Beyond Graphs: Can Large Language Models Comprehend Hypergraphs?

## Abstract

Abstract Existing benchmarks like NLGraph and GraphQA evaluate LLMs on graphs by focusing mainly on pairwise relationships, overlooking the high-order correlations found in real-world data. Hypergraphs, which can model complex beyond-pairwise relationships, offer a more robust framework but are still underexplored in the context of LLMs. To address this gap, we introduce LLM4Hypergraph , the first comprehensive benchmark comprising 21,500 problems across eight low-order, five high-order, and two isomorphism tasks, utilizing both synthetic and real-world hypergraphs from citation networks and protein structures. We evaluate six prominent LLMs, including GPT-4o, demonstrating our benchmark’s effectiveness in identifying model strengths and weaknesses. Our specialized prompting framework incorporates seven hypergraph languages and introduces two novel techniques, Hyper-BAG and Hyper-COT , which enhance high-order reasoning and achieve an average 4% (up to 9%) performance improvement on structure classification tasks. This work establishes a foundational testbed for integrating hypergraph computational capabilities into LLMs, advancing their comprehension. The source codes are at https://github.com/iMoonLab/LLM4Hypergraph .

## 1 Introduction

Large Language Models (LLMs) (Vaswani, ; Devlin, ; Brown, ; Ouyang et al., ) have made significant strides in domains such as dialogue systems (Bubeck et al., ) and image understanding (Zhao et al., ). However, they often produce untruthful or unsupported content, known as hallucinations (Wang et al., ). To mitigate this, Retrieval-Augmented Generation (RAG) (Vu et al., ) enhances prompts with relevant, factual, and up-to-date information (Khandelwal et al., ), thereby grounding outputs more effectively.
RAG typically retrieves structured data with complex relational dependencies (Guu et al., ), such as social networks or molecular structures, which are efficiently represented as graphs. Graph representations capture intricate interdependencies and provide a concise encapsulation of data relationships.
This has spurred research to improve LLMs’ understanding of graph-structured data (Guo et al., ), leading to benchmarks like NLGraph (Wang et al., ), GraphQA (Fatemi et al., ), and LLM4DyG (Zhang et al., ). These benchmarks evaluate and enhance LLMs’ capabilities in handling graph-related tasks, promoting the integration of graph-based representations in large language models.

However, real-world data often involve complex correlations beyond simple pairwise relationships (Zhou et al., ). For example, sentences within a document sharing common keywords may exhibit high-order correlations that traditional graph models fail to capture (PM et al., ). In multimodal scenarios (Kim et al., ; Feng et al., ), interactions across different data types further increase correlation complexity, exceeding the capabilities of conventional graphs, which are limited to pairwise correlations.
In contrast, hypergraphs can model high-order correlations, capturing intricate interdependencies among multiple entities simultaneously. This capability allows hypergraphs to more accurately represent the complex relationships in retrieved information, thereby enhancing prompting strategies in LLMs.
Despite these advantages, hypergraphs pose a challenge for LLMs, which are primarily designed to process linear textual data and cannot naturally ingest hypergraph structures (Feng et al., ). This limitation hinders the integration of hypergraph-based representations into LLM training and inference. Therefore, investigating whether LLMs can comprehend hypergraphs and developing prompts to improve their understanding of hypergraph structures remains an underexplored and promising research direction.

Figure: Figure 1: Performance of six leading LLMs on three types of hypergraph understanding task.
Refer to caption: /html/2410.10083/assets/x1.png

Here, we introduce the LLM4Hypergraph Benchmark, the first comprehensive testbed designed to evaluate LLMs’ understanding and reasoning of hypergraphs.
Unlike existing graph benchmarks that focus on low-order, pairwise vertex correlation tasks, our benchmark presents high-order tasks involving vertex sets and isomorphism tasks that assess models’ ability to recognize and interpret global structural correlations within hypergraphs.
Our benchmark includes both synthetic and real-world hypergraphs, such as citation and protein networks, ensuring diverse and representative evaluations across domains. It comprises 21,500 problems, covering eight low-order tasks, five high-order tasks, and two isomorphism tasks. Low-order tasks extend pairwise correlation reasoning to more complex vertex interactions, while high-order tasks involve reasoning about relationships among vertex sets, capturing the rich multi-vertex correlations inherent to hypergraphs. Isomorphism tasks evaluate the models’ ability to identify and interpret global structural patterns and symmetries within hypergraphs.
To assess our benchmark’s effectiveness, we evaluate six mainstream LLMs (including ERNIE-Lite-8K, ERNIE-Speed-128K, Qwen-Long, LLaMA3-8B, GPT-3.5-Turbo, and GPT-4o) on the LLM4Hypergraph Benchmark, as shown in [Figure 1](#S1.F1). The evaluation revealed four key findings:

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In-context learning enhances LLMs’ comprehension and performance in hypergraph tasks.
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Current mainstream LLMs struggle with isomorphism recognition, especially in larger hypergraphs, though improvements are possible.
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The proposed High-order languages (specified for hypergraphs) outperform low-order languages in enabling LLMs to understand beyond-pairwise correlations in hypergraphs.
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The proposed Hyper-COT and Hyper-BAG significantly boost LLMs’ performance on more challenging hypergraph tasks like structure classification.

Additionally, we design a specialized prompting framework for hypergraphs with seven languages for low-order and high-order structures, enabling nuanced interactions with LLMs. We introduce two instruction-based prompting techniques: Hyper-BAG and Hyper-COT, which guide LLMs to visualize high-order correlations and better understand multi-vertex relationships. Experimental results show that Hyper-COT improves performance by $4\%$ on average (up to $9\%$) in challenging structure classification tasks. Despite these advancements, challenges in high-order reasoning and encoding remain. Our work paves the way for future research to overcome these limitations and further integrate hypergraph capabilities into LLMs.

## 2 LLM4Hypergraph Benchmark

In this section, we introduce the LLM4Hypergraph Benchmark, designed to evaluate LLMs’ ability to comprehend the intricate higher-order structures of hypergraphs. To ensure a comprehensive assessment, the benchmark encompasses a diverse array of tasks and hypergraph configurations, as shown in [Figure 2](#S2.F2). It includes Low-order tasks (basic hypergraph properties and pairwise relational inferences), High-order tasks (understanding complex multi-vertex interactions and higher-order dependencies), and Isomorphic tasks (recognizing structural equivalences despite hypergraph variations).
Additionally, the LLM4Hypergraph Benchmark incorporates hypergraphs of varying scales and types, ranging from small-scale hypergraphs with limited entities and relationships to large-scale, intricate hypergraphs that emulate real-world complexity. By addressing these dimensions, the benchmark provides a holistic framework for assessing the versatility and depth of LLMs in comprehending and processing hypergraphs.

Figure: Figure 2: Overview of the LLM4Hypergraph Benchmark.
Refer to caption: /html/2410.10083/assets/x2.png

Our LLM4Hypergraph Benchmark comprises a diverse collection of hypergraphs, including both synthetic and real-world instances. Synthetic hypergraphs are generated using random and regular-structured methods, while real-world hypergraphs are sourced from citation and protein networks. These hypergraphs are categorized by scale to ensure a balanced complexity for comprehensive evaluation. For a detailed description of the hypergraph generation process, please refer to [Appendix A](#A1).

### 2.1 Task Design

In this subsection, we briefly introduce the 14 tasks included in our LLM4Hypergraph Benchmark. Detailed descriptions and methodologies for these tasks are provided in [Appendix B](#A2).

Isomorphism Tasks.
Isomorphic tasks are a fundamental category within the LLM4Hypergraph Benchmark, designed to evaluate LLMs’ ability to recognize structural equivalences and classify hypergraphs based on their overall architectural patterns. These tasks are crucial for applications in specialized fields such as molecular and protein structure analysis, where distinguishing between subtly different structural configurations is essential. We introduce two key isomorphic tasks:

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Isomorphism Recognition. This task assesses the model’s ability to determine whether two hypergraph representations correspond to the same underlying structure.
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Structure Classification. This task evaluates the model’s proficiency in distinguishing hypergraphs based on their macro-level architectural frameworks.

Low-Order Tasks.
In addition to isomorphic tasks, our LLM4Hypergraph Benchmark incorporates a series of low-order tasks designed to test LLMs’ understanding of fundamental hypergraph properties and vertex relationships. These tasks focus on basic hypergraph attributes and simple connectivity patterns essential for more complex structural analyses.

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Hyperedge Count. Counts the total number of hyperedges.
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Vertex Count. Counts the total number of vertices.
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Vertex Degree. Counts the hyperedges connected to a specific vertex.
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Vertex Connection. Checks if two vertices are directly connected by a hyperedge.
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Connected Vertices. Lists all vertices connected to a given vertex.
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Disconnected Vertices. Lists all vertices not connected to a given vertex.
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Shortest Path. Finds the shortest path between two vertices.
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Reachability Check. Determines if one vertex can be reached from another.

High-Order Tasks.
Building upon low-order tasks, LLM4Hypergraph Benchmark introduces high-order tasks designed to assess LLMs’ comprehension of complex correlations between vertex sets within hypergraphs. Unlike low-order tasks that focus on individual vertices and hyperedges, high-order tasks evaluate the model’s ability to understand and manipulate relationships involving groups of vertices and their interactions through hyperedges.

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Hyperedge Degree. Determines the number of vertices in a given hyperedge.
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Vertex Set Connection (VS Connection). Checks if two vertex sets are jointly contained within at least one hyperedge.
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Vertex-Set-in-Hyperedge Check (VS-in-He Check). Determines if a set of vertices is entirely contained within any hyperedge.
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Hyperedge-in-Hyperedge Check (He-in-He Check). Assesses if one hyperedge is completely contained within another hyperedge.
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Shared-Vertices between Hyperedges. Identifies and outputs the set of vertices shared between two hyperedges.

**Table 1: Statistics of the LLM4Hypergraph Benchmark.**
| Task | Task Name | Task Type | Difficulty | #Sample |
| --- | --- | --- | --- | --- |
| Low-Order | Hyperedge Count | Counting Problems | Easy | 1500 |
| Vertex Count | Counting Problems | Easy | 1500 |  |
| Vertex Degree | Counting Problems | Easy | 1500 |  |
| Vertex Connection Check | Decision Problems | Medium | 1500 |  |
| Reachability Check | Decision Problems | Medium | 1500 |  |
| Shortest Path | Computational Problems | Hard | 1500 |  |
| Connected Vertices | Descriptive Problems | Hard | 1500 |  |
|  | Disconnected Vertices | Descriptive Problems | Hard | 1500 |
| High-Order | Hyperedge Degree | Counting Problems | Easy | 1500 |
| Vertex Set Connection Check | Decision Problems | Medium | 1500 |  |
| Vertex-Set-in-Hyperedge Check | Decision Problems | Medium | 1500 |  |
| Hyperedge-in-Hyperedge Check | Decision Problems | Medium | 1500 |  |
| Shared-Vertices between Hyperedges | Descriptive Problems | Hard | 1500 |  |
| Isomorphism | Isomorphism Recognition | Computational Problems | Hard | 1500 |
| Structure Classification | Classification Problems | Hard | 500 |  |
| Total | 15 | 5 | 3 | 21,500 |

### 2.2 Benchmark Statistics

Our LLM4Hypergraph Benchmark comprises 14 tasks totaling 21,500 problems, as detailed in [Table 1](#S2.T1). These tasks are categorized into five types: Counting Problems (counting specific elements), Computational Problems (numerical answers), Decision Problems (yes/no responses), Descriptive Problems (listing sets of vertices or hyperedges), and Classification Problems (categorical labels). Each type includes 1,500 samples, providing a comprehensive assessment of LLMs’ abilities in numerical computations, binary decisions, descriptive generation, and hypergraph classification.

## 3 Prompt Engineering for Hypergraphs

Prompt design is essential for accurately evaluating LLMs in our LLM4Hypergraph Benchmark, as prompts guide the models’ understanding of hypergraph structures. Here, we integrate existing strategies such as CoT (Wei et al., ; Kojima et al., ) and Few-Shot Prompting (Brown, ; Zhou et al., ) to develop a prompt framework tailored for hypergraph-related tasks.
Additionally, we introduce a hypergraph language for textual descriptions and adapt these techniques with modifications like Hyper-BAG and Hyper-COT to better accommodate hypergraphs.

Figure: Figure 3: The proposed prompt structure for hypergraphs.
Refer to caption: /html/2410.10083/assets/x3.png

### 3.1 Prompt Framework

Prompt design is crucial for accurately evaluating LLMs in our LLM4Hypergraph Benchmark, as it directs the models’ understanding of hypergraph structures. [Figure 3](#S3.F3) illustrates our prompt framework, which consists of six components: Example, Hypergraph Description, Hyper-BAG, Question, Tail, and Hyper-COT. The Example section includes question-answer pairs and may optionally contain COT-for-Answer for detailed reasoning. Optional modules are indicated by brackets in the figure. More details of the framework are provided in [Appendix C](#A3). In the following, we briefly introduce the following prompt configurations:

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ZERO-SHOT: Includes only Hypergraph Description, Question, and Tail, without examples or additional reasoning prompts.
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ZERO-HYPER-COT: Extends Zero-Shot by adding Hyper-COT after the Tail, introducing hypergraph-specific reasoning.
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FEW-SHOT: Adds Example content with Hypergraph Description, Question, Tail, and Answer to provide concrete instances for guidance.
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COT: Enhances Few-Shot by including COT-for-Answer in each example, offering detailed reasoning steps for more accurate responses.
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COT-HYPER-BAG: Builds on the CoT setup by inserting Hyper-BAG between Hypergraph Description and Question, contextualizing the model’s focus on hypergraph construction and comprehension.

### 3.2 Hypergraph Language

Unlike traditional graphs, hypergraphs allow hyperedges to connect any number of vertices, requiring more complex textual descriptions. Abstract high-order descriptions may hinder LLM comprehension, while low-order descriptions enhance understanding but omit essential information. To address this, we designed two hypergraph languages: Low-Order Structure Language and High-Order Structure Language, as shown in [Figure 4](#S3.F4). The former focuses on pairwise relationships, while the latter preserves complex multi-vertex correlations. More details can refer to [Appendix D](#A4).

Low-Order Structure Language.
To facilitate LLMs’ understanding of hypergraph structures through pairwise correlations, we introduce three low-order structure languages as follows:

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Low-Order-Incidence Description (LO-Inc): Describes pairwise connections between vertices, e.g., “Vertex $v_{1}$ is connected to vertices $v_{2}$ and $v_{3}$.”
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Neighbor-Pair Description (N-Pair): Lists all pairs of vertices that share a hyperedge, e.g., “$(v_{1},v_{2}),(v_{1},v_{3})$.”
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Raw Adjacency Matrix Description (Adj-Mat): Uses a numerical adjacency matrix where binary values indicate connections between vertex pairs.

High-Order Structure Language.
To further enhance LLMs’ comprehension of hypergraph structures through higher-order correlations, we introduce four high-order structure languages as follows:

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High-Order Neighbor Description (HO-Neigh): Describes hypergraph relationships in two stages, detailing connections between vertices and hyperedges, then hyperedges and their vertices.
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High-Order Incidence Description (HO-Inc): Extends LO-Inc by including higher-order correlations, such as “Vertex $v_{1}$ is connected to vertices $v_{2}$ and $v_{3}$ with hyperedge $e_{1}$.”
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Neighbor-Set Description (N-Set): Lists entire sets of vertices connected by each hyperedge, for example, “$(v_{1},v_{2},v_{3})$.”
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Raw Incidence Matrix Description (Inc-Mat): Uses an incidence matrix where each entry indicates the inclusion of a vertex in a hyperedge.

Figure: Figure 4: The proposed “Hypergraph Language” for Hypergraph Description section.
Refer to caption: /html/2410.10083/assets/x4.png

### 3.3 Specifical Prompting for Hypergraphs

Given that hypergraphs are difficult to describe using conventional language, we introduce two specialized prompting techniques to enhance LLMs’ comprehension of hypergraph structures: Hyper-BAG and Hyper-COT. More details can be found in [Appendix E](#A5).

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Build-a-Hypergraph Prompting (Hyper-BAG) facilitates mental visualization of hypergraphs by guiding LLMs to imagine their architecture, helping the model form a coherent representation of hypergraph connections (Wang et al., ).
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Chain-of-Thought for Hypergraphs (Hyper-COT) adapts traditional CoT prompting by incorporating step-by-step reasoning tailored for hypergraphs. It includes specific prompts that guide the model to break down complex hypergraph tasks into manageable steps:
–
v1: “Let’s think step by step. Make sure the data is calculated and recorded accurately at each step.”
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v2: “Let’s analyze the connectivity by considering hyperedges linked to vertices and vertices linked through hyperedges.”
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v3: “Let’s think hyperedges connected by vertices then vertices connected by hyperedges.”

## 4 Experiments

Based on the LLM4Hypergraph Benchmark, we aim to investigate whether language models can comprehend hypergraphs by evaluating LLMs and different prompting settings.

**Table 2: Results of different prompting frameworks and hypergraph languages.**
| Difficulty | Esay | Medium | Hard |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Settings | Hyperdge | Vertex | Vertex | Vertex | Reachability | Shortest | Connected | Disconnection |  |
| Count | Count | Degree | Connection | Check | Path | Vertices | Vertices |  |  |
| ZERO-SHOT | N-Pair | 18.4% | 99.4% | 41.2% | 93.0% | 91.6% | 60.2% | 51.8% | 34.0% |
| LO-Inc | 59.2% | 100.0% | 40.4% | 97.4% | 93.8% | 67.4% | 79.4% | 40.8% |  |
| Adj-Mat | 57.6% | 100.0% | 23.0% | 92.2% | 73.2% | 47.6% | 19.8% | 14.0% |  |
| Avg. | 45.1% | 99.8% | 34.9% | 94.2% | 86.2% | 58.4% | 50.3% | 29.6% |  |
| N-Set | 85.8% | 99.6% | 90.4% | 89.6% | 88.8% | 58.6% | 46.2% | 37.8% |  |
| HO-Inc | 88.2% | 100.0% | 53.8% | 84.0% | 91.8% | 70.4% | 67.4% | 39.8% |  |
| Inc-Mat | 95.8% | 99.8% | 50.6% | 79.8% | 67.8% | 33.8% | 9.4% | 8.6% |  |
| HO-Neigh | 90.2% | 100.0% | 85.8% | 87.6% | 93.4% | 56.4% | 33.0% | 25.0% |  |
| Avg. | 90.0% | 99.9% | 70.2% | 85.3% | 85.5% | 54.8% | 39.0% | 27.8% |  |
| ZERO-HYPER-COT | N-Pair | 20.6% | 75.0% | 49.0% | 94.6% | 95.6% | 62.9% | 70.6% | 39.4% |
| LO-Inc | 53.0% | 99.8% | 53.8% | 99.8% | 98.4% | 69.8% | 75.6% | 57.2% |  |
| Adj-Mat | 56.4% | 99.2% | 21.4% | 94.0% | 73.2% | 51.8% | 16.6% | 7.2% |  |
| Avg. | 43.3% | 91.3% | 41.4% | 96.1% | 89.1% | 61.5% | 54.3% | 34.6% |  |
| N-Set | 96.2% | 54.6% | 99.2% | 91.4% | 91.6% | 63.2% | 73.4% | 46.4% |  |
| HO-Inc | 97.2% | 100.0% | 97.4% | 100.0% | 99.2% | 72.0% | 73.8% | 58.6% |  |
| Inc-Mat | 76.0% | 99.4% | 67.8% | 79.4% | 69.4% | 33.0% | 16.4% | 1.8% |  |
| HO-Neigh | 91.4% | 89.4% | 97.6% | 96.0% | 95.6% | 58.6% | 58.6% | 47.4% |  |
| Avg. | 90.2% | 85.9% | 90.5% | 91.7% | 89.0% | 56.7% | 55.6% | 38.6% |  |
| FEW-SHOT | N-Pair | 65.0% | 99.2% | 49.0% | 93.8% | 91.0% | 74.2% | 65.6% | 30.4% |
| LO-Inc | 95.6% | 99.0% | 45.4% | 96.0% | 93.6% | 79.4% | 92.8% | 47.2% |  |
| Adj-Mat | 79.2% | 100.0% | 33.8% | 87.2% | 85.6% | 64.8% | 34.2% | 23.8% |  |
| Avg. | 79.9% | 99.4% | 42.7% | 92.3% | 90.1% | 72.8% | 64.2% | 33.8% |  |
| N-Set | 84.0% | 99.6% | 78.0% | 94.0% | 88.8% | 75.2% | 55.8% | 31.0% |  |
| HO-Inc | 98.6% | 100.0% | 80.0% | 98.2% | 87.4% | 74.8% | 75.6% | 45.4% |  |
| Inc-Mat | 98.8% | 99.8% | 58.0% | 73.0% | 62.2% | 46.9% | 12.0% | 14.8% |  |
| HO-Neigh | 97.8% | 98.0% | 96.4% | 90.4% | 88.6% | 63.2% | 40.4% | 23.4% |  |
| Avg. | 94.8% | 99.4% | 78.1% | 88.9% | 81.8% | 65.0% | 46.0% | 28.7% |  |
| COT | N-Pair | 90.6% | 99.6% | 48.0% | 93.8% | 91.6% | 66.4% | 63.2% | 32.2% |
| LO-Inc | 97.4% | 99.6% | 45.6% | 95.6% | 94.0% | 73.4% | 92.8% | 48.2% |  |
| Adj-Mat | 92.6% | 99.8% | 38.8% | 87.2% | 87.6% | 64.6% | 35.4% | 24.2% |  |
| Avg. | 93.5% | 99.7% | 44.1% | 92.2% | 91.1% | 68.1% | 63.8% | 34.9% |  |
| N-Set | 82.6% | 99.8% | 81.6% | 94.0% | 89.4% | 71.6% | 59.4% | 31.4% |  |
| HO-Inc | 97.2% | 100.0% | 87.2% | 97.6% | 87.6% | 74.0% | 78.2% | 44.8% |  |
| Inc-Mat | 97.8% | 99.4% | 60.4% | 74.2% | 66.6% | 44.4% | 13.0% | 15.0% |  |
| HO-Neigh | 96.6% | 97.6% | 96.4% | 92.2% | 88.6% | 60.4% | 42.8% | 24.0% |  |
| Avg. | 93.6% | 99.2% | 81.4% | 89.5% | 83.1% | 62.6% | 48.4% | 28.8% |  |
| COT-HYPER-BAG | N-Pair | 87.8% | 100.0% | 44.4% | 95.0% | 92.0% | 70.8% | 62.0% | 32.4% |
| LO-Inc | 95.8% | 99.8% | 47.0% | 96.4% | 94.0% | 78.4% | 90.8% | 46.2% |  |
| Adj-Mat | 89.2% | 100.0% | 36.4% | 88.4% | 87.8% | 67.2% | 35.2% | 25.0% |  |
| Avg. | 90.9% | 99.9% | 42.6% | 93.3% | 91.3% | 72.1% | 62.7% | 34.5% |  |
| N-Set | 93.6% | 100.0% | 79.4% | 93.6% | 86.8% | 74.6% | 55.8% | 29.8% |  |
| HO-Inc | 98.4% | 100.0% | 84.8% | 97.8% | 87.0% | 76.8% | 77.2% | 43.2% |  |
| Inc-Mat | 97.6% | 99.6% | 60.2% | 77.2% | 72.0% | 47.2% | 11.8% | 16.0% |  |
| HO-Neigh | 97.2% | 98.8% | 96.2% | 90.0% | 87.0% | 67.2% | 37.4% | 22.6% |  |
| Avg. | 96.7% | 99.6% | 80.2% | 89.7% | 83.2% | 66.5% | 45.6% | 27.9% |  |
| Overall Avg. | 83.4% | 97.3% | 63.4% | 91.0% | 86.6% | 63.5% | 52.1% | 31.7% |  |

Experimental Settings.
We evaluate six LLMs: ERNIE-Lite-8K (Zhang et al., ), ERNIE-Speed-128K (Sun et al., ), Qwen-Long (Bai et al., ), LLaMA3-8B (Touvron et al., ), GPT-3.5-Turbo (Brown, ), and GPT-4 (Achiam et al., ), selected for their diverse architectures.
In our experiments, we employ Qwen-Long as the LLM due to its robust performance and scalability.
In the “Few-Shot”, “CoT”, and “CoT-Hyper-BAG” settings, we provide two examples by default. For Decision Problems, we maintain a balanced positive-to-negative ratio of 1:1 to ensure fairness. Other tasks assess consistency by comparing LLM outputs to ground truth.

**Table 3: Results of different prompting frameworks and hypergraph languages.**
|  | High-Order Tasks | Isomorphism Tasks |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Difficulty | Easy | Medium | Hard | Hard |  |  |  |  |
| Settings | Hyperedge | Vertex Set | VS-in-He | He-in-He | Shared-Vertices | Ismorphism | Structure |  |
| Degree | Connection | Check | Check | between He | Recognition | Classification |  |  |
| ZERO-SHOT | N-Pair | 51.6% | 65.4% | 73.0% | 33.4% | 30.3% | 52.0% | 25.2% |
| LO-Inc | 22.6% | 77.4% | 82.2% | 54.6% | 25.1% | 47.0% | 31.9% |  |
| Adj-Mat | 19.6% | 14.6% | 10.6% | 51.4% | 35.7% | 50.6% | 29.2% |  |
| Avg. | 31.3% | 52.5% | 55.3% | 46.5% | 30.4% | 49.9% | 28.8% |  |
| N-Set | 99.6% | 92.6% | 85.8% | 67.0% | 51.7% | 46.2% | 41.2% |  |
| HO-Inc | 82.0% | 83.8% | 91.2% | 61.6% | 31.3% | 50.6% | 18.2% |  |
| Inc-Mat | 39.2% | 4.8% | 1.4% | 41.6% | 45.4% | 64.6% | 36.3% |  |
| HO-Neigh | 98.2% | 82.2% | 74.8% | 57.2% | 32.0% | 51.2% | 31.2% |  |
| Avg. | 79.8% | 65.9% | 63.3% | 56.9% | 40.1% | 53.2% | 31.7% |  |
| ZERO-HYPER-COT | N-Pair | 54.2% | 64.2% | 79.6% | 33.6% | 29.7% | 53.2% | 27.0% |
| LO-Inc | 21.2% | 83.6% | 83.8% | 51.4% | 28.8% | 51.6% | 31.4% |  |
| Adj-Mat | 19.2% | 13.0% | 4.6% | 53.2% | 31.7% | 48.6% | 25.7% |  |
| Avg. | 31.5% | 53.6% | 56.0% | 46.1% | 30.1% | 51.1% | 28.0% |  |
| N-Set | 52.8% | 93.8% | 92.0% | 60.8% | 91.5% | 47.8% | 32.3% |  |
| HO-Inc | 97.0% | 96.6% | 98.0% | 52.0% | 96.9% | 52.8% | 31.0% |  |
| Inc-Mat | 52.0% | 6.4% | 2.6% | 41.0% | 44.4% | 64.8% | 26.1% |  |
| HO-Neigh | 100.0% | 97.8% | 98.6% | 47.2% | 94.4% | 56.4% | 38.5% |  |
| Avg. | 75.5% | 73.7% | 72.8% | 50.3% | 81.8% | 55.5% | 32.0% |  |
| FEW-SHOT | N-Pair | 56.4% | 78.6% | 81.2% | 37.2% | 22.6% | 45.0% | 30.5% |
| LO-Inc | 33.2% | 84.4% | 90.8% | 62.2% | 20.7% | 44.9% | 42.0% |  |
| Adj-Mat | 26.2% | 68.8% | 76.8% | 53.2% | 23.0% | 41.4% | 38.5% |  |
| Avg. | 38.6% | 77.3% | 82.9% | 50.9% | 22.1% | 43.8% | 37.0% |  |
| N-Set | 99.6% | 83.4% | 87.2% | 82.6% | 89.2% | 43.3% | 82.7% |  |
| HO-Inc | 84.8% | 87.8% | 93.6% | 65.2% | 51.7% | 44.2% | 43.8% |  |
| Inc-Mat | 40.6% | 44.0% | 53.2% | 48.2% | 23.0% | 47.8% | 53.5% |  |
| HO-Neigh | 99.8% | 81.4% | 87.2% | 65.2% | 62.7% | 47.7% | 65.5% |  |
| Avg. | 81.2% | 74.2% | 80.3% | 65.3% | 56.7% | 45.8% | 61.4% |  |
| COT | N-Pair | 49.8% | 80.8% | 76.4% | 40.0% | 25.7% | 47.2% | 28.3% |
| LO-Inc | 34.0% | 84.4% | 82.4% | 59.0% | 18.5% | 46.6% | 41.8% |  |
| Adj-Mat | 25.4% | 74.6% | 77.4% | 62.4% | 29.3% | 44.2% | 40.3% |  |
| Avg. | 36.4% | 79.9% | 78.7% | 53.8% | 24.5% | 46.0% | 36.8% |  |
| N-Set | 100.0% | 84.0% | 87.2% | 77.4% | 91.1% | 47.2% | 82.3% |  |
| HO-Inc | 86.6% | 88.4% | 88.4% | 69.8% | 55.2% | 46.0% | 41.6% |  |
| Inc-Mat | 42.6% | 47.8% | 53.6% | 57.4% | 24.1% | 60.0% | 52.2% |  |
| HO-Neigh | 100.0% | 84.2% | 82.6% | 66.6% | 62.0% | 49.2% | 66.4% |  |
| Avg. | 82.3% | 76.1% | 78.0% | 67.8% | 58.1% | 50.6% | 60.6% |  |
| COT-HYPER-BAG | N-Pair | 53.2% | 77.4% | 75.0% | 38.8% | 24.9% | 47.0% | 27.4% |
| LO-Inc | 32.8% | 86.4% | 82.2% | 65.4% | 23.9% | 46.5% | 39.4% |  |
| Adj-Mat | 24.0% | 49.8% | 66.4% | 64.6% | 33.6% | 47.4% | 38.9% |  |
| Avg. | 36.7% | 71.2% | 74.5% | 56.3% | 27.5% | 47.0% | 35.2% |  |
| N-Set | 100.0% | 84.6% | 87.6% | 81.4% | 89.4% | 46.6% | 87.2% |  |
| HO-Inc | 86.6% | 90.4% | 88.2% | 77.2% | 53.5% | 48.2% | 38.9% |  |
| Inc-Mat | 44.6% | 27.2% | 43.0% | 58.2% | 28.4% | 58.8% | 49.6% |  |
| HO-Neigh | 100.0% | 83.0% | 81.6% | 70.4% | 61.8% | 56.6% | 65.0% |  |
| Avg. | 82.8% | 71.3% | 75.1% | 71.8% | 58.3% | 52.6% | 60.2% |  |
| Overall Avg. | 60.8% | 69.9% | 72.0% | 57.4% | 45.2% | 49.8% | 42.3% |  |

### 4.1 Results of Different Hypergraph Languages

We evaluated different hypergraph languages under various settings using synthetic hypergraphs, with the experimental results presented in [Table 2](#S4.T2) and [Table 3](#S4.T3).

Observation 1: Low-order structure languages enhance LLMs’ understanding of vertex correlations within hypergraphs. As shown in [Table 2](#S4.T2), high-order structure languages achieve higher accuracy in simple low-order tasks by effectively describing high-order structures. However, in medium and difficult low-order tasks, low-order structure languages outperform high-order ones due to the redundancy of high-order descriptions when focusing on pairwise vertex correlations. Therefore, low-order languages are more effective for analyzing individual vertex relationships, thereby improving LLMs’ comprehension of hypergraphs.

Observation 2: High-order structure languages enhance LLMs’ comprehension of vertex set correlations within hypergraphs. As shown in [Table 3](#S4.T3), high-order structure languages significantly outperform low-order structure languages in high-order and isomorphism tasks across all difficulty levels. This improvement stems from high-order tasks focusing on vertex set correlations, which low-order languages find challenging to describe. High-order languages effectively represent complex multi-vertex structures, enabling LLMs to perform accurate computations and provide precise answers for intricate relationships. Consequently, using high-order structure languages significantly boosts LLMs’ understanding of vertex set correlations within hypergraphs.

Observation 3: Current LLMs struggle with isomorphism recognition tasks, but example-based prompting can partially bridge this gap. In binary classification isomorphism tasks, LLMs achieve around $50\%$ accuracy ([Table 3](#S4.T3)), showing that existing hypergraph textualizations are inadequate. In three-class structure classification, low-order structure languages produce approximately $30\%$ accuracy, failing to distinguish macro structures, and high-order structure languages alone offer little improvement. However, combining high-order structure languages with example-based prompting increases accuracy to over $60\%$, suggesting that providing examples can aid LLMs in better understanding high-order structures and distinguishing between different macro-structures.

Observation 4: In-context learning methods enhance LLMs’ understanding of hypergraph structures and task logic, thereby improving accuracy. Our experiments in [Table 2](#S4.T2) and [Table 3](#S4.T3) show that providing examples, such as Few-Shot and CoT prompts, significantly boost LLM performance compared to scenarios without examples. These examples help LLMs grasp the task logic, follow instructions more effectively, and produce accurate results. Specifically, settings with examples exhibit notable accuracy increases, highlighting the crucial role of in-context learning in understanding complex hypergraph structures and task requirements.

In the following, we selected two typical tasks from each of the three hypergraph categories to study the LLMs’ understanding of hypergraphs in depth.

### 4.2 Results of Different LLMs

We evaluate six mainstream LLMs on hypergraph tasks, as shown in [Figure 1](#S1.F1). Larger models like GPT-4 outperform smaller ones such as GPT-3.5, highlighting the importance of model capacity in understanding complex hypergraph relationships. While LLMs handle low-order and high-order tasks—analyzing local and multi-vertex associations—they struggle with isomorphism recognition, which requires a comprehensive understanding of entire hypergraph structures beyond their current visualization and global pattern recognition abilities.

Observation 5: Enhancing LLM capabilities improves hypergraph understanding, but current models generally cannot solve isomorphism recognition tasks for macro-structures. Our evaluation shows that increasing model power aids hypergraph comprehension, yet existing LLMs fail to recognize and distinguish complex global structures within hypergraphs. This limitation underscores the need for future research to enhance LLMs’ ability to visualize and internally represent high-order relational dependencies essential for effective isomorphism recognition. Consequently, only low-order and high-order tasks are used for further evaluation.

### 4.3 Results of Different Hyper-COT Promptings

Observation 6: Hyper-COT enhances LLMs’ hypergraph comprehension. Hypergraphs’ high-order correlations challenge LLMs more than low-order structures. We introduce Hyper-COT, adding step-by-step prompts. Tested on structure classification tasks ([Table 4](#S4.T4)), Hyper-COT consistently outperforms the “Naive COT”. Note that “Naive COT” refers to “Let’s think step by step”. Specifically, Hyper-COT v3 boosts performance by $4\%$ on average across seven encodings and up to $9\%$ in N-Pair and N-Set encodings by guiding LLMs to interpret connections hierarchically.

**Table 4: Resutls of different Hyper-COT prompts on the structure classification task.**
|  | Low-Order Language | High-Order Language | Avg. |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| N-Pair | LO-Inc | Adj-Mat | N-Set | HO-Inc | Inc-Mat | HO-Neigh | Rank |  |
| Naive COT | 24.3% | 30.1% | 27.0% | 27.0% | 25.2% | 23.0% | 32.3% | 2.86 |
| Hyper-COT v1 | 22.1% | 27.4% | 26.1% | 29.6% | 25.8% | 23.9% | 31.4% | 2.86 |
| Hyper-COT v2 | 22.1% | 23.5% | 27.1% | 28.8% | 27.0% | 23.5% | 31.9% | 2.71 |
| Hyper-COT v3 | 33.6% | 36.7% | 27.9% | 36.0% | 20.9% | 29.2% | 32.4% | 1.43 |

### 4.4 Results of Hyper-BAG prompting

Observation 7: Hyper-BAG enhances LLMs’ performance on high-order tasks. While Build-A-Graph (BAG) prompting Wang et al. () is effective for graph understanding, we introduce Hyper-BAG tailored for hypergraphs. Experiments in [Table 5](#S4.T5) show that BAG does not significantly improve simple low-order tasks. However, both BAG and Hyper-BAG enhance performance by over $2\%$ in high-order tasks, with Hyper-BAG achieving a $2.8\%$ improvement in the Vertex Set Connection task compared to no BAG usage. These findings demonstrate that Hyper-BAG effectively aids LLMs in understanding and performing high-order hypergraph tasks.

Table: Table 5: The influence of using additional Hyper-BAG prompting under different settings.

| Settings | Low-Order Tasks | High-Order Tasks |  |  |
| --- | --- | --- | --- | --- |
| Vertex | Rea. | VS | VS-in-He |  |
| Con. | Check | Connection | Check |  |
| w.o. | 94.0% | 88.8% | 83.4% | 87.2% |
| BAG | 93.0% | 90.2% | 85.2% | 89.2% |
| COT-BAG | 93.6% | 87.8% | 84.6% | 87.2% |
| Hyper-BAG | 93.6% | 89.4% | 86.2% | 89.8% |

Table: Table 6: The influence of the Hyper-COT with k-shot step-by-step demonstration.

| #Shot | Low-Order Tasks | High-Order Tasks |  |  |
| --- | --- | --- | --- | --- |
| Vertex. | Rea. | VS | VS-in-He |  |
| Con. | Check | Connection | Check |  |
| 0 | 83.8% | 91.4% | 84.4% | 84.4% |
| 1 | 82.4% | 79.2% | 90.2% | 86.4% |
| 2 | 96.2% | 84.4% | 87.8% | 93.4% |
| 3 | 97.8% | 85.8% | 90.4% | 92.6% |

### 4.5 Results of Hyper-COT with Different Number of Shots

Observation 8: Providing example solutions enhances LLMs’ understanding of hypergraphs. In structure classification tasks, supplying example solutions improves LLMs’ comprehension of hypergraphs. Experiments in [Table 6](#S4.T6) show that increasing the number of examples does not enhance performance in low-order tasks. However, in high-order tasks, more examples lead to significant performance gains of approximately $6\%$ to $9\%$ compared to no examples. Specifically, example prompts help LLMs better understand and distinguish complex hypergraph structures, thereby enabling more accurate classifications.

### 4.6 Results of Different Hypergraph Size on Real-World Hypergraphs

Observation 9: Larger hypergraphs pose greater challenges for LLMs’ comprehension. We investigated the impact of hypergraph size on LLMs’ understanding, with results shown in [Table 7](#S4.T7). Using hypergraphs sampled from citation networks and protein structures (see [Appendix A](#A1)), we found that as hypergraph size increases, LLM performance deteriorates. This decline is more pronounced in high-order tasks, with up to a $13\%$ decrease in citation data. The reduced performance is attributed to the increased complexity of larger hypergraphs, which hinders LLMs’ ability to comprehend and reason effectively.

**Table 7: Influence of hypergraph size on real-world citation and protein datasets.**
|  | Low-Order Tasks | High-Order Tasks |  |  |  |
| --- | --- | --- | --- | --- | --- |
| Vertex | Reachability | Vertex Set | Vertex-Set-in-Hyperedge |  |  |
| Connection | Check | Connection | Check |  |  |
| Citation | Small | 96.0% | 90.0% | 95.0% | 94.0% |
| Middle | 100.0% | 82.0% | 93.0% | 90.0% |  |
| Large | 97.0% | 77.0% | 91.0% | 81.0% |  |
| Protein | Small | 86.0% | 92.0% | 96.0% | 92.0% |
| Middle | 90.0% | 91.0% | 94.5% | 86.0% |  |
| Large | 90.0% | 88.0% | 94.0% | 88.0% |  |

## 5 Related Works

LLMs for Graphs.
LLMs’ ability to address graph-based problems has garnered significant attention. Benchmarks like NLGraph Wang et al. (), GraphInstruct Luo et al. (), GraphQA Fatemi et al. (), and LLM4DyG Zhang et al. () demonstrate LLMs’ potential in handling structured and dynamic graph data, emphasizing the importance of encoding strategies. Additionally, in structured commonsense reasoning, LLMs generate various graph structures from natural language inputs (Tandon et al., ; Madaan et al., ; Saha et al., ). However, LLMs’ ability to perform reasoning within hypergraphs remains underexplored. To bridge this gap, we introduce the benchmark, designed to enhance LLMs’ understanding of hypergraph tasks.

LLMs for Few-Shot Reasoning.
Extensive research has assessed LLMs’ few-shot reasoning across arithmetic, logical, and commonsense domains. Arithmetic tasks use datasets like AQUA (Ling et al., ), GSM8K (Cobbe et al., ), and SVAMP (Patel et al., ). For mathematical reasoning, NaturalProofs (Welleck et al., ) evaluates LLMs’ ability to generate proof steps and complete proofs. Logical reasoning includes symbolic problems such as Coin Flip and Last Letter Concatenation (Wei et al., ), and the Logic Grid Puzzle from BIG-BENCH (Srivastava et al., ). Commonsense reasoning utilizes datasets from Talmor et al. () and Geva et al. (). Additionally, methods to enhance and evaluate LLMs’ algorithmic reasoning have been developed (Zhou et al., ; Lee & Kim, ; Liu et al., ). Despite these efforts, achieving reliable and deep reasoning remains challenging. To address this, we introduce the LLM4Hypergraph, aimed at improving the assessment of LLMs’ reasoning abilities in hypergraph-based tasks.

## 6 Conclusion

In this paper, we present LLM4Hypergraph, the first benchmark to evaluate and enhance LLMs’ understanding of hypergraph data, comprising 21,500 low-order, high-order, and isomorphism tasks using synthetic and real-world hypergraphs. By assessing six leading LLMs, the benchmark effectively identifies model strengths and weaknesses. This work paves the way for integrating hypergraph capabilities into LLMs, enabling more advanced data comprehension.

## Appendix A Hypergraph Generation

We create a diverse series of hypergraphs that cover both synthetic and real-world instances. The LLM4Hypergraph Benchmark encompasses a wide spectrum of hypergraph types and scales, thereby providing a robust and versatile framework for assessing the proficiency of large language models in understanding and processing hypergraph-based data structures.

### A.1 Synthetic Hypergraphs

We first introduce the construction of synthetic hypergraphs. The synthetic hypergraphs can be categorized into two distinct types: random hypergraphs and regular-structured hypergraphs. Random hypergraphs are generated using the ‘hypergraph_Gnm()’ function(^1^11[dhg.random.hypergraph_Gnm()](https://deephypergraph.readthedocs.io/en/latest/api/random.html#dhg.random.hypergraph_Gnm)) from the DHG toolkit(^2^22[www.deephypergraph.com](https://github.com/iMoonLab/DeepHypergraph)). Importantly, we configure these hypergraphs with a “low-order first” structure. This configuration is grounded in the principle of Occam’s razor, which posits that simpler, lower-order associations are more prevalent and thus more representative of real-world scenarios where low-order relationships typically dominate over high-order ones. Consequently, random hypergraphs are extensively utilized across various tasks within the benchmark, with the exception of isomorphic tasks where structured classification is paramount.
In contrast, regular-structured hypergraphs are specifically tailored for structural classification tasks. We select three classical hypergraph structures Feng et al. () known for their distinct and regular patterns: the Hyper Pyramid, Hyper Checked Table, and Hyper Wheel. To introduce variability and enhance the robustness of the benchmark, we systematically modify the hyperparameters of these base structures. For instance, we vary the number of layers in the Hyper Pyramid and the number of vertices per blade in the Hyper Wheel, thereby generating a multitude of structurally diverse yet regularly patterned hypergraphs.

### A.2 Real-World Hypergraphs

As for the real-world instances, we source data from citation networks and protein structure datasets. The construction process involves randomly sampling sub-hypergraphs by selecting a central vertex and performing a random walk through selected hyperedges. During this walk, the vertices encountered are randomly retained until the sub-hypergraph reaches the predetermined vertex count. This methodology ensures that the sampled hypergraphs retain the intricate and authentic correlations inherent in real-world data.

### A.3 Multi-Scale Settings

To ensure scalability and applicability across a range of scenarios, both synthetic and real-world hypergraphs are categorized based on their size, defined by the number of vertices: small-scale hypergraphs (Containing between 5 to 10 vertices), medium-scale hypergraphs (Containing between 10 to 15 vertices), and large-scale hypergraphs (containing between 15 to 20 vertices).
Additionally, to maintain structural generality, synthetic hypergraphs adhere to a hyperedge-to-vertex ratio ranging from 0.2 to 1.5 times the number of vertices. This ratio ensures a balanced complexity that is neither too sparse nor excessively dense, thereby facilitating meaningful evaluations of LLM capabilities. In contrast, real hypergraphs do not impose such restrictions on the number of hyperedges, allowing them to naturally reflect the inherent connectivity patterns of their source datasets.

## Appendix B Details of Task Design

In this section, we will introduce the proposed three types of tasks in detail, respectively.

### B.1 Isomorphism Tasks

In the LLM4Hypergraph Benchmark, we prioritize isomorphic tasks as a fundamental component of our evaluation framework. This prioritization is grounded in the belief that a comprehensive understanding of the overall structural architecture is essential for LLMs to be effectively utilized in practical applications. Isomorphic tasks serve as a theoretical cornerstone for numerous model capabilities, particularly in specialized fields such as molecular structure analysis and protein analysis. In these domains, proteins with similar global structures but differing local configurations can exhibit entirely distinct properties. This phenomenon underscores the necessity for models to possess a heightened sensitivity to structural isomorphism, enabling them to accurately recognize and differentiate between nuanced structural variations.
To evaluate this capability, we introduce two representative isomorphic tasks within our benchmark:

- •
Isomorphism Recognition. This task assesses the model’s ability to determine whether two different representations accurately reflect the same underlying hypergraph structure. Given two hypergraphs ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and ${\mathcal{G}}^{\prime}=\{{\mathcal{V}}^{\prime},{\mathcal{E}}^{\prime}\}$, the target is to find whether a bijective mapping $g:={\mathcal{V}}\rightarrow{\mathcal{V}}^{\prime}$ exists. The mapping $g$ is called the isomorphism function, such that $(v_{1},v_{2},\ldots,v_{m})\in{\mathcal{E}}\iff(g(v_{1}),g(v_{2}),\ldots,g(v_{m}))\in{\mathcal{E}}^{\prime}$. Note that if $|{\mathcal{V}}|\neq|{\mathcal{V}}^{\prime}|$, we can directly have ${\mathcal{G}}$ and ${\mathcal{G}}^{\prime}$ are not isomorphism.
- •
Structure Classification. This task evaluates the model’s proficiency in distinguishing hypergraphs based on their macro-level architectural frameworks. It requires the model to identify and understand differences in the overall structural layout, even when local configurations may appear similar. Let ${\mathbb{S}}$ denote an equivalence class set of hypergraphs, encompassing several distinct types of hypergraphs $\{{\mathcal{T}}_{1},{\mathcal{T}}_{2},\ldots,{\mathcal{T}}_{n}\}$. Each subtype ${\mathcal{T}}_{i}$ within ${\mathbb{S}}$ adheres to a specific topological rule or structural pattern that defines its global architecture. These topological rules encapsulate the macroscopic organizational principles that distinguish one hypergraph type from another within the equivalence class. Prior to the task, the LLMs is provided with the fundamental topological rules governing each hypergraph subtype in ${\mathcal{T}}$. The model is expected to internalize these structural paradigms to facilitate accurate classification. This task emphasizes the model’s ability to comprehend and differentiate hypergraphs based on their overarching structural frameworks, rather than merely local or pairwise relationships. Accurate performance on the Macroscopic Architecture Differentiation task indicates that the LLM possesses a nuanced understanding of hypergraph topology, which is critical for applications in domains such as molecular structure analysis and protein modeling.

### B.2 Low-Order Tasks

In addition to isomorphic tasks, our LLM4Hypergraph Benchmark incorporates a series of low-order tasks designed to enhance LLMs’ understanding of fundamental hypergraph properties and the relationships between vertices. These tasks facilitate the comprehension of basic hypergraph attributes and simple connectivity patterns, which are essential for more complex structural analyses.

- •
Hyperdge Count. This task aims to evaluate the model’s ability to determine the total number of hyperedges within a given hypergraph. Given a hypergraph ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$, the target is to return $|{\mathcal{E}}|$, the cardinality of the hyperedge set ${\mathcal{E}}$.
- •
Vertex Count. This task assesses the model’s capability to ascertain the total number of vertices in a hypergraph. Given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$, the target is to return $|{\mathcal{V}}|$, the cardinality of the vertex set ${\mathcal{V}}$.
- •
Vertex Degree. This task assesses the model’s capability to count the number of hyperedges that connects a specific vertex. Given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and vertex $v$, the target is to return $|\{e\mid v\in e\}|$.
- •
Vertex Connection. This task is designed to evaluate whether a specific pair of vertices is directly connected by at least one hyperedge. Given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and a pair of vertices $(u,v)$, the target is to determine if there exists a hyperedge $e\in{\mathcal{E}}$ such that $u\in e$ and $v\in e$.
- •
Connected Vertices. This task examines, given a vertex $u$, the model’s ability to output the set of vertices directly connected to $u$ by at least one hyperedge. Formally, given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and a vertex $u\in{\mathcal{V}}$, the target is to output the subset $\mathcal{W}\subseteq{\mathcal{V}}$ such that for each $w\in\mathcal{W}$, there exists a hyperedge $e\in{\mathcal{E}}$ where $u\in e$ and $w\in e$.
- •
Disconnected Vertices. This task requires, given a vertex $u$, the identification and output of the set of vertices that are not connected to $u$ by any hyperedge. Formally, given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and a vertex $u\in{\mathcal{V}}$, the target is to output the subset ${\mathcal{Z}}\subseteq{\mathcal{V}}$ where for each $z\in{\mathcal{Z}}$, there does not exist any hyperedge $e\in{\mathcal{E}}$ such that $u\in e$ and $z\in e$.
- •
Shortest Path. This task aims to determine, given two vertices $u$ and $v$, the minimal sequence of vertices representing the shortest path connecting them. Formally, given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and a pair of vertices $(u,v)$, the target is to find the shortest sequence $[v_{1},v_{2},\ldots,v_{n}]$ such that $u=v_{1}$, $v=v_{n}$, and for each consecutive pair $(v_{i},v_{i+1})$, there exists a hyperedge $e\in{\mathcal{E}}$ containing both $v_{i}$ and $v_{i+1}$.
- •
Reachability Check. This task assesses whether one vertex can be reached from another through a sequence of hyperedges. Formally, given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and a pair $(u,v)$, the target is to determine if there exists a sequence of hyperedges $e_{1},e_{2},\ldots,e_{k}\in{\mathcal{E}}$ that connects $u$ to $v$.

These low-order tasks collectively ensure that LLMs develop a robust understanding of basic hypergraph properties and simple relational structures. By accurately performing tasks such as counting hyperedges and vertices, determining direct and indirect connections, and assessing reachability, models build a solid foundation for tackling more complex hypergraph-related challenges. The inclusion of these tasks within the LLM4Hypergraph Benchmark not only provides a comprehensive assessment of a model’s foundational capabilities but also facilitates the identification of specific areas requiring further enhancement, thereby contributing to the advancement of hypergraph computational intelligence.

### B.3 High-Order Tasks

Building upon the foundation of low-order tasks, our LLM4Hypergraph Benchmark introduces a series of high-order tasks tailored to assess LLMs’ comprehension of complex correlations between vertex sets within hypergraphs. Unlike low-order tasks that focus on individual vertices and hyperedges, high-order tasks evaluate the model’s ability to understand and manipulate relationships involving groups of vertices and their interactions through hyperedges.

- •
Hyperedge Degree. This task aims to determine the degree of a given hyperedge, which is defined as the number of vertices it encompasses. Formally, given a hypergraph ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and a hyperedge $e\in{\mathcal{E}}$, the target is to compute $\text{deg}(e)=|\{v\in{\mathcal{V}}\mid v\in e\}|$.
- •
Vertex Set Connection. This task evaluates whether two distinct sets of vertices ${\mathcal{A}}$ and ${\mathcal{B}}\subseteq{\mathcal{V}}$ are jointly contained within at least one hyperedge, thereby determining if ${\mathcal{A}}$ and ${\mathcal{B}}$ are connected through a common hyperedge. Formally, given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and two vertex subsets ${\mathcal{A}},{\mathcal{B}}\subseteq{\mathcal{V}}$, the target is to ascertain whether there exists a hyperedge $e\in{\mathcal{E}}$ such that ${\mathcal{A}}\cup{\mathcal{B}}\subseteq e$.
- •
Vertex-Set-in-Hyperedge Check. This task involves determining whether a specific subset of vertices ${\mathcal{S}}\subseteq{\mathcal{V}}$ is entirely contained within any hyperedge of the hypergraph. Formally, given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and a vertex subset ${\mathcal{S}}\subseteq{\mathcal{V}}$, the target is to evaluate if there exists a hyperedge $e\in{\mathcal{E}}$ such that ${\mathcal{S}}\subseteq e$.
- •
Hyperedge-in-Hyperedge Check. This task assesses whether one hyperedge is completely contained within another, thereby identifying hierarchical or nested structures within the hypergraph. Formally, given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and two hyperedges $e_{1},e_{2}\in{\mathcal{E}}$, the target is to determine if $e_{1}\subseteq e_{2}$.
- •
Shared-Vertices between Hyperedges. This task requires the model to identify and output the set of vertices shared between any two given hyperedges, thus revealing the overlapping structures and potential intersections within the hypergraph. Formally, given ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$ and two hyperedges $e_{1},e_{2}\in{\mathcal{E}}$, the target is to return ${\mathcal{S}}=e_{1}\cap e_{2}$, where ${\mathcal{S}}\subseteq{\mathcal{V}}$.

Collectively, these high-order tasks compel LLMs to engage in more sophisticated relational reasoning and structural analysis, enabling a deeper and more comprehensive understanding of hypergraph architectures. By mastering these tasks, models can better handle complex scenarios encountered in real-world applications such as molecular chemistry, social network analysis, and biological systems, where understanding intricate multi-way relationships is paramount. The inclusion of high-order tasks within the LLM4Hypergraph Benchmark thus ensures a thorough evaluation of an LLM’s ability to navigate and interpret the multifaceted connectivity inherent in hypergraph data structures, promoting advancements in hypergraph computational intelligence.

### B.4 Comparison with other Benchmarks

In this subsection, we compare our LLM4Hypergraph Benchmark with existing graph benchmarks, categorizing them into two main types: purely synthetic graph structure benchmarks and real-world graph structure benchmarks. Prominent examples of purely synthetic benchmarks include NLGraph Wang et al. () and CLEAR Chen et al. (), while real-world benchmarks encompass GPT4Graph Guo et al. () and GraphArena Tang et al. (). These existing benchmarks predominantly focus on tasks that assess the understanding of local, low-order structural properties within graphs. In contrast, our LLM4Hypergraph Benchmark not only includes low-order structural understanding tasks but also introduces high-order tasks specifically designed to evaluate models’ comprehension of associations between sets of vertices. This dual focus enables a more comprehensive exploration of LLMs’ ability to grasp complex multi-way relationships inherent in hypergraphs.
Moreover, our benchmark uniquely incorporates isomorphic tasks, marking the first instance of such an inclusion in this domain. Isomorphic tasks serve to assess the models’ proficiency in understanding and discerning global structural properties of graphs and hypergraphs, thereby providing a more holistic evaluation of their structural comprehension capabilities. Additionally, our benchmark integrates both synthetic hypergraph data and real-world hypergraph data, ensuring a diverse and representative dataset that mirrors the complexities found in practices.

By encompassing both low-order and high-order tasks, along with the novel isomorphic tasks, our LLM4Hypergraph Benchmark extends beyond the scope of existing benchmarks. It not only evaluates foundational structural understanding but also delves into advanced relational reasoning and global structural interpretation. This comprehensive approach facilitates a thorough assessment of LLMs’ abilities to comprehend and process hypergraph structures, thereby advancing the field of hypergraph computational intelligence. Consequently, our benchmark provides a robust framework for identifying both the strengths and limitations of current models, guiding future improvements in model architecture and training methodologies.

## Appendix C Details of Prompt Framework

The overall prompt framework tailored for hypergraph structures is illustrated in [Figure 3](#S3.F3). A complete prompt comprises six components: Example, Hypergraph Description, Hyper-BAG, Question, Tail, and Hyper-COT. The Example section includes question-answer (QA) pairs for each hypergraph and may additionally contain COT-for-Answer, which provides a standard chain-of-thought (CoT) explanation detailing the step-by-step reasoning process to derive the answer. In the depicted figure, certain modules within the prompt are enclosed in brackets “[]”, indicating their optional nature depending on the specific prompt configuration employed. We introduce a set of distinct prompt configurations as follows:

- •
ZERO-SHOT: This configuration includes only the Hypergraph Description, Question, and Tail sections, providing the model with the essential information required to address the query without any illustrative examples or additional reasoning prompts.
- •
ZERO-HYPER-COT: Building upon the Zero-Shot setup, this configuration incorporates the Hyper-COT component after the Tail, introducing a hypergraph-specific chain-of-thought prompt that facilitates structured reasoning tailored to hypergraph-related tasks.
- •
FEW-SHOT: Extending the Zero-Shot approach, the Few-Shot configuration precedes the main prompt with Example content. These examples consist solely of Hypergraph Description, Question, Tail, and Answer, thereby providing the model with concrete instances to guide its understanding and response generation.
- •
COT: Enhancing the Few-Shot configuration, the CoT setup adds COT-for-Answer to each example within the Example section. This addition entails detailed explanations of how each answer is systematically derived, enabling the model to follow a logical reasoning pathway for more accurate and transparent responses.
- •
COT-HYPER-BAG: This advanced configuration builds upon the CoT setup by inserting the Hyper-BAG component as a transitional sentence between the Hypergraph Description and the Question. The Hyper-BAG serves to contextualize the model’s focus on hypergraph construction and comprehension, thereby guiding it into the appropriate analytical framework for handling hypergraph-specific queries.

By integrating these varied prompt structures, the proposed framework effectively leverages existing prompting strategies—such as Chain-of-Thought and Few-Shot Prompting—while introducing hypergraph-specific enhancements. This comprehensive prompt architecture ensures that large language models are adequately guided through diverse hypergraph scenarios, encompassing both synthetic and real-world complexities. Consequently, the framework optimizes the models’ interpretative and reasoning capabilities, facilitating a more precise and nuanced evaluation of their hypergraph comprehension within the LLM4Hypergraph Benchmark. The deliberate inclusion of optional modules allows for flexible prompt configurations tailored to different evaluation requirements, thereby enhancing the robustness and versatility of the benchmarking process.

## Appendix D Details of Hypergraph Language

Unlike traditional graphs, where an edge connects exactly two vertices, hypergraphs allow hyperedges to connect any number of vertices, thereby necessitating more intricate textual descriptions. However, directly employing abstract language to describe high-order associative structures in hypergraphs can impede LLMs’ comprehension of hypergraph architectures, subsequently diminishing performance on downstream tasks. Conversely, utilizing language that describes low-order structures facilitates better understanding by LLMs but may result in the loss of essential high-order information. Inspired by graph description languages, we have designed two distinct types of languages for describing hypergraph structures: Low-Order Structure Language and High-Order Structure Language, as shown in [Figure 4](#S3.F4). The former primarily adopts a graph-like approach, leveraging low-order structural descriptions to represent hypergraphs by focusing on pairwise relationships and individual connections. This method enhances LLMs’ ability to parse and interpret the hypergraph by simplifying its representation into more familiar graph terminology. In contrast, the High-Order Structure Language directly conveys the complex, multi-vertex correlations inherent in hypergraphs, encapsulating high-order relational information without reducing them to mere pairwise interactions.
This approach preserves the richness of hypergraph structures, allowing LLMs to grasp the full extent of multi-way relationships that define hypergraph topology. By employing these two complementary descriptive languages, our framework ensures that LLMs can effectively balance the simplicity of low-order descriptions with the comprehensive detail of high-order structures, thereby optimizing their ability to understand and perform accurately on hypergraph-related tasks within the LLM4Hypergraph Benchmark.

### D.1 Low-Order Structure Language

To effectively facilitate LLMs’ understanding of hypergraph structures through pairwise correlations, we introduce three distinct low-order structure languages as follows: Low-Order Incidence Description (LO-Inc), Neighbor-Pair Description (N-Pair), and Raw Adjacency Matrix Description (Adj-Mat). Each method provides a unique approach to textualizing hypergraph relationships, enhancing the models’ ability to interpret and process hypergraph data.

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Low-Order-Incidence Description (LO-Inc): This method directly describes how a given vertex is connected to other vertices through hyperedges by explicitly stating the connections in a pairwise manner. For example, consider a hypergraph where vertex $v_{1}$ is connected to vertices $v_{2}$ and $v_{3}$. The LO-Inc representation would be: Vertex $v_{1}$ is connected to vertices $v_{2}$ and $v_{3}$. This description clearly indicates the direct correlations involving $v_{1}$, facilitating the model’s understanding of its immediate connections within the hypergraph.
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Neighbor-Pair Description (N-Pair): This method focuses on enumerating all pairs of vertices that share a hyperedge, thereby explicitly listing the connected vertex pairs within the hypergraph. Using the same hypergraph example where vertex $v_{1}$ is connected to vertices $v_{2}$ and $v_{3}$, the N-Pair representation would be: $(v_{1},v_{2}),(v_{1},v_{3})$. This method emphasizes pairwise relationships without referencing the broader connectivity of individual vertices, providing a straightforward enumeration of connected pairs.
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Raw Adjacency Matrix Description (Adj-Mat): This method utilizes an adjacency matrix to represent the hypergraph structure numerically. In this matrix-based approach, the presence or absence of a connection between any two vertices is indicated by binary values. For the hypergraph where vertex $v_{1}$ is connected to vertices $v_{2}$ and $v_{3}$, the Adj-Mat representation would be: $[[0,1,1,0],[1,0,1,1],[1,1,0,0],[0,1,0,0]]$. In this adjacency matrix, each row and column corresponds to a vertex ($v_{1}$ to $v_{4}$). A value of ‘1’ at position $(i,j)$ indicates that vertex $v_{i}$ is connected to vertex $v_{j}$ via a hyperedge, while a ‘0’ signifies no direct connection. For instance, the entry at $(1,2)$ is ‘1’, indicating a connection between $v_{1}$ and $v_{2}$.

These low-order structure languages—LO-Inc, N-Pair, and Adj-Mat—provide varied methods for representing hypergraph structures through pairwise correlations. LO-Inc offers descriptive clarity by outlining direct vertex connections, N-Pair emphasizes the enumeration of connected vertex pairs, and Adj-Mat delivers a comprehensive numerical representation through an adjacency matrix. By employing these diverse descriptive techniques, our framework enhances LLMs’ ability to accurately interpret and analyze the fundamental properties and local relationships within hypergraphs, thereby supporting robust performance on subsequent high-order and isomorphic tasks within the LLM4Hypergraph Benchmark.

### D.2 High-Order Structure Language

To further enhance LLMs’ comprehension of hypergraph structures through higher-order correlations, we introduce four distinct high-order structure languages as follows:
High-Order Neighbor Description (HO-Neigh), High-Order Incidence Description (HO-Inc), Neighbor-Set Description (N-Set), and Raw Incidence Matrix Description (Inc-Mat).

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High-Order Neighbor Description (HO-Neigh): This method leverages the neighbor definitions from HGNN^+, describing hypergraph relationships in a two-stage manner by first detailing the connections between vertices and hyperedges, and subsequently outlining the connections between hyperedges and their constituent vertices. For instance, it may state, “Vertex $v_{1}$ is connected to hyperedge $e_{1}$”, followed by “Hyperedge $e_{1}$ is connected to vertices $v_{1}$, $v_{2}$, and $v_{3}$”, thereby providing a comprehensive view of the local neighborhood structure.
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High-Order Incidence Description (HO-Inc): This method extends the LO-Inc by incorporating higher-order correlations between vertices. An example of HO-Inc would be, “Vertex $v_{1}$ is connected to vertices $v_{2}$ and $v_{3}$ with hyperedge $e_{1}$”, which succinctly captures the multi-vertex association facilitated by a single hyperedge.
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Neighbor-Set Description (N-Set): This method builds upon the N-Pair by enumerating entire sets of vertices connected by each hyperedge, thus providing a more holistic representation of the hypergraph’s connectivity. For example, it may list, “$(v_{1},v_{2},v_{3})$” and “$(v_{2},v_{4})$”, explicitly indicating the groups of vertices that jointly form hyperedges.
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Raw Incidence Matrix Description (Inc-Mat): This method employs an incidence matrix to numerically represent the hypergraph structure, where each entry indicates the presence or absence of a connection between vertices and hyperedges. An example of an incidence matrix is $[[1,0],[1,1],[1,0],[0,1]]$. In this matrix, rows correspond to vertices ($v_{1}$ to $v_{4}$) and columns correspond to hyperedges ($e_{1}$ and $e_{2}$), where a value of $1$ denotes the inclusion of a vertex in a hyperedge and 0 0 denotes absence.

Collectively, these high-order structure languages (HO-Neigh, HO-Inc, N-Set, and Inc-Mat) provide varied and sophisticated methods for representing hypergraph structures through multi-vertex associations. By utilizing these descriptive techniques, our framework facilitates a deeper and more nuanced understanding of hypergraph topologies, thereby supporting advanced analytical capabilities in subsequent high-order and isomorphic tasks within the LLM4Hypergraph Benchmark.

## Appendix E Details of Specifical Prompting for Hypergraphs

Given that hypergraphs do not naturally lend themselves to description using conventional natural language constructs, we introduce two specialized instruction-based prompting techniques designed to enhance LLMs’ ability to comprehend hypergraph structures: Hyper-BAG and Hyper-COT.

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Build-a-Hypergraph Prompting (Hyper-BAG) addresses the inherent complexity of hypergraphs by facilitating a mental visualization of their structures (Wang et al., ). Recognizing that certain aspects of hypergraph comprehension are more effectively understood through visualization, Hyper-BAG employs instruction-based enhancements that explicitly guide LLMs to imagine the hypergraph’s architecture. This approach provides a conceptual buffer, allowing the model to assimilate hypergraph information by mapping it into a structured conceptual space. For instance, an instruction might prompt the model to ”imagine a hypergraph where vertex $v_{1}$ is connected to hyperedge $e_{1}$,” thereby aiding the model in forming a coherent mental representation that better prepares it for subsequent queries related to the hypergraph.
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Chain-of-Thought for Hypergraphs (Hyper-COT) builds upon the traditional CoT prompting methodology (Wei et al., ) to address the observed difficulties that LLMs encounter in understanding hypergraph structures. Inspired by the step-by-step reasoning inherent in CoT, Hyper-COT integrates a tailored reasoning process specifically designed for hypergraphs. By appending “step-by-step thinking” instructions, Hyper-COT encourages the model to decompose complex hypergraph-related tasks into manageable inference steps, thereby enhancing its comprehension in zero-shot scenarios. Through our experiments, we identified four hypergraph-friendly prompts that effectively guide the model’s reasoning process.
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v1: “Let’s think step by step. Make sure the data is calculated and recorded accurately at each step.”
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v2: “Let’s analyze the connectivity by considering hyperedges linked to vertices and vertices linked through hyperedges.”
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v3: “Let’s think hyperedges connected by vertices then vertices connected by hyperedges.”
These prompts are strategically designed to align the model’s reasoning with the multi-way relationships characteristic of hypergraphs, thereby facilitating a deeper and more accurate understanding of hypergraph connectivity and structure.

By implementing Hyper-BAG and Hyper-COT within our prompting framework, we aim to significantly bolster the LLMs’ proficiency in interpreting and processing hypergraph structures. These instruction-based techniques address the unique challenges posed by hypergraphs’ multi-vertex associations, ensuring that models can effectively navigate both low-order and high-order structural information. Consequently, these strategies play a crucial role in advancing the capabilities of LLMs within the LLM4Hypergraph Benchmark, promoting more accurate and sophisticated analyses of hypergraph data.

## Appendix F Prompt Examples

### F.1 Examples of Hypergraph Languages

#### F.1.1 Low-Order Structure Languages

#### F.1.2 High-Order Structure Languages

### F.2 Examples of Question for Different Tasks

#### F.2.1 Low-Order Tasks

#### F.2.2 High-Order Tasks

#### F.2.3 Isomorphism Tasks

### F.3 Examples of Prompts under Different Settings

## Peer Reviews

confidence: 5
contribution: 3
initial_rating: 8
presentation: 3
questions: How do the prompting techniques generalize to other complex data structures beyond hypergraphs?
Could the authors elaborate on the potential scalability issues of the prompting techniques with increasingly large and complex hypergraphs?
soundness: 4
strengths: Originality: The paper proposes a new benchmark and prompting techniques tailored for hypergraphs, addressing a gap in the assessment of LLMs' capabilities.
Quality: The benchmark is comprehensive, covering a wide range of tasks and hypergraph types, which strengthens the validity of the findings.
Clarity: The paper is well-organized, with clear explanations of the hypergraph languages and prompting frameworks.
Significance: The work is significant as it pushes the boundaries of LLMs' understanding of complex data structures, which has implications for various real-world applications.
summary: The paper introduces LLM4Hypergraph, a benchmark designed to evaluate large language models' (LLMs) understanding of hypergraphs, which can capture complex, multi-way relationships beyond pairwise correlations found in traditional graphs. The benchmark includes 21,500 problems across low-order, high-order, and isomorphism tasks using both synthetic and real-world hypergraphs. The study evaluates six prominent LLMs and introduces novel prompting techniques to enhance LLMs' performance on hypergraph tasks.
weaknesses: The paper could benefit from a deeper analysis of the limitations of the current LLMs in handling hypergraphs, beyond performance metrics.
While the benchmark is comprehensive, it may lack diversity in terms of the types of real-world hypergraphs used, which could affect the generalizability of the findings.

---

confidence: 4
contribution: 1
initial_rating: 3
presentation: 2
questions: 1. Why is prompting a promising research direction for hypergraph understanding in light of other techniques such as function calling?
2. What are the empirical graphs used in the experiments? What are the selection criteria?
3.  Some tasks involve computing tasks whose answers are numbers. How does the accuracy is computed for these tasks? Is it an exact match? Or allow some error under a certain threshold?
4. Does the BAG and CoT outperform beyond statistical variations attributed to the variations of individual graph data and stochatic behaviors of LLMs?
soundness: 2
strengths: The paper is easy to read and the experiments are comprehensive and thorough.
summary: The paper provided a new benchmark to evaluate the LLM's ability to understand hypergraphs and developed a new prompting framework to improve the hypergraph comprehension. The prompting framework demonstrated that CoT and BAG, adapted to hypergraphs, can improve the LLM's performance on hypergraph tasks, especially for high-order tasks such as Vertex Set Connection Checks and Vertex-Set--in-Hypergraph Checks using synthetic hypergraphs.
weaknesses: **Main arguments**:
1. The paper adapts existing benchmarks and prompting techniques for hypergraphs. While the results offer some insights into the extent to which LLMs understand hypergraphs, they largely mirror findings for simple graphs---specifically, that CoT and BAG can enhance LLM performance. The only notable point is that using suitable language to describe hypergraphs can aid LLM comprehension, which is novel but trivial.
Given that the proposed techniques are naive adaptations of existing techniques and new insights specific to hypergraphs are not found, the contribution of the paper is incremental and not significant.
2. The paper lays out a main motivation by the underexploration of (i) investigating the LLM's ability to understand hypergraphs and (ii) developing prompting framework for hypergraph understanding and argue that they are promising research directions. This is not a strong motivation, i.e.,  "underexploration" alone does not justify the promising research directions. More specific question is: why is prompting a promissing research direction for hypergraph understanding in light of other techniques such as function calling?
3. Unsupported claim 1: In abstract, ``our specialized prompting framework incorporates seven hypergraph languages and introduces two novel techniques, Hyper-BAG and Hyper-COT, which enhance high-order reasoning and achieve an average 4% (up to 9%) performance improvement on structure classification tasks.'' This is not sufficiently supported by the empirical results. The performance improvement for Hyper-COT is 4\% on average. However, for the Hyper-BAG, it is 2\% for low-order hypergraphs and 2.8\% for high-order hypergraphs.

**Minor arguments**:
1. Give the stochastic nature of the LLMs and the graph data, it is crucial to report the variation of the results across different runs (e.g., confidence intervals, standard deviations), given the performance gain of the proposed prompting techniques (Hyper-BAG and Hyper-COT) is slim.
2. Unsupported claim 2: The paper claimed in the supporting information (B.4) that the benchmark represents the first instance that includes isomorphism checks. This is not precise. Isomorphism checks are a special case of Maximum Common Subgraph (MCS) problem, which is included in the existing benchmark cited in the paper (GraphArena Tang et al. (2024)). The author used "in this domain" to limit the scope of their claim, and it is crucial to spell out the "domain" (e.g., general graphs, or hypergraphs specifically) to be more precise.
3. The paper did not provide descriptions about real-world graphs used in the experiments and their selection criteria.

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: - How does the performance of LLMs depend on the hypergraph domains (e.g., emails, coauthorship)?
soundness: 3
strengths: - This paper proposes the first benchmark for evaluating LLMs on hypergraphs.
- The authors thoroughly address questions about hypergraphs.
- The problems are well-structured and clearly categorized according to their objectives.
- The code is released for reproducibility.
summary: This paper introduces LLM4Hypergraph, the first benchmark aimed at evaluating the ability of LLMs to understand hypergraph data. The authors design a series of tasks of varying difficulty levels and evaluate six different LLMs. Then, they identify their strengths and weaknesses. While this work represents a first step and provides a comprehensive study, there are several areas where improvement is needed.
weaknesses: - The motivations for this research are not sufficiently discussed. Why is it important to enable LLMs to understand hypergraph structures? Are there potential practical use cases? Are there any motivations beyond the fact that similar research has been done with graphs?
- The datasets used in the study are not comprehensive. To be specific:
  - The definition of "hypergraph size" is unclear. Is it referring to the number of nodes, the number of hyperedges, or the sum of hyperedge sizes?
  - The specific sizes of the hypergraphs (both real-world and synthetic) are not mentioned in the main content. How large are the synthetic hypergraphs used for evaluation?
  - According to the appendix, even the so-called "large-scale hypergraphs" only contain 15 to 20 vertices, which is too small to meaningfully capture higher-order structures typically expected in hypergraphs.
  - The synthetic hypergraphs are not sufficiently representative. There are other synthetic hypergraph models (e.g., configuration models) available.
  - It is unclear how the random walk approach for sampling sub-hypergraphs from real-world hypergraphs (Appendix A.2) ensures that the sampled hypergraphs "retain the intricate and authentic correlations inherent in the original data."
- The definition of task "difficulty" is unclear.
- The authors may consider discussing/citing the recent work "When LLM Meets Hypergraph: A Sociological Analysis on Personality via Online Social Networks" (CIKM 2024) in the related work.

**In summary**, this paper makes a valuable contribution to LLMs and hypergraph analysis. However, the benchmark datasets lack comprehensiveness and have room to consider additional synthetic hypergraph generators. Also, the paper lacks detailed statistics on real-world hypergraphs. Scalability is also a concern; if large-scale hypergraph handling poses challenges for LLMs, these limitations should be clearly discussed.

Meta-review: The authors present the first comprehensive benchmark for LLM evaluation on hypergraph reasoning. This benchmark covers eight tasks, evaluates six LLMs, and incorporates both synthetic and real-world datasets. Furthermore, the authors propose novel prompt engineering techniques for enhancing hypergraph reasoning capabilities.

The reviewers widely recognized the originality and breadth of the benchmark, and the clarity of the paper.

However, they identified several areas for improvement:
- Clarifying the motivation for hypergraph comprehension.
- Providing deeper analysis of experimental results.
- Expanding datasets to include larger-scale hypergraphs, including synthetic hypergraphs from advanced generation and sampling methods and real-world hypergraphs from various domains.
- Covering a closely related study.

While the reviews were mixed, the meta-reviewer believes this paper marks an important first step in a promising research direction. Given its potential impact, the merits of acceptance outweigh the drawbacks of rejection.

Decision: Accept (Poster)