# Understanding Chain-of-Thought in Large Language Models via Topological Data Analysis

arXiv: https://arxiv.org/abs/2512.19135 · HTML: https://mlsanity.com/paper/2512.19135/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Prompting and chain-of-thought reasoning, one of 17 directions in LLM reasoning.
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Placed in

- LLM reasoning / [Prompting and chain-of-thought reasoning](https://mlsanity.com/lr/prompting-and-chain-of-thought-reasoning/)

## Builds on (most cited first, 18 of 18)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Plan-and-Solve Prompting: Improving Zero-Shot Chain-of-Thought Reasoning by Large Language Models](https://mlsanity.com/paper/2305.04091/) (2023)
- [L1: Controlling How Long A Reasoning Model Thinks With Reinforcement Learning](https://mlsanity.com/paper/2503.04697/) (2025)
- [Dynamic Prompt Learning via Policy Gradient for Semi-structured Mathematical Reasoning](https://mlsanity.com/paper/2209.14610/) (2022)
- [Demystifying Long Chain-of-Thought Reasoning in LLMs](https://mlsanity.com/paper/2502.03373/) (2025)
- [Navigate through Enigmatic Labyrinth A Survey of Chain of Thought Reasoning: Advances, Frontiers and Future](https://mlsanity.com/paper/2309.15402/) (2023)
- [Take a Step Back: Evoking Reasoning via Abstraction in Large Language Models](https://mlsanity.com/paper/2310.06117/) (2023)
- [A Survey of Efficient Reasoning for Large Reasoning Models: Language, Multimodality, and Beyond](https://mlsanity.com/paper/2503.21614/) (2025)
- [MME-CoT: Benchmarking Chain-of-Thought in Large Multimodal Models for Reasoning Quality, Robustness, and Efficiency](https://mlsanity.com/paper/2502.09621/) (2025)
- [Efficient Reasoning Models: A Survey](https://mlsanity.com/paper/2504.10903/) (2025)
- [Algorithm of Thoughts: Enhancing Exploration of Ideas in Large Language Models](https://mlsanity.com/paper/2308.10379/) (2023)
- [Towards Better Chain-of-Thought Prompting Strategies: A Survey](https://mlsanity.com/paper/2310.04959/) (2023)
- [Deconstructing Long Chain-of-Thought: A Structured Reasoning Optimization Framework for Long CoT Distillation](https://mlsanity.com/paper/2503.16385/) (2025)
- [Landscape of Thoughts: Visualizing the Reasoning Process of Large Language Models](https://mlsanity.com/paper/2503.22165/) (2025)
- [Benchmarking Multimodal CoT Reward Model Stepwise by Visual Program](https://mlsanity.com/paper/2504.06606/) (2025)
- [Beyond In-Distribution Success: Scaling Curves of CoT Granularity for Language Model Generalization](https://mlsanity.com/paper/2502.18273/) (2025)

## Built on by (most cited first, 1 of 1)

- [On the Role of Reasoning Patterns in the Generalization Discrepancy of Long Chain-of-Thought Supervised Fine-Tuning](https://mlsanity.com/paper/2604.01702/) (2026)

## Abstract (arXiv)

With the development of large language models (LLMs), particularly with the introduction of the long reasoning chain technique, the reasoning ability of LLMs in complex problem-solving has been significantly enhanced. While acknowledging the power of long reasoning chains, we cannot help but wonder: Why do different reasoning chains perform differently in reasoning? What components of the reasoning chains play a key role? Existing studies mainly focus on evaluating reasoning chains from a functional perspective, with little attention paid to their structural mechanisms. To address this gap, this work is the first to analyze and evaluate the quality of the reasoning chain from a structural perspective. We apply persistent homology from Topological Data Analysis (TDA) to map reasoning steps into semantic space, extract topological features, and analyze structural changes. These changes reveal semantic coherence, logical redundancy, and identify logical breaks and gaps. By calculating homology groups, we assess connectivity and redundancy at various scales, using barcode and persistence diagrams to quantify stability and consistency. Our results show that the topological structural complexity of reasoning chains correlates positively with accuracy. More complex chains identify correct answers sooner, while successful reasoning exhibits simpler topologies, reducing redundancy and cycles, enhancing efficiency and interpretability. This work provides a new perspective on reasoning chain quality assessment and offers guidance for future optimization.