 2 , . . . , uk } and e 2 = { u 1 , u 2 , . . . , uk , uk + 1 } . Although we want to encode the position of these two hyperedges, we also want these two hyperedges to have almost the same encoding as they share many vertices. Accordingly, we suggest viewing a hyperedge as a set of vertices. We first encode the position of vertices, and then we aggregate the position encodings of nodes that are connected by a hyperedge to compute the positional encoding of the hyperedge. 2 However, since we focus on undirected hypergraphs, the order of a hyperedge's vertices in the aggregation process should not a ff ect the hyperedge positional encodings. Therefore, we need a permutation invariant pooling strategy. 3 While several existing studies used simple pooling functions such as M ean ( . ) or S um ( . ) [26], these pooling functions do not capture the higher-order dependencies between obtained nodes' position encodings, missing the advantage of higher-order interactions. That is, a pooling function such as M ean ( . ) is a non-parametric method that sees the positional encoding of each node in a hyperedge separately. Therefore, it is unable to aggregate them in a non-linear manner, which, depending on the data, can miss information. To address challenges 2 and 3 , we design S et M ixer , a permutation invariant pooling strategy that uses MLPs to learn how to aggregate positional encodings of vertices in a hyperedge to compute the hyperedge positional encoding.

A.2 Random Walk on Hypergraphs

Chung [98] presents some of the earliest research on the hypergraph Laplacian, defining the Laplacian of the k -uniform hypergraph. Following this line of work, Zhou et al. [18] defined a two-step CEbased random walk-based Laplacian for general hypergraphs. Given a node u , in the first step, we uniformly sample a hyperedge e including node u , and in the second step, we uniformly sample a node in e . Following this idea, several studies developed more sophisticated (weighted) CE-based random walks on hypergraphs [20]. However, Chitra and Raphael [40] shows that random walks on hypergraphs with edge-independent node weights are limited to capturing pair-wise interactions, making them unable to capture higher-order information. To address this limitation, they designed an edge-dependent sampling procedure of random walks on hypergraphs. Carletti et al. [37] and Carletti

et al. [99] argued that to sample more informative walks from a hypergraph, we must consider the degree of hyperedges in measuring the importance of vertices in the first step. Concurrently, some studies discuss the dependencies among hyperedges and define the s -th Laplacian based on simple walks on the dual hypergraphs [41, 100]. Finally, more sophisticated random walks with non-linear Laplacians have been designed [23, 101-103].

S et W alk s addresses three main drawbacks from existing methods: 1 None of these methods are designed for temporal hypergraphsm so they cannot capture temporal properties of the network. Also, natural attempts to extend them to temporal hypergraphs and let the walker uniformly walk over time ignore the fact that recent hyperedges are more informative than older ones (see Table 2). To address this issue, S et W alk uses a temporal bias factor in its sampling procedure (Equation 1). 2 Existing hypergraph random walks are unable to capture either higher-order interactions of vertices or higher-order dependencies of hyperedges. That is, random walks with edge-independent weights [37] are not able to capture higher-order interactions and are equivalent to simple random walks on the CE of the hypergraph [40]. The expressivity of random walks on hypergraphs with edge-dependent walks is also limited when we have a limited number of sampled walks (see Theorem 1). Finally, defining a hypergraph random walk as a random walk on the dual hypergraph also cannot capture the higher-order dependencies of hyperedges (see Appendix C and Appendix D). S et W alk by its nature is able to walk over hyperedges (instead of vertices) and time and can capture higher-order interactions. Also, with a structural bias factor in its sampling procedure, which is based on hyperedge-dependent node weights, it is more informative than a simple random walk on the dual hypergraph, capturing higher-order dependencies of hyperedges. See Appendix C for further discussion.

A.3 MLP-Mixer

MLP-M ixer [44] is a family of models, based on multi-layer perceptions (MLPs), that are simple, amenable to e ffi cient implementation, and robust to long-term dependencies (unlike R nn s, attention mechanisms, and Transformers [83]) with a wide array of applications from computer vision [104] to neuroscience [105]. The original architecture is designed for image data, where it takes image tokens as inputs. It then encodes them with a linear layer, which is equivalent to a convolutional layer over the image tokens, and updates their representations with a sequence of feed-forward layers applied to image tokens and features. Accordingly, we can divide the architecture of MLP-M ixer into two main parts: 1 Token Mixer: The main intuition of the token mixer is to clearly separate the cross-location operations and learn the cross-feature (cross-location) dependencies. 2 Channel Mixer: The intuition behind the channel mixer is to clearly separate the per-location operations and provide positional invariance, a prominent feature of convolutions. In both M ixer and S et M ixer we use the channel mixer as designed in MLP-M ixer . Next, we discuss the token mixer and its limitation in mixing features in a permutation variant manner:

Token Mixer. Let E be the input of the MLP-M ixer , then the token mixer phase is defined as:

where σ ( . ) is nonlinear activation function (usually GeLU [80]). Since it feeds the input's columns to an MLP, it mixes the cross-feature information, which results in the MLP-M ixer being sensitive to permutation. Although natural attempts to remove the token mixer or its linear layer can produce a permutation invariant method, it misses cross-feature dependencies, which are the main motivation for using the MLP-M ixer architecture. To address this issue, S et M ixer uses the Softmax (.) function over features. Using Softmax over features can be seen as cross-feature normalization, which can capture their dependencies. While Softmax (.) is a non-parametric method that can bind token-wise information, it is also permutation equivariant, and as we prove in Appendix E.3, makes the S et M ixer permutation invariant.

B Additional Related Work

B.1 Learning (Multi)Set Functions

(Multi)set functions are pooling architectures for (multi)sets with a wide array of applications in many real-world problems including few-shot image classification [106], conditional regression [107], and causality discovery [108]. Zaheer et al. [97] develop D eep S ets , a universal approach to parameterize

the (multi)set functions. Following this direction, some works design attention mechanisms to learn multiset functions [109], which also inspired Baek et al. [110] to adopt attention mechanisms designed for (multi)set functions in graph representation learning. Finally, Chien et al. [25] build the connection between learning (multi)set functions with propagations on hypergraphs. To the best of our knowledge, S et M ixer is the first adaptive permutation invariant pooling strategy for hypergraphs, which views each hyperedge as a set of vertices and aggregates node encodings by considering their higher-order dependencies.

B.2 Simplicial Complexes Representation Learning

Simplicial complexes can be considered a special case of hypergraphs and are defined as a collection of polytopes such as triangles and tetrahedra, which are called simplices [111]. While these frameworks can be used to represent higher-order relations, simplicial complexes require the downward closure property [112]. That is, every substructure or face of a simplex contained in a complex K is also in K . Recently, to encode higher-order interactions, representation learning on simplicial complexes has attracted much attention [6, 113-119]. The first group of methods extend node2vec [67] to simplicial complexes with random walks on interactions through Hasse diagrams and simplex connections inside p -chains [113, 115]. With the recent advances in message-passing-based methods, several studies focus on designing neural networks on simplicial complexes [116-119]. Ebli et al. [116] introduced Simplicial neural networks (SNN), a generalization of spectral graph convolutio,n to simplicial complexes with higher-order Laplacian matrices. Following this direction, some works propose simplicial convolutional neural networks with di ff erent simplicial filters to exploit the relationships in upper- and lower-neighborhoods [117, 118]. Finally, the last group of studies use an encoder-decoder architecture as well as message-passing to learn the representation of simplicial complexes [114, 120].

CA t -W alk is di ff erent from all these methods in three main aspects: 1 Contrary to these methods, CA t -W alk is designed for temporal hypergraphs and is capable of capturing higher-order temporal properties in a streaming manner, avoiding the drawbacks of snapshot-based methods. 2 CA t -W alk works in the inductive setting by extracting underlying dynamic laws of the hypergraph, making it generalizable to unseen patterns and nodes. 3 All these methods are designed for simplicial complexes, which are special cases of hypergraphs, while CA t -W alk is designed for general hypergraphs and does not require any assumption of the downward closure property.

B.3 How Does CA t -W alk Di ff er from Existing Works? (Contributions)

As we discussed in Appendix A.2, existing random walks on hypergraphs are unable to capture either 1 higher-order interactions between nodes or 2 higher-order dependencies of hyperedges. Moreover, all these walks are for static hypergraphs and are not able to capture temporal properties. To this end, we design S et W alk a higher-order temporal walk on hypergraphs. Naturally, S et W alk s are capable of capturing higher-order patterns as a S et W alk is defined as a sequence of hyperedges. We further design a new sampling procedure with temporal and structural biases, making S et W alk s capable of capturing higher-order dependencies of hyperedges. To take advantage of complex information provided by S et W alk s as well as training the model in an inductive manner, we design a two-step anonymization process with a novel pooling strategy, called S et M ixer . The anonymization process starts with encoding the position of vertices with respect to a set of sampled S et W alk s and then aggregates node positional encodings via a non-linear permutation invariant pooling function, S et M ixer , to compute their corresponding hyperedge positional encodings. This two-step process lets us capture structural properties while we also care about the similarity of hyperedges. Finally, to take advantage of continuous-time dynamics in data and avoid the limitations of sequential encoding, we design a neural network for temporal walk encoding that leverages a time encoding module to encode time as well as a M ixer module to encode the structure of the walk.

C S et W alk and Random Walk on Hypergraphs

We reviewed existing random walks in Appendix A.2. Here, we discuss how these concepts are di ff erent from S et W alk s and investigate whether S et W alk s are more expressive than these methods.

As we discussed in Sections 1 and 3.2, there are two main challenges for designing random walks on hypergraphs: 1 Random walks are a sequence of pair-wise interconnected vertices, even though

edges in a hypergraph connect sets of vertices. 2 A sampling probability of a walk on a hypergraph must be di ff erent from its sampling probability on the CE of the hypergraph [37-43]. To address these challenges, most existing works on random walks on hypergraphs ignore 1 and focus on 2 to distinguish the walks on simple graphs and hypergraphs, and 1 is relatively unexplored. To this end, we answer the following questions:

Q1: Can 2 alone be su ffi cient to take advantage of higher-order interactions? First, semantically, decomposing hyperedges into sequences of simple pair-wise interactions (CE) loses the semantic meaning of the hyperedges. Consider the collaboration network in Figure 1. When decomposing the hyperedges into pair-wise interactions, both ( A , B , C ) and ( H , G , E ) have the same structure (a triangle), while the semantics of these two structures in the data are completely di ff erent. That is, ( A , B , C ) have all published a paper together, while each pair of ( H , G , E ) separately have published a paper. One might argue that although the output of hypergraph random walks and simple random walks on the CE might be the same, the sampling probability of each walk is di ff erent and with a large number of samples, our model can distinguish these two structures. In Theorem 1 (proof in Appendix E) we theoretically show that when we have a finite number of hypergraph walk samples, M , there is a hypergraph G such that with M hypergraph walks, the G and its CE are not distinguishable. Note that in reality, the bottleneck for the number of sampled walks in machine learning-based methods is memory. Accordingly, even with tuning the number of samples for each dataset, the size of samples is bounded by a small number. This theorem shows that with a limited budget for walk sampling, 2 alone is not enough to capture higher-order patterns.

Q2: Can addressing 1 alone be su ffi cient to take advantage of higher-order interactions? To answer this question, we use the extended version of the edge-to-vertex dual graph concept for hypergraphs:

Definition 4 (Dual Hypergraph) . Given a hypergraph G = ( V , E ) , the dual hypergraph of G is defined as ˜ G = ( ˜ V , ˜ E ) , where ˜ V = E and a hyperedge ˜ e = { e 1 , e 2 , . . . , ek } ∈ ˜ E shows that T k i = 1 ei , ∅ .

To address 1 , we need to see walks on hypergraphs as a sequence of hyperedges (instead of a sequence of pair-wise connected nodes). One can interpret this as a hypergraph walk on the dual hypergraph. That is, each hypergraph walk on the dual graph is a sequence of G 's hyperedges: e 1 → e 2 → ·· · → ek . However, as shown by Chitra and Raphael [40], each walk on hypergraphs with edge-independent weights for sampling vertices is equivalent to a simple walk on the (weighted) CE graph. To this end, addressing 2 alone can be equivalent to sample walks on the CE of the dual hypergraph, which misses the higher-order interdependencies of hyperedges and their intersections.

Based on the above discussion, both 1 and 2 are required to capture higher-order interaction between nodes as well as higher-order interdependencies of hyperedges. The definition of S et W alk s (Definition 3) with structural bias , introduced in Equation 1, satisfies both 1 and 2 . In the next section, we discuss how a simple extension of S et W alk s can not only be more expressive than all existing walks on hypergraphs and their CEs, but its definition is universal, and all these methods are special cases of extended S et W alk .

C.1 Extension of S et W alk s

Random walks on hypergraphs are simple but less expressive methods for extracting network motifs, while S et W alk s are more complex patterns that provide more expressive motif extraction approaches. One can model the trade-o ff of simplicity and expressivity to connect all these concepts in a single notion of walks. To establish a connection between S et W alk s and existing walks on hypergraphs, as well as a universal random walk model on hypergraphs, we extend S et W alk s to r -S et W alk s, where parameter r controls the size of hyperedges that appear in the walk:

Definition 5 ( r -S et W alk ) . Given a temporal hypergraph G = ( V , E , X ) , and a threshold r ∈ Z + , a rS et W alk with length ℓ on temporal hypergraph G is a randomly generated sequence of hyperedges (sets):

where ei ∈ E , | ei | ≤ r, t ei + 1 < t ei , and the intersection of ei and ei + 1 is not empty, ei ∩ ei + 1 , ∅ . In other words, for each 1 ≤ i ≤ ℓ -1 : ei + 1 ∈ E t i ( ei ) . We use S w [ i ] to denote the i-th hyperedge-time pair in the S et W alk . That is, S w [ i ][0] = ei and S w [ i ][1] = t ei .

The only di ff erence between this definition and Definition 3 is that r -S et W alk limits hyperedges in the walk to hyperedges with size at most r . The sampling process of r -S et W alk s is the same as that of S et W alk (introduced in Section 3.2 and Appendix D), while we only sample hyperedges with size at most r . Now to establish the connection of r -S et W alk s and existing walks on hypergraphs, we define the extended version of the clique expansion technique:

Definition 6 ( r -Projected Hypergraph) . Given a hypergraph G = ( V , E ) and an integer r ≥ 2 , we construct the (weighted) r-projected hypergraph of G as a hypergraph ˆ G r = ( V , ˆ E r ) , where for each e = { u 1 , u 2 , . . . , uk } ∈ E :

if k ≤ r: add e to the ˆ E r ,

Each of steps 1 or 2 can be done in a weighted manner. In other words, we approximate each hyperedge with a size of more than r with GLYPH<16> k r GLYPH<17> (weighted) hyperedges with size r . For example, when r = 2, the 2-projected graph of G is equivalent to its clique expansion, and r = ∞ is the hypergraph itself. Furthermore, we define the Union Projected Hyperaph (UP hypergraph) as the union of all r -projected hypergraphs, i.e., G ∗ = ( V , S ∞ r = 2 ˆ E r ). Note that the UP hypergraph has the downward closure property and is equivalent to the simplicial complex representation of the hypergraph G . The next proposition establishes the universality of the r -S et W alk concept.

Proposition 2. Edge-independent random walks on hypergraphs [37], edge-dependent random walks on hypergraphs [40], and simple random walks on the CE of hypergraphs are all special cases of rS et W alk , when applied to the 2 -projected graph, UP hypergraph, and 2 -projected graph, respectively. Furthermore, all the above methods are less expressive than rS et W alk s.

The proof of this proposition is in Appendix E.5.

D E ffi cient Hyperedge Sampling

For sampling S et W alk s, inspired by Wang et al. [33], we use two steps: 1 Online score computation: we assign a set of scores to each incoming hyperedge. 2 Iterative sampling: we use assigned scores in the previous step to sample hyperedges in a S et W alk .

Algorithm 1 Online Score Computation

Online Score Computation. The first part essentially works in an online manner and assigns each new incoming hyperedge e a four-tuple of scores:

Iterative Sampling. In the iterative sampling algorithm, we use pre-computed scores by Algorithm 1 and sample a hyperedge ( e , t ) given a previously sampled hyperedge ( ep , t p ). In the next proposition,

Algorithm 2 Iterative S et W alk Sampling

Input: Given a hypergraph G = ( V , E ), α ∈ [0 , 1], and previously sampled hyperedge ( ep , t p ) Output: Next sampled hyperedge ( e , t )

1: for ( e , t ) ∈ E t p ( ep ) with an decreasing order of t do

2: Sample b ∼ U niform (0 , 1);

3: Get Pe , t [0] , Pep , t p [1] , Pep , t p [2] and φ ( e , ep ) from the output of Algorithm 1;

4: P ← Normalize Pe , t [0] Pep , t p [1] × exp ( φ ( e , ep ) ) Pep , t p [2] ;

5: if b < P then return ( e , t ); return ( eX , tX );

▷ ( eX , tX ) is a dummy empty hyperedge signaling the end of algorithm.

...

we show that this sampling algorithm samples each hyperedge with the probability mentioned in Section 3.2.

Proposition 3. Algorithm 2 sample a hyperedge ( e , t ) after ( ep , t p ) with a probability proportional to P [( e , t ) | ( ep , t p )] (Equation 1).

How can this sampling procedure capture higher-order patterns? As discussed in Appendix C,

S et W alk s on G can be interpreted as a random walk on the dual hypergraph of G , ˜ G . However, a simple (or hyperedge-independent) random walk on the dual hypergraph is equivalent to the walk on the CE of the dual hypergraph [40, 41], missing the higher-order dependencies of hyperedges. Inspired by Chitra and Raphael [40], we use hyperedge-dependent weights Γ : V×E → R ≥ 0 and sample hyperedges with a probability proportional to exp GLYPH<16>P u ∈ e ∩ ep Γ ( u , e ) Γ ( u , e ′ ) GLYPH<17> , where ep is the previously sampled hyperedge. In the dual hypergraph ˜ G = ( E , V ), we assign a score ˜ Γ : E × V → R ≥ 0 to each pair of ( e , u ) as ˜ Γ ( e , u ) = Γ ( u , e ). Now, a S et W alk with this sampling procedure is equivalent to the edge-dependent hypergraph walk on the dual hypergraph of G with edge-dependent weight ˜ Γ ( . ). Chitra and Raphael [40] show that an edge-dependent hypergraph random walk can capture some information about higher-order interactions and is not equivalent to a simple walk on the weighted CE of the hypergraph. Accordingly, even on the dual hypergraph, S et W alk with this sampling procedure can capture higher-order dependencies of hyperedges and is not equivalent to a simple walk on the CE of the dual hypergraph ˜ G . We conclude that, unlike existing random walks on hypergraphs [37, 38, 41, 79], S et W alk can capture both higher-order interactions of nodes, and, based on its sampling procedure, higher-order dependencies of hyperedges.

E Theoretical Results

E.1 Proof of Theorem 1

Theorem 1. A random S et W alk is equivalent to neither the hypergraph random walk, the random walk on the CE graph, nor the random walk on the SE graph. Also, for a finite number of samples of each, S et W alk is more expressive than existing walks.

Proof. In this proof, we focus on the hypergraph random walk and simple random walk on the CE. The proof for the SE graph is the same and also it has been proven that the SE graph and the CE of a hypergraph have close (or equal in uniform hypergraphs) Laplacian and have the same expressiveness power in the representation of hypergraphs [121-123].

First, note that each S et W alk can be approximately decomposed to a set of either hypergraph walks, simple random walks, or walk on the SE. Moreover, each of these walks can be mapped to

a corresponding S et W alk (but not a bijective mapping), by sampling hyperpedges corresponding to each consecutive pair of nodes in these walks. Accordingly, S et W alk s includes the information provided by these walks and so its expressiveness is not less than these methods. To this end, next, we discuss two examples in two di ff erent tasks for which S et W alk s are successful while other walks fail.

1 In the first task, we want to see if there exists a pair of hypergraphs with di ff erent semantics that S et W alk s can distinguish, but other walks cannot. We construct such hypergraphs. Let G = ( V , E ) be a hypergraph with V = { u 1 , u 2 , . . . , uN } and E = { ( e , t i ) } T i = 1 , where e = { u 1 , u 2 , . . . , uN } and t 1 < t 2 < · · · < tT . Also, let A be an edge-independent hypergraph random walk (or random walk on the CE) sampling algorithm. Chitra and Raphael [40] show that each of these walks is equivalent to a random walk on the weighted CE. Assume that ξ ( . ) is a function that assigns weights to edges in G ∗ = ( V , E ∗ ), the weighted CE of the G , such that a hypergraph random walk on G is equivalent to a walk on this weighted CE graph. Next, we construct a weighted hypergraph G ′ = ( V , E ′ ) with the same set of vertices but with E ′ = S T k = 1 { (( ui , uj ) , t k ) } ui , uj ∈V , such that each edge ei , j = ( ui , uj ) is associated with a weight ξ ( ei , j ). Clearly, sampling procedure A on G and G ′ are the same, while they have di ff erent semantics. For example, assume that both are collaboration networks. In G , all vertices have published a single paper together, while in G ′ , each pair of vertices have published a separate paper together. The proof for the hypergraph random walk with hyperedge-dependent weights is the same, while we construct weights of the hypergraph G ′ based on the sampling probability of hyperedges in the hypergraph random walk procedure.

2 Next, in the second task, we investigate the expressiveness of these walks for reconstructing hyperedges. That is, we want to see that given a perfect classifier, can these walks provide enough information to detect higher-order patterns in the network. To this end, we show that for a finite number of samples of each walk, S et W alk is more expressive than all of these walks in detecting higher-order patterns. Let M be the maximum number of samples and L be the maximum length of walks, we show that for any M ≥ 2 and L ≥ 2 there exists a pair of hypergraphs G , with higher-order interactions, and G ′ , with pairwise interactions, such that S et W alk s can distinguish them, while they are indistinguishable by any of these walks. We construct a temporal hypergraph G = ( V , E ) as a hypergraph with V = { u 1 , u 2 , . . . , uM ( L + 1) + 1 } and E = { ( e , t i ) } L i = 1 , where e = { u 1 , u 2 , . . . , uM ( L + 1) + 1 } and t 1 < t 2 < · · · < tL . We further construct G ′ = ( V , E ′ ) with the same set of vertices but with E ′ = S L k = 1 { (( ui , uj ) , t k ) } ui , uj ∈V . Figure 6 illustrates G and its projected graphs at a given timestamp t ∈ { t 1 , t 2 , . . . , tL } .

S et W alk with only one sample, S w , can distinguish interactions in these two hypergraphs. That is, let S w : ( e , tL ) → ( e , tL -1 ) →··· → ( e , t 1) be the sample S et W alk from G (note that masking the time, this is the only S et W alk on G , so in any case the sampled S et W alk is S w ). Since all interactions in G ′ are pairwise, any sampled S et W alk on G ′ , S w ′ , includes only pairwise interactions, so S w , S w ′ , in any case. Accordingly, S et W alk can distinguish interactions in these two hypergraphs.

Since the output of hypergraph random walks, simple walks on the CE, and walks on the SE include only pairwise interactions, it seems that they are unable to detect higher-order patterns, so are unable to distinguish these two hypergraphs. However, one might argue that by having a large number of sampled walks and using a perfect classifier, which learned the distribution of sampled random walks and can detect whether a set of sampled walks is from a higher-order interaction, we might be able to detect higher-order interactions. To this end, we next assume that we have a perfect classifier C ( . ) that can detect whether a set of sampled hypergraph walks, simple walks on the CE, or walks on the SE are sampled from a higher-order structure or pair-wise patterns. Next, we show that hypergraph random walks cannot provide enough information about every vertex for C ( . ) to detect whether all vertices in V shape a hyperedge. To this end, assume that we sample S = { W 1 , W 2 , . . . , WM } walks from hypergraph G and S ′ = { W ′ 1 , W ′ 2 , . . . , W ′ M } walks from hypergraph G ′ . In the best case scenario, since C ( . ) is a perfect classifier, it can detect that G ′ includes only pair-wise interactions based on sampled walk S ′ . To distinguish these two hypergraphs, we need C ( . ) to detect sampled walks from G (i.e., S ) that come from a higher-order pattern. For any M sampled walks with length L from G , we observe at most M × ( L + 1) vertices, so we have information about at most M × ( L + 1) vertices, unable to capture any information about the neighborhood of at least one vertex. Due to the symmetry of vertices, without loss of generality, we can assume that this vertex is u 1. This means that with these M sampled hypergraph random walks with length L , we are not able to provide any information about node u 1 at any timestamp for C ( . ). Therefore, even a perfect classifier C ( . ) cannot verify whether u 1

is a part of higher-order interaction or pair-wise interaction, which completes the proof. Note that the proof for the simple random walk is completely the same.

□

Remark 1. Note that while the first task investigates the expressiveness of these methods with respect to their sampling procedure, the second tasks discuss the limitation and di ff erence in their outputs.

Remark 2. Note that in reality, we can have neither an unlimited number of samples nor an unlimited walk length. Also, the upper bound for the number of samples or walk length depends on the RAM of the machine on which the model is being trained. In our experiments, we observe that usually, we cannot sample more than 125 walks with a batch size of 32.

E.2 Proof of Theorem 2

Before discussing the proof of Theorem 2 we first formally define what missing information means in this context.

Definition 7 (Missing Information) . We say a pooling strategy like Ψ ( . ) misses information if there is a model M such that using Ψ ( . ) on top of the M (call it ˜ M ) decreases the expressive power of M . That is, ˜ M has less expressive power than M .

Theorem 2. Given an arbitrary positive integer k ∈ Z + , let Ψ ( . ) be a pooling function such that for any set S = { w 1 , . . . , wd } :

where f is some function. Then the pooling function can cause missing information, limiting the expressiveness of the method to applying to the projected (hyper)graph of the hypergraph.

Proof. The main intuition of this theorem is that a pooling function needs to capture higher-order dependencies of its input's elements and if it can be decomposed to a summation of functions that capture lower-order dependencies, it misses information. We show that, in the general case for a given k ∈ Z + , the pooling function Ψ ( . ) when applied to a hypergraph G is at most as expressive as Ψ ( . ) when applied to the k -projected hypergraph of G (Definition 6). Let G = ( V , E ) be a hypergraph with V = { u 1 , u 2 , . . . , uk + 1 } and E = { ( V , t ) } = { ( { u 1 , u 2 , . . . , uk + 1 } , t ) } for a given time t , and ˆ G = ( V , ˆ E ) be its k -projected graph, i.e., ˆ E = { ( e 1 , t ) , . . . , ( e ( k + 1 k ) , t ) } , where ei ⊂ { u 1 , u 2 , . . . , uk + 1 } such that | ei | = k . Applying pooling function Ψ ( . ) on the hypergraph G is equivalent to applying Ψ ( . ) to the hyperedge ( V , t ) ∈ E , which provides Ψ ( V ) = P k + 1 i = 1 f ( ei ). On the other hand, applying Ψ ( . ) on projected graph ˆ G means applying it on each hyperedge ei ∈ ˆ E . Accordingly, since for each hyperedge ei ∈ ˆ E we have Ψ ( ei ) = f ( ei ), all captured information by pooling function Ψ ( . ) on ˆ G is the set of S = { f ( ei ) } k + 1 i = 1 . It is clear that Ψ ( V ) = P k + 1 i = 1 f ( ei ) is less informative than S = { f ( ei ) } k + 1 i = 1 as it is the summation of elements in S (in fact, Ψ ( V ) cannot capture the non-linear combinations of positional encodings of vertices, while S can). Accordingly, the provided information by applying Ψ ( . ) on G cannot be more informative than applying Ψ ( . ) on the G 's k -projected hypergraph. □

Remark 3. Note that the pooling function Ψ ( . ) is defined on a (hyper)graph and gets only (hyper)edges as input.

Remark 4. Although Ψ ( . ) = M ean ( . ) cannot be written as Equation 13, we can simply see that the above proof works for this pooling function as well.

E.3 Proof of Theorem 3

Theorem 3. S et M ixer is permutation invariant and is a universal approximator of invariant multi-set functions. That is, S et M ixer can approximate any invariant multi-set function.

Proof. Let π ( S ) be a given permutation of set S , we aim to show that Ψ ( S ) = Ψ ( π ( S )). We first recall the S et M ixer and its two phases: Let S = { v 1 , . . . , v d } , where v i ∈ R d 1 , be the input set and V = [ v 1 , . . . , v d ] T ∈ R d × d 1 be its matrix representation:

where

Let π ( V ) = [ v π (1) , . . . , v π ( d ) ] T be a permutation of the input matrix V . In the token mixer phase, None of LayerNorm , Softmax , and activation function σ ( . ) can a ff ect the order of elements (note that Softmax is applied row-wise). Accordingly, we can see the output of the token mixer is permuted by π ( . ):

Next, in the channel mixer, by using Equation 14 we have:

In the last step, we use the fact that M ean ( . ) is permutation invariant. Based on Equation 15 we can see that S et M ixer is permutation invariant.

Since the token mixer is just normalization it is inevitable and cannot a ff ect the expressive power of S et M ixer . Also, channel mixer is a 2-layer MLP, which is the universal approximator of any function. Therefore, S et M ixer is a universal approximator. □

E.4 Proof of Theorem 4

Theorem 4. The set-based anonymization method is more expressive than any existing anonymization strategies on the CE of the hypergraph. More precisely, there exists a pair of hypergraphs G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) with di ff erent structures (i.e., G 1 GLYPH<29> G 2 ) that are distinguishable by our anonymization process and are not distinguishable by the CE-based methods.

Proof. To the best of our knowledge, there exist two anonymization processes for random walks by Wang et al. [33] and Micali and Zhu [45]. Both of these methods are designed for graphs and to adapt them to hypergraphs we need to apply them to the (weighted) CE. Here, we focus on the process designed by Wang et al. [33], which is more informative than the other. The proof for the Micali and Zhu [45] process is the same. Note that the goal of this theorem is to investigate whether a method can distinguish a hypergraph from its CE. Accordingly, this theorem does not provide any information about the expressivity of these methods in terms of the isomorphism test.

The proposed 2-step anonymization process can be seen as a positional encoding for both vertices and hyperedges. Accordingly, it is expected to assign di ff erent positional encodings to vertices and hyperedges of two non-isomorphism hypergraphs. To this end, we construct the same hypergraphs as in the proof of Theorem 1. Let M be the number of sampled S et W alk s with length L . We construct a temporal hypergraph G = ( V , E ) as a hypergraph with V = { u 1 , u 2 , . . . , uM ( L + 1) + 1 } and E = { ( e , t i ) } L i = 1 , where e = { u 1 , u 2 , . . . , uM ( L + 1) + 1 } and t 1 < t 2 < · · · < tL . We further construct G ′ = ( V , E ′ ) with the same set of vertices but with E ′ = S L k = 1 { (( ui , uj ) , t k ) } ui , uj ∈V . As we have seen in Theorem 1, random walks on the CE of the hypergraph cannot distinguish these two hypergraphs. Since CAW [33] also uses simple random walks, it cannot distinguish these two hypergraphs. Accordingly, after its anonymization process, it again cannot distinguish these two hypergraphs.

The main part of the proof is to show that in our method, the assigned positional encodings are di ff erent in these hypergraphs. The first step is to assign each node a positional encoding. Masking

the timestamps, there is only one S et W alk in the G . Accordingly, the positional encodings of nodes in G are the same and non-zero. Given a S et W alk with length L we might see at most L × ( d max -1) + 1 nodes, where d max is the maximum size of hyperedges in the hypergraph. Accordingly, with M samples on G ′ , which d max = 2, we can see at most M × ( L + 1) vertices. Therefore, in any case, we assign a zero vector to at least one vertex. This proves that the positional encodings by S et W alk s are di ff erent in these two hypergraphs, and if the assigned hidden identities to counterpart nodes are di ff erent, clearly, feeding them to the S et M ixer results in di ff erent hyperedge encodings.

Note that each S et W alk can be decomposed into a set of causal anonymous walks [33]. Accordingly, it includes the information provided by these walks, so its expressiveness is not less than the CAW method on hypergraphs, which completes the proof of the theorem. □

Although the above statement completes the proof, next we discuss that even given the same positional encodings for vertices in these two hypergraphs, S et M ixer can capture higher-order interactions by capturing the size of the hyperedge. Recall token mixer phase in S et M ixer :

where V = [ v 1 , . . . , v M ( L + 1) + 1 ] T ∈ R ( M ( L + 1) + 1) × d 1 and v i , 01 × d 1 represents the positional encoding of ui in G . We assumed that the positional encoding of ui in G ′ is the same. The input of the token mixer phase on G is V as all of them are connected by a hyperedge. Then we have:

On the other hand, when applied to hypergraph G ′ and ( uk 1 , uk 2 ). We have:

Since we use zero padding, for any i ≥ 3, ( H token ) i , j , 0 and ( H ′ token ) i , j = 0. These zero rows, which capture the size of the hyperedge, result in di ff erent encodings for each connection.

Remark 5. To the best of our knowledge, the only anonymization process that is used on hypergraphs is by Liu et al. [78], which uses simple walks on the CE and is the same as Wang et al. [33]. Accordingly, it also su ff ers from the above limitation. Also, note that this theorem shows the limitation of these anonymization procedures when simply adopted to hypergraphs.

E.5 Proof of Proposition 2

Proposition 2. Edge-independent random walks on hypergraphs [37], edge-dependent random walks on hypergraphs [40], and simple random walks on the CE of hypergraphs are all special cases of rS et W alk , when applied to the 2 -projected graph, UP graph, and 2 -projected graph, respectively. Furthermore, all the above methods are less expressive than rS et W alk s.

Proof. For the first part, we discuss each walk separately:

1 Simple random walks on the CE of the hypergraphs: We perform 2-S et W alk s on the (weighted) 2-projected hypergraph with Γ ( . ) = 1. Accordingly, for every two adjacent edges in the 2-Projected graph like e and e ′ , we have φ ( e , e ′ ) = 1. Therefore, it is equivalent to a simple random walk on the CE (2-projected graph).

2 Edge-independent random walks on hypergraphs: As is shown by Chitra and Raphael [40], each edge-independent random walk on hypergraphs is equivalent to a simple random walk on the (weighted) CE of the hypergraph. Therefore, as discussed in 1 , these walks are a special case of r -S et W alk s, when r = 2 and applied to (weighted) 2-Projected hypergraph.

3 Edge-dependent random walks on hypergraphs: Let Γ ′ ( e , u ) be an edge-dependent weight function used in the hypergraph random walk sampling. For each node u in the UP hypergraph, we store the set of Γ ′ ( e , u ) that e is a maximal hyperedge that u belongs to. Note that there might be several maximal hyperedges that u belongs to. Now, we perform 2-S et W alk sampling on the UP hypergraph with these weights and in each step, we sample each hyperedge with weight Γ ( u , e ) = Γ ′ ( e , u ). It is

straightforward to show that given this procedure, the sampling probability of a hyperedge is the same in both cases. Therefore, edge-dependent random walks on hypergraphs are equivalent to 2-S et W alk s when applied to the UP hypergraph.

As discussed above, all these walks are special cases of r -S et W alk s and cannot be more expressive than r -S et W alk s. Also, as discussed in Theorem 1, all these walks are less expressive than S et W alk s, which are also special cases of r -S et W alk s, when r = ∞ . Accordingly, all these methods are less expressive than r -S et W alk s. □

F Experimental Setup Details

F.1 Datasets

We use 10 publicly available 3 benchmark datasets, whose descriptions are as follows:

NDC Class [6]: The NDC Class dataset is a temporal higher-order network, in which each hyperedge corresponds to an individual drug, and the nodes contained within the hyperedges represent class labels assigned to these drugs. The timestamps, measured in days, indicate the initial market entry of each drug. Here, hyperedge prediction aims to predict future drugs.

NDC Substances [6]: The NDC Substances is a temporal higher-order network, where each hyperedge represents an NDC code associated with a specific drug, while the nodes represent the constituent substances of the drug. The timestamps, measured in days, indicate the initial market entry of each drug. The hyperedge prediction task is the same as NDC Classes dataset.

High School [6, 92]: The High School is a temporal higher-order network dataset constructed from interactions recorded by wearable sensors in a high school setting. The dataset captures a high school contact network, where each student / teacher is represented as a node and each hyperedge shows Face-to-face contact among individuals. Interactions were recorded at a resolution of 20 seconds, capturing all interactions that occurred within the previous 20 seconds. Node labels in this data are the class of students, and we focus on the node class "PSI" in our classification tasks.

Primary School [6, 93]: The primary school dataset resembles the high school dataset, di ff ering only in terms of the school level from which the data is collected. Node labels in this data are the class of students, and we focus on the node class "Teachers" in our classification tasks.

Congress Bill [6, 94, 95]: Each node in this dataset represents a US Congressperson. Each hyperedge is a legislative bill in both the House of Representatives and the Senate, connecting the sponsors and co-sponsors of each respective bill. The timestamps, measured in days, indicate the date when each bill was introduced.

Email Enron [6]: In this dataset nodes are email addresses at Enron and hyperedges are formed by emails, connecting the sender and recipients of each email. The timestamps have a resolution of milliseconds.

Email Eu [6, 96]: In this dataset, the nodes represent email addresses associated with a European research institution. Each hyperedge consists of the sender and all recipients of the email. The timestamps in this dataset are measured with a resolution of 1 second.

Question Tags M (Math sx) [6]: This dataset consists of nodes representing tags and hyperedges representing sets of tags applied to questions on math.stackexchange.com. The timestamps in the dataset are recorded at millisecond resolution and have been normalized to start at 0.

Question Tags U (Ask Ubuntu) [6]: In this dataset, the nodes represent tags, and the hyperedges represent sets of tags applied to questions on askubuntu.com. The timestamps in the dataset are recorded with millisecond resolution and have been normalized to start at 0.

Users-Threads [6]: In this dataset, the nodes represent users on askubuntu.com, and a hyperedge is formed by users participating in a thread that lasts for a maximum duration of 24 hours. The timestamps in the dataset denote the time of each post, measured in milliseconds but normalized such that the earliest post begins at 0.

The statistics of these datasets can be found in Table 3.

F.2 Baselines

We compare our method to eight previous state-of-the-art methods and baselines on the hyperedge prediction task:

CHESHIRE [11]: Chebyshev spectral hyperlink predictor (CHESHIRE), is a hyperedge prediction methods that initializes node embeddings by directly passing the incidence matrix through a one-layer neural network. CHESHIRE treats a hyperedge as a fully connected graph (clique) and uses a Chebyshev spectral GCN to refine the embeddings of the nodes within the hyperedge. The Chebyshev spectral GCN leverages Chebyshev polynomial expansion and spectral graph theory to learn localized spectral filters. These filters enable the extraction of local and composite features from graphs that capture complex geometric structures. The model with code provided is here.

H yper SAGCN [26]: Self-attention-based graph convolutional network for hypergraphs (HyperSAGCN) utilizes a Spectral Aggregated Graph Convolutional Network (SAGCN) to refine the embeddings of nodes within each hyperedge. HyperSAGCN generates initial node embeddings by hypergraph random walks and combines node embeddings by M ean (.) pooling to compute the embedding of hyperedge. The model with code provided is here.

NHP [87]: Neural Hyperlink Predictor (NHP), is an enhanced version of HyperSAGCN. NHP initializes node embeddings using Node2Vec on the CE graph and then uses a novel maximum minimum-based pooling function that enables adaptive weight learning in a task-specific manner, incorporating additional prior knowledge about the nodes. The model with code provided is here.

HPLSF [89]: Hyperlink Prediction using Latent Social Features (HPLSF) is a probabilistic method. It leverages the homophily property of the networks and introduces a latent feature learning approach, incorporating the use of entropy in computing hyperedge embedding. The model with code provided is here.

HPRA [88]: Hyperlink Prediction Using Resource Allocation (HPRA) is a hyperedge prediction method based on the resource allocation process. HPRA calculates a hypergraph resource allocation (HRA) index between two nodes, taking into account direct connections and shared neighbors. The HRA index of a candidate hyperedge is determined by averaging all pairwise HRA indices between the nodes within the hyperedge. The model with code provided is here.

CE-CAW: This model is a baseline that we apply CAW [33] on the CE of the hypergraph. CAWis a temporal edge prediction method that uses causal anonymous random walks to capture the dynamic laws of the network in an inductive manner. The model with code provided is here.

CE-E volve GCN: This is a snapshot-based temporal graph learning method that we apply E volve GCN [90], which uses R nn s to estimate the GCN parameters for the future snapshots, on the CE of the hypergraph. The model with code provided is here.

CE-GCN: We apply Graph Convolutional Networks [91] to the CE of the hypergraph to obtain node embeddings. Next, we use MLP to predict edges. The implementation is provided in the Pytorch Geometric library.

For node classification, we use additional five state-of-the-art deep hypergraph learning methods and a CE-based baseline:

H yper GCN [19]: This is a generalization of GCNs to hypergraphs, where it uses hypergraph Laplacian to define convolution.

A ll D eep S ets and A ll S et T ransformer [25]: These two methods are two variants of the general message passing framework, Allset, on hypergraphs, which are based on the aggregation of messages from nodes to hyperedges and from hyperedges to nodes.

U ni GCNII [28]: Is an advanced variant of U ni G nn , a general framework for message passing on hypergraphs.

ED-HNN [124]: Inspired by hypergraph di ff usion algorithms, this method uses star expansions of hypergraphs with standard message passing neural networks.

CE-GCN: We apply Graph Convolutional Networks [91] to the CE of the hypergraph to obtain node embeddings. Next, we use MLP to predict the labels of nodes. The implementation is provided in the Pytorch Geometric library. [91]

For all the baselines, we set all sensitive hyperparameters (e.g., learning rate, dropout rate, batch size, etc.) to the values given in the paper that describes the technique. Following [60], for deep learning methods, we tune their hidden dimensions via grid search to be consistent with what we did for CA t -W alk . We exclude HPLSF [89] and HPRA [88] from inductive hyperedge prediction as it does not apply to them.

F.3 Implementation and Training Details

In addition to hyperparameters and modules (activation functions) mentioned in the main paper, here, we report the training hyperparameters of CA t -W alk : On all datasets, we use a batch size of 64 and set learning rate = 10 -4 . We also use an early stopping strategy to stop training if the validation performance does not increase for more than 5 epochs. We use the maximum training epoch number of 30 and dropout layers with rate = 0 . 1. Other hyperparameters used in the implementation can be found in the README file in the supplement.

Also, for tuning the model's hyperparameters, we systematically tune them using grid search. The search domains of each hyperparameter are reported in Table 4. Note that, the last column in Table 4 reports the search domain for hidden dimensions of modules in CA t -W alk , including S et M ixer , MLP-M ixer , and MLPs. Also, we tune the last layer pooling strategy with two options: S et M ixer or M ean (.) whichever leads to a better performance.

We implemented our method in Python 3.7 with PyTorch and run the experiments on a Linux machine with nvidia RTX A4000 GPU with 16GB of RAM.

G Additional Experimental Results

G.1 Results on More Datasets

Due to the space limit, we report the AUC results on only eight datasets in Section 4. Table 6 reports both AUC and average precision (AP) results on all 10 datasets in both inductive and transductive hyperedge prediction tasks.

G.2 Node Classification

In the main text, we focus on the hyperedge prediction task. Here we describe how CA t -W alk can be used for node classification tasks.

For each node u 0 in the training set, we sample max { deg( u 0 ) , 10 } hyperedges such as e 0 = { u 0 , u 1 , . . . , uk } . Next, for each sampled hyperedge we sample M S et W alk s with length m starting from each ui ∈ e 0 to construct S ( ui ). Next, we anonymize each hyperedge that appears in at least one S et W alk in S k i = 0 S ( ui ) by Equation 3 and then use the MLP-M ixer module to encode each S w ∈ S k i = 0 S ( ui ). To encode each node ui ∈ e 0, we use M ean ( . ) pooling over S et W alk s in S ( ui ). Finally, for node classification task, we use a 2-layer perceptron over the node encodings to make the final prediction.

Table 5 reports the results of dynamic node classification tasks on High School and Primary School datasets. CA t -W alk achieves the best or on-par performance on dynamic node classification tasks. While all baselines are specifically designed for node classification tasks, CA t -W alk achieves superior results due to 1 its ability to incorporate temporal properties (both from S et W alk s and our time encoding module), which helps to learn underlying dynamic laws of the network, and 2 its two-step set-based anonymization process that hides node identities from the model. Accordingly, CA t -W alk can learn underlying patterns needed for the node classification task, instead of using node identities, which might cause memorizing vertices.

G.3 Performance in Average Precision

In addition to the AUC, we also compare our model with baselines with respect to Average Precision (AP). Table 6 reports both AUC and AP results on all 10 datasets in inductive and transductive hyperedge prediction tasks. As discussed in Section 4, CA t -W alk due to its ability to capture both temporal and higher-order properties of the hypergraphs, achieves superior performance and outperforms all baselines in both transductive and inductive settings with a significant margin.

G.4 More Results on R nn v.s. MLP-M ixer in Walk Encoding

Most existing methods on (temporal) random walk encoding see a walk as a sequence of vertices and uses sequence encoders like R nn s or T ransformer s to encode each walk. The main drawback of these methods is that they fail to directly process temporal walks with irregular gaps between timestamps. That is, sequential encoders can be seen as discrete approximations of dynamic systems; however, this discretization often fails if we have irregularly observed data [125].nper This is the main motivation of recent studies to develop methods on continuous-time temporal networks [34, 126]. Most of these methods are too complicated and sometimes fail to generalize [127]. In CA t -W alk , we suggest a simple architecture to encode temporal walks by a time-encoding module along with a M ixer module (see Section 3.4 for the details). In this part, we evaluate the power of our M ixer module and compare its performance when we replace it with R nn s [82]. Figure 7 reports the results on all datasets. We observe that using MLP-M ixer with the time-encoding module in CA t -W alk can always outperform CA t -W alk when we replace MLP-M ixer with a R nn , and mostly this improvement is more on datasets with high variance in their timestamps. We relate this superiority to the importance of using continuous-time encoding instead of sequential encoders.

NDC Class

Congress Bill

Question Tags M

NDC Substances

Question Tags U

High School

Primary School

Users in Threads

40

60

80

100

AUC

MLP-MIXER

RNN

H Broader Impacts

Temporal hypergraph learning methods, such as CA t -W alk , benefit a wide array of real-world applications, including but not limited to social network analysis, recommender systems, brain network analysis, drug discovery, stock price prediction, and anomaly detection (e.g. bot detection in social media or abnormal human brain activity). However, there might be some potentially negative impacts, which we list as: 1 Learning underlying biased patterns in the training data, which may result in stereotyped predictions. Since CA t -W alk learns underlying dynamic laws in the training data, given biased training data, the predictions of CA t -W alk can be biased. 2 Also, powerful dynamic hypergraph models can be used for manipulation in the abovementioned applications (e.g., stock price manipulation). Accordingly, to prevent the potential risks in sensitive tasks, e.g., decision-making from graph-structured data in health care, interpretability and explainability of machine learning models on hypergraphs is a critical area for future work.

Furthermore, this work does not perform research on human subjects as part of the study and all used datasets are anonymized and publicly available.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes an hyperedge learning method suitable for modelling temporal hyperedges. The approach consists of a novel hyperedge-centric random walk, a permutation-invariant pooling method and a technique for anonymizing both the identity of the nodes and  the identity of the hyperedges. \n\nDifferent than standard random walks for hypergraphs that samples sequences of nodes, the SetWalk generates sequences of hyperedges, making it strictly more powerful than clique-expantion based methods. The pooling method used to encode the hyperedges represents a variation of MLP-mixer and the authors proved empirically that it is more powerful than other pooling approaches such as mean pooling and RNNs. Finally, the anonymization startegy incorporates positional information from hyperedges, while still allowing the model to be applied in the inductive setup. The experimental setup validates the benefits of applying the CAT-Walk for hyperedge prediction. Additionally, they also validate the importance of each component in a series of ablation studies.\n\nSTRENGTHS:\n* The proposed method contains a series of components that could be useful beyond the hyperedge prediction task presented in this paper. The idea of producing hyperedge-centric random walk is interesting and it is worth being explored in other contexts where random walk is required. Moreover the set-based pooling is a general technique that could be useful in other architectures where a set-level aggregator is needed.\n\n* The paper is very nicely presented. Even if the proposed architecture contains a  complex pipeline, containing several components, each one with its own notations, the paper is easy to follow, with a nice structure and good visualisations.\n\n* The paper introduces completely novel concepts and results presented in the experimental part clearly show the advantages of incorporating them in the method.\n\nWEAKNESSES:\n* SetWalk constitutes an hypergraph random walk technique  that, instead of producing sequences of nodes, it produces sequences of hyperedges. Given that, I am wondering what is the connection between the SetWalk method and a classical graph-based random walk applied on the line expansion of the hypergraph?\n\n* The paper mentions that “[the proposed method] removes self-attention and RNNs from the walk encoding procedure, avoiding their limitations.” A detailed discussion about what are these limitations and how the proposed method overcomes them is required.\n\n* The universal approximation results from [1] show that, by applying another function on top of the summation presented in Eq.4  $g(\\sum_i f(x_i))$ leads to a universal approximator for multisets. How does SetMixer pooling compare to that in terms of expressivity? What are the advantages of using SetMixer instead of this classical approach?\n\n* The paper mentions several times that a common drawbacks brought by various pooling methods is that they are losing important information. While being more expressive than some of them, it is not clear what are the limitations of SetMixer (if any). Is SetMixer universal approximator? If not, what are some examples of sets that SetMixer is not able to distinguish due to losing information.\n\n* In Eq.5, shouldn’t the multiplication with W_s^2 be a right side multiplication. The way it is defined at this moment is not a permutation invariant function (which would contradict the results in Theorem 3).\n\n* Theorem 4  states that the anonymization technique introduced in the paper is more expressive than any existing anonymization  strategies based on CE, mentioning that it can distinguish at least a pair of graphs that is not distinguished by those. However, being “more expressive than” should also imply that it does not lose any expressivity compared to CE.  However, the statement from Theorem 4 is unclear in this respect, since it is not obvious if the proposed method is able to distinguish between every pair of graphs that the previous methods are able to distinguish. This aspect should be more clearly mention in thetext.\n\n* A couple of additional ablation studies should be included in order to clearly emphasize the advantages brought by CAT-Walk. A few suggestions include:\n\n1. replacing the SetMIXER not only with RNNs but also self-attention/transformer as a more recent and powerful baseline.\n2. Replacing the pooling with the sum-based universal approximator for sets  g(sum_i f(x_i)) where f and g are 2-layers MLPs seems a fairer comparison than the basic MEAN-pooling operator.\n\n* In the ablation studies paragraph, which type of the hypergraph random walk is used for comparison? A citation or description of the method would be useful,\n\n[1] How Powerful are Graph Neural Networks? Xu et al\n\nQUESTIONS:\nPlease refer to the questions mentioned in the Weaknesses section.\n\nLIMITATIONS:\nA couple of limitations are mentioned in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

comments: SUMMARY:
The paper proposes an inductive approach to temporal hypergraph learning. The method includes two main components: a hypergraph random walk called SET-WALK and a pooling strategy termed SETMIXER, for extracting higher-order causal patterns. In addition, the authors design a simple neural network for hyperedge encoding.

STRENGTHS:
- The paper is well-organized.
- The authors provide a new direction for the random walk on hypergraph, though the random walk seems more complicated than the CE-based random walk.
- The paper gives many intuitions and theoretical analyses to demonstrate the superior performance of set-walk compared to existing walks.
- The set-walk can be regarded as a generalization of node-walk on hypergraph, which seems interesting and useful.
- The experiments in Table 1 suggest the proposed methods significantly outperform the baselines.

WEAKNESSES:
1. The method contains several components: including Time Encoding, SetMixer, Setwalk, and MLP-Mixer. The experiments only provide three datasets in Table 2 to evaluate each component. It would be more convincing if each component could be evaluated in all datasets and all settings in Table 1.

2. Some definitions are not very easy to understand. It would be better to add some schematic for illustrating those concepts,  e.g. Def. 1 and Def. 2.

3. The experiments on node classification are inadequate. There are many important baselines and benchmarks concerning node classification, such as [1][2].

[1] Equivariant Hypergraph Diffusion Neural Operators

[2] Hypergraph Convolutional Networks via Equivalency Between Hypergraphs and Undirected Graphs

QUESTIONS:
Is any hypergraph random walk can replace the set-walk in the CAT-Walk network?

LIMITATIONS:
Yes.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a walk-based learning method for temporal hyperedge prediction. Several conceptional contributions are made to walks in (temporal) hypergraphs which are compared to existing techniques such as clique or star expansion to represent hypergraphs by simple (weighted) graphs. A corresponding neural architecture is proposed and shown to outperform existing methods in an extensive experimental comparison.\n\nSTRENGTHS:\n* The newly introduced concepts and ideas on walks in hypergraphs are nicely motivated by the shortcomings of existing methods. The authors provide illustrative examples, which are discussed in sufficient detail.\n* The paper is well-written and comprehensive. The appendix provides interesting additional results and answers most of my questions.\n* The experimental comparison is extensive and shows good results for the proposed method.\n\nWEAKNESSES:\n* While the conceptional contributions are nicely motivated, this is not the case to the same extent for the design choices of the neural architecture. Some arguments remain vague or are not clear to me. Theorem 2 makes a statement on set functions (or pooling) and suggests that applying a function $f$ to the individual elements (or subsets of fixed size $k$) and computing their sum \"can cause missing information.\" The statement is not precise, and the meaning of \"$f$ is some function\" is unclear. The proof in the appendix also is not helpful and essentially states that it is clear that the sum is less informative than the set itself. In light of the following papers [[Deep Sets](https://arxiv.org/abs/1703.06114), [Some Might Say All You Need Is Sum](https://arxiv.org/pdf/2302.11603.pdf)] this is not clear. \n* The approach shows great results for the task of \"temporal hyperedge prediction\", which is very specific. For the standard task of node classification, it still works well but the improvement is less significant.\n\nQUESTIONS:\n* Can you make Theorem 2 more precise and address the above-mentioned concerns?\n* For some datasets, it is not clear to me how hyperedges were derived. For example, face-to-face contacts in the high school dataset naturally give rise to simple graphs. Can you explain this?\n* The considered task of \"temporal hyperedge prediction\" is very specific. Which competitors were designed for this task?\n\nLIMITATIONS:\nA detailed discussion of the prediction tasks the approach is (most) suitable for would strengthen the paper.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a novel way of defining random walks on hypergraph. The random walks consist in a sequence of hyperedges, where the transition probabilities take into account both the temporal closeness of the hyperedges and the importance of the nodes through which the hyperedges are connected. \nThe resulting random walk is more expressive (in the sense that transition probabilities capture more features of the hypergraph) than the Clique expansion and star expansion random graphs. These random walks are used to build an encoding of the nodes using a permutation invariant procedure. Finally these encodings are fed into a 2 layer MLP that outputs hyperedge predictions. The model compares favorably to other approaches used in the literature.\n\nSTRENGTHS:\nThe paper is clearly written. Although I’m not completely familiar with the relevant literature, the method seems novel and aims to solve a relevant problem in hypergraph link prediction. The theoretical reasons for introducing a new type of random walk on hypergraphs are compelling, and the method indeed manages to solve the expressivity problems associated to earlier random walk approaches. The experimental results are well cured and show a significant improvement in performance over previous algorithms.\n\nWEAKNESSES:\nThe theorems 1 and 2 are stated in an informal and imprecise way. While I understand the need not to flood the main text with mathematical details, one would have expected to find a rigorous formulation at least in the appendix. In partiuclar the definition of «expressiveness», «missing information» are not made explicit.\n\nQUESTIONS:\nCan you comment on the number of hyperparameters (such as $\\alpha,M,m$) used in CAT-WALK compared to those needed in other algorithms? Is there more hyperparameter tuning to do when using CAT-WALK?\n\nLIMITATIONS:\nThe authors have addressed the limitations of their algorithm.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-14"}