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

TL;DR

This work addresses the problem of evaluating reasoning chain quality in large language models from a structural perspective by framing reasoning steps as a semantic point cloud and applying Topological Data Analysis (TDA). It combines semantic embeddings with structure-aware positional encodings for CoT, ToT, and GoT, builds Vietoris–Rips complexes across multiple scales, and analyzes persistent homology via $H_0$ (connectivity) and $H_1$ (loops) to quantify coherence and redundancy. The authors demonstrate that networked reasoning (GoT) yields richer topological structures and higher accuracy, while successful final paths tend to converge to simpler topologies, offering a balance between exploration and consolidation. Overall, the paper provides new, topology-based metrics for comparing reasoning chains and offers guidance for optimizing reasoning strategies in LLMs with practical implications for interpretability and efficiency.

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.

Figures (9)

• Figure 1: Overall framework. It presents the complete topological analysis pipeline from reasoning text to a semantic point cloud, then to a Vietoris--Rips complex, and finally to persistent homology (barcode/diagram).
• Figure 2: Persistent barcode & persistence diagram: CoT/ToT/GoT. Birth–death patterns of connected components ($H_0$) and loops ($H_1$) across scales for CoT, ToT, and GoT. In each column, the top panel shows two-layer barcodes for $H_0$ and $H_1$ (horizontal bars), and the bottom panel shows a persistence diagram with a diagonal reference line, where points are colored by homology dimension and include $+\infty$ markers.
• Figure 3: Full-graph correlation heatmaps. Three correlation heatmaps (for CoT/ToT/GoT) show the correlation structure between accuracy and metrics such as Token, Time, $H_0$, and $H_1$ under the full-graph view.
• Figure 4: Topological 2D visualization. It visualizes a reasoning process via a 2D embedding and annotates the numbers of $H_0$ components and $H_1$ cycles.
• Figure 5: TDA persistence barcode.