# Topology of Reasoning: Understanding Large Reasoning Models through Reasoning Graph Properties

arXiv: https://arxiv.org/abs/2506.05744 · HTML: https://mlsanity.com/paper/2506.05744/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Placed in Knowledge- and tool-augmented reasoning, one of 17 directions in LLM reasoning.
- Builds on 32 of its 75 references that are in the map.
- First posted on arXiv 6 Jun 2025.

## Placed in

- LLM reasoning / [Knowledge- and tool-augmented reasoning](https://mlsanity.com/lr/knowledge-and-tool-augmented-reasoning/)

## Builds on (most cited first, 30 of 32)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [Beyond the Imitation Game: Quantifying and extrapolating the capabilities of language models](https://mlsanity.com/paper/2206.04615/) (2022)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [Stop Overthinking: A Survey on Efficient Reasoning for Large Language Models](https://mlsanity.com/paper/2503.16419/) (2025)
- [Training Language Models to Self-Correct via Reinforcement Learning](https://mlsanity.com/paper/2409.12917/) (2024)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [Demystifying Long Chain-of-Thought Reasoning in LLMs](https://mlsanity.com/paper/2502.03373/) (2025)
- [VinePPO: Refining Credit Assignment in RL Training of LLMs](https://mlsanity.com/paper/2410.01679/) (2024)
- [Thoughts Are All Over the Place: On the Underthinking of o1-Like LLMs](https://mlsanity.com/paper/2501.18585/) (2025)
- [The Danger of Overthinking: Examining the Reasoning-Action Dilemma in Agentic Tasks](https://mlsanity.com/paper/2502.08235/) (2025)
- [LLMs Can Easily Learn to Reason from Demonstrations Structure, not content, is what matters!](https://mlsanity.com/paper/2502.07374/) (2025)
- [Small Models Struggle to Learn from Strong Reasoners](https://mlsanity.com/paper/2502.12143/) (2025)
- [Common 7B Language Models Already Possess Strong Math Capabilities](https://mlsanity.com/paper/2403.04706/) (2024)
- [Towards System 2 Reasoning in LLMs: Learning How to Think With Meta Chain-of-Thought](https://mlsanity.com/paper/2501.04682/) (2025)
- [OverThink: Slowdown Attacks on Reasoning LLMs](https://mlsanity.com/paper/2502.02542/) (2025)
- [Strategic Reasoning with Language Models](https://mlsanity.com/paper/2305.19165/) (2023)
- [Missing Premise exacerbates Overthinking: Are Reasoning Models losing Critical Thinking Skill?](https://mlsanity.com/paper/2504.06514/) (2025)
- [Physics of Language Models: Part 2.2, How to Learn From Mistakes on Grade-School Math Problems](https://mlsanity.com/paper/2408.16293/) (2024)
- [Understanding Aha Moments: from External Observations to Internal Mechanisms](https://mlsanity.com/paper/2504.02956/) (2025)
- [Self-Explore: Enhancing Mathematical Reasoning in Language Models with Fine-grained Rewards](https://mlsanity.com/paper/2404.10346/) (2024)

## Built on by (most cited first, 29 of 29)

- [RL Squeezes, SFT Expands: A Comparative Study of Reasoning LLMs](https://mlsanity.com/paper/2509.21128/) (2025)
- [What Characterizes Effective Reasoning? Revisiting Length, Review, and Structure of CoT](https://mlsanity.com/paper/2509.19284/) (2025)
- [Attention Illuminates LLM Reasoning: The Preplan-and-Anchor Rhythm Enables Fine-Grained Policy Optimization](https://mlsanity.com/paper/2510.13554/) (2025)
- [The Geometry of Reasoning: Flowing Logics in Representation Space](https://mlsanity.com/paper/2510.09782/) (2025)
- [Reasoning or Retrieval? A Study of Answer Attribution on Large Reasoning Models](https://mlsanity.com/paper/2509.24156/) (2025)
- [The Shape of Reasoning: Topological Analysis of Reasoning Traces in Large Language Models](https://mlsanity.com/paper/2510.20665/) (2025)
- [DAGGER: Distractor-Aware Graph Generation for Executable Reasoning in Math Problems](https://mlsanity.com/paper/2601.06853/) (2026)
- [Emergent Slow Thinking in LLMs as Inverse Tree Freezing](https://mlsanity.com/paper/2509.23629/) (2025)
- [Towards a Mechanistic Understanding of Large Reasoning Models: A Survey of Training, Inference, and Failures](https://mlsanity.com/paper/2601.19928/) (2026)
- [ReJump: A Tree-Jump Representation for Analyzing and Improving LLM Reasoning](https://mlsanity.com/paper/2512.00831/) (2025)
- [Characterizing, Evaluating, and Optimizing Complex Reasoning](https://mlsanity.com/paper/2602.08498/) (2026)
- [Modeling Hierarchical Thinking in Large Reasoning Models](https://mlsanity.com/paper/2510.22437/) (2025)
- [Student Guides Teacher: Weak-to-Strong Inference via Spectral Orthogonal Exploration](https://mlsanity.com/paper/2601.06160/) (2026)
- [How Does Reasoning Flow? Tracing Attention-Induced Information Flow for Targeted RL in LLMs](https://mlsanity.com/paper/2606.10646/) (2026)
- [Transformers in the Dark: Navigating Unknown Search Spaces via Bandit Feedback](https://mlsanity.com/paper/2603.24780/) (2026)
- [SARL: Label-Free Reinforcement Learning by Rewarding Reasoning Topology](https://mlsanity.com/paper/2603.27977/) (2026)
- [Soundness-Aware Level: A Microscopic Signature that Predicts LLM Reasoning Potential](https://mlsanity.com/paper/2510.15216/) (2025)
- [Dynamic Thinking-Token Selection for Efficient Reasoning in Large Reasoning Models](https://mlsanity.com/paper/2601.18383/) (2026)
- [Boule or Baguette? A Study on Task Topology, Length Generalization, and the Benefit of Reasoning Traces](https://mlsanity.com/paper/2602.14404/) (2026)
- [Reasoning Models Know What's Important, and Encode It in Their Activations](https://mlsanity.com/paper/2604.18307/) (2026)
- [ThoughtFold: Folding Reasoning Chains via Introspective Preference Learning](https://mlsanity.com/paper/2606.03503/) (2026)
- [Diverse Thinking Schemata Elicit Better Reasoning in Large Language Models](https://mlsanity.com/paper/2606.08974/) (2026)
- [GSM-Agent: Understanding Agentic Reasoning Using Controllable Environments](https://mlsanity.com/paper/2509.21998/) (2025)
- [Step-Tagging: Toward controlling the generation of Language Reasoning Models through step monitoring](https://mlsanity.com/paper/2512.14332/) (2025)
- [Topology of Reasoning: Retrieved Cell Complex-Augmented Generation for Textual Graph Question Answering](https://mlsanity.com/paper/2602.19240/) (2026)
- [Quantifying and Understanding Uncertainty in Large Reasoning Models](https://mlsanity.com/paper/2604.13395/) (2026)
- [TRACES: Tagging Reasoning Steps for Adaptive Cost-Efficient Early-Stopping](https://mlsanity.com/paper/2604.21057/) (2026)
- [Selecting Diverse SFT Traces Improves Post-RL Generalization](https://mlsanity.com/paper/2609.33780/) (2026)
- [Topology-Consistent Task Planning over Cellular Workflow Complexes for LLM-based Agents](https://mlsanity.com/paper/2610.07004/) (2026)

## Abstract (arXiv)

Recent large-scale reasoning models have achieved state-of-the-art performance on challenging mathematical benchmarks, yet the internal mechanisms underlying their success remain poorly understood. In this work, we introduce the notion of a reasoning graph, extracted by clustering hidden-state representations at each reasoning step, and systematically analyze three key graph-theoretic properties: cyclicity, diameter, and small-world index, across multiple tasks (GSM8K, MATH500, AIME 2024). Our findings reveal that distilled reasoning models (e.g., DeepSeek-R1-Distill-Qwen-32B) exhibit significantly more recurrent cycles (about 5 per sample), substantially larger graph diameters, and pronounced small-world characteristics (about 6x) compared to their base counterparts. Notably, these structural advantages grow with task difficulty and model capacity, with cycle detection peaking at the 14B scale and exploration diameter maximized in the 32B variant, correlating positively with accuracy. Furthermore, we show that supervised fine-tuning on an improved dataset systematically expands reasoning graph diameters in tandem with performance gains, offering concrete guidelines for dataset design aimed at boosting reasoning capabilities. By bridging theoretical insights into reasoning graph structures with practical recommendations for data construction, our work advances both the interpretability and the efficacy of large reasoning models.