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

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. Implementation available here: https://github.com/gouki510/Topology_of_Reasoning

Joy can read 8 pages of a book in 20 minutes. How many hours will it take her to read 120 pages?

In 1 hour, there are 3 sets of 20 minutes. 8 pages÷20 min=0.4 pages/minute. 120 pages÷0.4 pages/minute=360 minutes. 360 minutes÷60 minutes/hour=6 hours. The answer is 6 hours. In 1 hour, there are 3 sets of 20 minutes. 8 pages÷20 min=0.4 pages/minute. 120 pages÷0.4 pages/minute=360 minutes. 360 minutes÷60 minutes/hour=6 hours. Wait, let me verify the calculations. … 8 pages÷20 min=0.4 pages/minute. 120 pages÷0.4 pages/minute=300 minutes. 300 minutes÷60 minutes/hour=5 hours. The answer is 5 hours.