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

2025-12-22

AI Summary

Prior work on chain-of-thought (CoT) reasoning primarily focuses on functional evaluation, neglecting the underlying structural mechanisms governing reasoning quality. Method: This study introduces topological data analysis (TDA) to CoT modeling—constructing a metric space from semantic embeddings of reasoning steps and applying persistent homology to quantify semantic coherence, logical redundancy, and structural breakpoints. Contributions/Results: (1) Topological complexity correlates positively with inference speed, yet high-accuracy reasoning exhibits simpler, less redundant, and more topologically stable structures; (2) For the first time, CoT topological features are shown to quantitatively predict reasoning accuracy; (3) Barcode diagrams and persistence diagrams are proposed as interpretable, structure-aware evaluation tools. Collectively, this work establishes a novel paradigm for probing the internal reasoning mechanisms of large language models through geometric and topological lenses.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Reasoning under Uncertainty: Causality

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

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.

Problem

• Analyzes reasoning chain quality from a structural perspective
• Identifies semantic coherence and logical breaks in reasoning steps
• Correlates topological complexity with reasoning accuracy and efficiency

Innovation

• Using persistent homology to map reasoning steps
• Extracting topological features to analyze structural changes
• Correlating topological complexity with reasoning accuracy