Higher-order Structure Boosts Link Prediction on Temporal Graphs

Bingheng Li; Harry Shomer; Jiliang Tang; Jingzhe Liu; Kaveh Hassani; Yan Xie; Yu Song; Zhigang Hua

REVIEW 4 major objections 6 minor 1 cited by

Higher-order Structure Boosts Link Prediction on Temporal Graphs

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By treating short-lived group interactions as hyperedges, HTGN claims both higher dynamic link prediction accuracy and up to 50% lower memory use than pairwise temporal GNNs.

desk verdict Interesting architecture, but the experimental tables are unreliable and the theory doesn't hold; not ready for review. read the letter →

arxiv 2505.15746 v1 pith:53TNOXFH submitted 2025-05-21 cs.LG cs.AI

Jingzhe Liu , Zhigang Hua , Yan Xie , Bingheng Li , Harry Shomer , Yu Song , Kaveh Hassani , Jiliang Tang This is my paper

classification cs.LGcs.AI

keywords temporalgraphneuralnetworksdynamiclinkpredictionhigher-orderstructureshypergraphconstructionmaximalcliquesmemoryefficiencyexpressivenessgraphs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that temporal graph neural networks can be both more accurate at dynamic link prediction and cheaper to run if they represent group interactions as hyperedges instead of expanding them into many pairwise edges. It introduces HTGN, which builds hyperedges from maximal cliques in short snapshots of the most recent edges, stores a single memory feature per hyperedge, and computes node embeddings by hypergraph convolution. The paper reports that HTGN ranks first or second on 9 of 10 real-world datasets and reduces GPU memory by 30–50 percent compared with memory-based pairwise baselines. It also proves that HTGN is strictly more expressive than pairwise message-passing temporal GNNs, giving a theoretical reason for the performance gain. The significance, if the claim holds, is that expressiveness and efficiency need not trade off in dynamic link prediction.

What carries the argument

The load-bearing mechanism is Algorithm 1, which collects incoming temporal edges into a snapshot and, whenever the snapshot reaches b edges, enumerates all maximal cliques with more than two nodes to define hyperedges, replacing any smaller hyperedges they contain. Each hyperedge then owns one memory unit instead of each node owning one, so the model's memory scales with the number of groups rather than the number of nodes. Hypergraph convolution (Bai et al., 2021) turns these hyperedge features into node embeddings, and an injective time-decay aggregation (Equation 3) merges old hyperedge features into new larger ones, which is how several hops of neighborhood information are compressed into a single representation.

What would settle it

Train HTGN and a strong pairwise baseline such as TGN or TNCN on a temporal network generated with no group interactions, using equal-sized embeddings; if HTGN still matches or beats the baseline, its edge is not coming from higher-order structure. Alternatively, run HTGN with snapshot sizes large enough that Theorem 1 predicts degraded hyperedge accuracy; if link prediction MRR does not drop, the hyperedge construction step is not driving the result.

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Extended reading notes

Core claim

HTGN's central claim is that higher-order structure can be identified cheaply from temporal locality: when edges are accumulated into short snapshots, maximal cliques within a snapshot are likely to be genuine group interactions, so they can be used as hyperedges. These hyperedges are stored in a purpose-built memory module whose features are updated by message passing and aggregated with a time-decaying injective function, and node embeddings are produced by hypergraph convolution over the hyperedges. The paper demonstrates superior Mean Reciprocal Rank on dynamic link prediction across both homogeneous and bipartite temporal graphs, and proves in Theorem 2 that HTGN distinguishes pairs of temporal computation trees that pairwise message-passing TGNNs cannot, making it strictly more expressive.

Load-bearing premise

The method assumes that maximal cliques found in short snapshots of the most recent b edges correspond to genuine group interactions, so that hyperedges built from them carry signal rather than noise for link prediction.

Editorial extensions

If this is right

Dynamic link prediction improves by about 5 percent MRR over the second-best method, averaged across the ten tested datasets.
GPU memory consumption drops by 30–50 percent relative to memory-based pairwise baselines like TGN and TNCN.
HTGN is strictly more expressive than pairwise message-passing temporal GNNs, meaning it can distinguish temporal computation trees those models cannot.
The same hyperedge construction approach works for both homogeneous graphs (maximal cliques) and bipartite graphs (one partition becomes hyperedges).
Memory use scales with the number of group structures rather than the number of nodes, avoiding the exponential expansion of multi-hop pairwise aggregation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: The snapshot mechanism could be replaced by a sliding time window to make hyperedge construction continuous rather than batch-based, which would make HTGN applicable to irregular streams without fixed batch sizes.
Editorial inference: The memory compression idea may transfer to static graph learning, where pooling cliques into hyperedges could substitute for expensive multi-hop message passing on dense subgraphs.
Editorial inference: The ablation result that common-neighbor information still helps suggests higher-order structure and pairwise closure signals are partly complementary; a hybrid model could push accuracy further.
Editorial inference: A direct test of the expressiveness claim would be to compare HTGN with a pairwise TGNN of matched parameter count on datasets with and without known group structure, predicting the gap is large only when group structure exists.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: HTGN is a plausible idea—build hyperedges from short-window snapshots, store hyperedge memory, and run hypergraph convolution—but the paper's own evidence undercuts it. The key tables are internally inconsistent, and the theoretical proofs don't work. I wouldn't trust the reported results.

What's genuinely new: the combination of temporal memory with hypergraph convolution to capture group interactions and reduce memory is not something I've seen in the TGNN literature. The hyperedge construction algorithm, while heuristic, is a reasonable way to get higher-order structure without enumerating all cliques globally. If it worked, it would be a useful practical contribution.

The problems are hard to ignore. Table 2's GraphMixer, TGAT, and DyRep rows are identical to Table 1's values for different datasets. That's not a typo; it's a direct copy-paste of baseline numbers onto datasets they weren't run on. The same happens for TCL and others. The average ranks and '5% improvement' claims are computed on top of these fabricated numbers. Table 3 is equally bad: tgbl-comment has 125,235 edges but 30,146,596 unique edges, which is impossible. The theorem proofs are also suspect. Appendix A's proof of Theorem 1 applies the PAC-Bayes bound in the wrong direction—Lemma A.1 gives E_Q[...] ≤ D_KL + ln E_P[...], but the proof introduces a minus sign. The statement of the theorem itself is odd, with an upper bound that could be negative. Theorem 2 is a sketch: it relies on the assumption that the constructed hyperedges correspond to real higher-order interactions, and the example in Figure 5 is not fully analyzed.

There's no code or data, so there's no way to check any of this independently.

I don't think this is a 'sound idea with minor flaws.' The flaws are load-bearing: the accuracy claims rest on broken tables, the memory claims are only shown in a log-scale figure with no numbers, and the theory is not proven. A serious editor would desk reject, or at minimum require a major overhaul with corrected tables, code, and a proper proof.

For a reader: if you're interested in higher-order temporal graph learning, the idea is worth a thought, but cite it with caution. I would not bring it to reading group as a model paper; maybe as a cautionary example.

Recommendation: do not send to peer review in its current form. If the authors can fix the tables and provide code with reproducible numbers, it might become a real submission.

Referee Report

4 major / 6 minor

Summary. The paper proposes HTGN, a temporal graph neural network that constructs hyperedges from temporal event streams (maximal cliques in short snapshots for homogeneous graphs, bipartite projections with a time threshold for heterogeneous graphs), stores hyperedge features in a dynamic memory module, and applies hypergraph convolution to compute node embeddings for dynamic link prediction. The authors claim that HTGN achieves superior link prediction performance on ten real-world datasets while reducing GPU memory by up to 50%, and they provide two theoretical results: Theorem 1, which asserts that hyperedge-construction accuracy decreases with snapshot duration, and Theorem 2, which asserts that HTGN is strictly more expressive than pairwise message-passing temporal GNNs.

Significance. If the claims were substantiated, the work would be a meaningful contribution to temporal graph learning: it would be the first temporal GNN to demonstrate that exploiting higher-order structures can simultaneously improve accuracy and reduce memory cost. The idea of constructing hyperedges dynamically from event streams and allocating memory units to hyperedges rather than to individual nodes is interesting and practically motivated, and the paper makes its algorithms explicit. However, the empirical evidence is compromised by apparent duplication of baseline numbers across different datasets and by impossible dataset statistics, and the theoretical proofs contain load-bearing gaps. As presented, the central claims are not established, so the current significance is mostly in the proposal's potential rather than in its validated results.

major comments (4)

[Section 6.2, Tables 1 and 2] The baseline rows for GraphMixer, TGAT, and DyRep in Table 2 reproduce exactly, to one decimal place, the values reported in Table 1 for UCI, Enron, and SocialEvo, even though the datasets in Table 2 (tgbl-wiki, Reddit, LastFM) are different. For example, GraphMixer is 17.6/20.4, 21.9/23.7, 35.5/35.3 in both tables, and TGAT is 20.4/23.5, 36.3/35.8, 40.4/41.6 in both tables. Such exact equality across disjoint datasets cannot arise from measurement, so the baseline results for heterogeneous graphs are not credible. Since HTGN's claimed rank of 1.3 and the average improvement of 5% are computed against these numbers, the central empirical claim of superior link prediction performance is unsupported.
[Table 3] The dataset statistics in Table 3 contain impossible entries. The tgbl-comment row reports # Edges = 125,235 and # Unique Edges = 30,146,596, which is impossible because unique edges cannot exceed total edges. The tgbl-flight row reports 67,169,570 edges and 18,143 nodes but only 3,125 unique edges, which is implausible for a network of this size and is identical to Enron's unique edge count, suggesting a copy-and-paste error. These inconsistencies indicate that the experimental data pipeline is unreliable and need to be corrected before any performance claims can be evaluated.
[Appendix A, proof of Theorem 1] The proof of Theorem 1 misapplies Lemma A.1 (the PAC-Bayes inequality from Germain et al.). The lemma states E_{h~Q}[ψ(h)] ≤ DKL(Q||P) + ln E_{h~P}[e^{ψ(h)}], which is an upper bound consisting of the KL divergence plus a log-partition term. In line (16), the proof instead claims E_Q[2^{-m}] ≤ DKL(Q||P) − E_P[m ln(e/2)], subtracting a positive quantity (since ln(e/2) > 0). This reverse direction is not justified by the lemma and invalidates the derivation. Consequently, Theorem 1 does not establish that shorter snapshots yield higher hyperedge-construction accuracy, so the heuristic in Algorithm 1 lacks the claimed theoretical support.
[Appendix B, proof of Theorem 2] The proof of Theorem 2 is not rigorous. It claims that HTGN uses an 'injective aggregation function' to combine hyperedge features, but Eq. (3) defines the aggregation as a sum of MLP outputs weighted by an exponential decay, which is not injective in general. The example in Figure 5 is underspecified: the node features and timestamps are not given in a way that verifies the claim that MP-TGN cannot distinguish the two graphs, and the argument that every link used by MP-TGN is also used by HTGN assumes a specific message-passing order that is not formalized. Thus the strict-expressiveness result is not established.

minor comments (6)

[Section 5.1, Eq. (3)] Equation (3) contains a typo: 'MLP(m[E]||)' has a dangling concatenation symbol and no second argument; it should likely be 'MLP(m[E])' or 'MLP(m[E] ∥ ...)' with the intended term specified.
[Definition 4.3] In Definition 4.3, the union symbol is missing in 'SK a=1 Ca = V'; it should read '∪_{a=1}^K C_a = V'.
[Section 6.1] The citation for negative sampling, Patania et al. (2017), appears unrelated to negative sampling in temporal link prediction; this reference should be replaced with the appropriate source.
[Appendix H, Algorithm 2] In Algorithm 2, line 7, 'Find the node in |E(v)|' should read 'Find the node in E[v]' since E(v) is a set, not a numeric quantity.
[Section 5.1, Eqs. (4)-(7)] The text refers to 'msgg()' while Eq. (5) defines 'msgd()'; the notation should be made consistent.
[Figure 4] The y-axis label 'MRR change' is ambiguous: it is unclear whether the numbers represent percentage-point differences or relative improvements, and the caption should state the normalization.

Circularity Check

1 steps flagged · score 3.0 of 10

Definitional circularity in the HT-SBM justification of clique-based hyperedge construction; the empirical claims are separate but undercut by non-circular data-integrity problems.

self definitional [Section 4.1, Definition 4.3 and Theorem 1]
"nodes within a hyperedge are more likely to interact collectively within a short time slot... This indicates that if we accumulate the dynamic links in a time interval to form a snapshot, the maximal cliques within the snapshot are likely to be real hyperedges. Next, we will formally illustrate this intuition. ... A hyperedge E ⊆ V of size k is said to exist at time t if every pair of nodes i, j∈ E is connected."

The formal model, HT-SBM, defines a hyperedge as any set of nodes whose every pair is connected, i.e., exactly the clique condition that Algorithm 1 enumerates. The 'accuracy' of the maximal-clique algorithm is therefore measured against a ground truth that is already the algorithm's output concept. Theorem 1 cannot provide independent evidence that 'maximal cliques within the snapshot are likely to be real hyperedges' because that claim is assumed by the definition of a hyperedge in the model. This step is load-bearing: Section 4.1 uses it to motivate Algorithm 1 and thus the entire HTGN design. The proof in Appendix A additionally applies a PAC-Bayes inequality with the wrong sign, but that is a correctness flaw rather than the circularity itself.

full rationale

The paper's central claimed contributions—superior dynamic link prediction and up to 50% memory reduction—are experimental or architectural outcomes, not quantities fitted to data and then renamed as predictions, so they are not circular in the rubric's sense. The one definitional circularity I can exhibit is the hyperedge-construction justification: HT-SBM defines a hyperedge as any fully connected set of nodes, so the maximal-clique algorithm's 'accuracy' is measured against a ground truth that already coincides with the algorithm's output concept. Theorem 1 therefore does not provide independent evidence that short-time cliques are genuine higher-order structures; it restates the modeling assumption. Theorem 2 has a definitional flavor ('The difference can be captured by HTGN'), but it is better classified as an under-specified proof rather than a circular reduction, because the paper does not show from its concrete embedding equations that the example TCTs map to different embeddings. I also note, outside the circularity score, that Table 2 duplicates Table 1 baseline values across disjoint heterogeneous datasets (the GraphMixer, TGAT, and DyRep rows are identical to the UCI, Enron, and SocialEvo columns), and Table 3 reports tgbl-comment as having 125,235 edges but 30,146,596 unique edges. These are data-integrity problems that undermine the empirical claims, but they are not circular reasoning. The score of 3 reflects one load-bearing self-definitional step while the core empirical method retains independent content.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the free hyperparameters of the hyperedge construction (snapshot size b, time threshold t', decay constants alpha and beta, and the receptive field settings) that are chosen by hand rather than derived. The theoretical results lean on the HT-SBM domain model, the temporal locality assumption for group interactions, the TCT expressiveness framework, and external PAC-Bayes and maximal-clique bounds, the first of which is applied incorrectly in the paper. No new physical entities are introduced.

free parameters (5)

snapshot size b (Algorithm 1) = 200
Controls how many recent edges are grouped before maximal cliques are mined; chosen by hand in Implementation Details and affects which hyperedges are created.
time threshold t' for heterogeneous hyperedges (Algorithm 2) = not reported in main text
Determines how long a node remains in a hyperedge; implementation mentions 'up to 15 distinct nodes' instead of giving t', so the exact value is unspecified.
aggregation decay hyperparameters alpha and beta (Eq. 3) = alpha=2, beta=1e-4
Shape the exponential time-decay used when merging hyperedge features; chosen by hand, not derived.
number of hyperedge message passing layers = 2
Depth of the hypergraph embedding; chosen by hand based on the receptive field ablation in Appendix F.
neighborhood hop count k and max neighbors = k=2, max neighbors=20
Receptive field size for the hypergraph convolution; selected to balance performance and efficiency per Appendix F.

assumptions (5)

domain assumption Hypergraph temporal stochastic block model (HT-SBM) with community connection probabilities evolving as Lambda_t = Lambda_0 * pi(t)
Definition 4.3 postulates that group interactions arise from community structure with a time-scaled connection matrix; this model is used to prove Theorem 1 but is not validated on real data.
domain assumption Nodes in a true hyperedge interact collectively within a short time window
Stated in Section 4.1, citing Benson et al. 2018; this is the premise that justifies mining maximal cliques in short snapshots.
standard math Expressiveness of TGNNs is defined by distinguishing temporal computation tree (TCT) pairs, following Souza et al. 2022
Appendix B adopts this definition to state and prove Theorem 2, so the theorem inherits the assumptions of that framework.
standard math PAC-Bayes lemma (Lemma 17 of Germain et al. 2015) and maximal-clique accuracy bound (Theorem 3 of Wang & Kleinberg 2024)
Appendix A invokes these external results; the first is applied incorrectly in the proof.
ad hoc to paper The heuristic Algorithm 1 recovers all true higher-order structures that matter for expressiveness
Theorem 2's proof in Appendix B assumes HTGN's hyperedge set reflects the actual group structure of the graph; the algorithm's snapshot threshold b is not guaranteed to find such structures, so the strict-expressiveness claim depends on this unverified premise.

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Cite this review

Pith. "Pith review of Higher-order Structure Boosts Link Prediction on Temporal Graphs." pith.science (2026). https://pith.science/paper/53TNOXFH

@misc{pith2026250515746,
  author       = {Pith},
  title        = {Pith review of: Higher-order Structure Boosts Link Prediction on Temporal Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/53TNOXFH}},
  note         = {Machine review of arXiv:2505.15746}
}

read the original abstract

Temporal Graph Neural Networks (TGNNs) have gained growing attention for modeling and predicting structures in temporal graphs. However, existing TGNNs primarily focus on pairwise interactions while overlooking higher-order structures that are integral to link formation and evolution in real-world temporal graphs. Meanwhile, these models often suffer from efficiency bottlenecks, further limiting their expressive power. To tackle these challenges, we propose a Higher-order structure Temporal Graph Neural Network, which incorporates hypergraph representations into temporal graph learning. In particular, we develop an algorithm to identify the underlying higher-order structures, enhancing the model's ability to capture the group interactions. Furthermore, by aggregating multiple edge features into hyperedge representations, HTGN effectively reduces memory cost during training. We theoretically demonstrate the enhanced expressiveness of our approach and validate its effectiveness and efficiency through extensive experiments on various real-world temporal graphs. Experimental results show that HTGN achieves superior performance on dynamic link prediction while reducing memory costs by up to 50\% compared to existing methods.

Figures

Figures reproduced from arXiv: 2505.15746 by the authors.

**Figure 1.** The transformation between heterogeneous bipartite graphs and hypergraphs. hyperedges, bipartite graphs are treated as explicit representations of higher-order structures (Kirkley, 2024). Specifically, nodes from one partition connected to the same node in the other partition form a hyperedge as shown in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** The pipeline of HTGN. Left: the higher-order structure memory module. It will be dynamically updated as the stream of interaction events is input into the model (a homogeneous temporal graph in this example). Middle: the hypergraph embedding module. It will compute the node embeddings with the hyperedge features. Right: the link predictor. It will predict whether two nodes will form a link at the given time accordin… view at source ↗

**Figure 3.** The efficiency of HTGN compared to the baselines. The x-axis indicates the running time of a single epoch, the y-axis indicates the model’s GPU memory costs. Lower x-axis and y-axis metrics indicate better efficiency. HTGN exhibits considerable efficiency advantages against the baselines, especially in GPU memory costs. w/o HM w/o HE w/ NCN 6.00 4.25 2.50 0.75 1.00 MRR change tgbl-wiki w/o HM w/o HE w/ NCN 6 4 2 0 2… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The change of MRR compared to original HTGN after removing/adding modules. HM indicates higher-order structure memory, HE indicates hypergraph embedding, and NCN indicates the common-neighbor information (Wang et al., 2023). ing its effectiveness. Obs. (2) The performa…

**Figure 5.** Figure 5: Two temporal graphs which HTGN can distinguish but MP-TGN can not. The colors represent node features. For MP-TGN, the embeddings of the corresponding nodes in the two figures are always the same (Souza et al., 2022). Hence, MP-TGN can not distinguish these two tempora…

**Figure 6.** Figure 6: The effects of model reception field. G. An Example of the Formation of False Hyperedges Here we provide an example to illustrate the formation of maximal cliques which are not hyperedges. As shown in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: An example temporal graph. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

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The colors represent node features

Two temporal graphs which HTGN can distinguish but MP-TGN can not. The colors represent node features. For MP-TGN, the embeddings of the corresponding nodes in the two figures are always the same (Souza et al., 2022). Hence, MP-TGN can not distinguish these two temporal graphs. However, from the perspective of higher-order structures, there are two hypere...

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URL https: //openreview.net/forum?id=xHNzWHbklj. Zhang, X., Wang, Y ., Wang, X., and Zhang, M. Efficient neural common neighbor for temporal graph link predic- tion. arXiv preprint arXiv:2406.07926,

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arXiv