# From Concentration to Differentiation and Back: Routing Effective Rank in MoE Reasoning Cohorts

arXiv: https://arxiv.org/abs/2609.06403 · HTML: https://mlsanity.com/paper/2609.06403/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 43 of its 52 references that are in the map.
- First posted on arXiv 6 Sep 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- Mixture-of-experts / [MoE large language models](https://mlsanity.com/moe/moe-large-language-models/)

## Builds on (most cited first, 30 of 43)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [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)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [GShard: Scaling Giant Models with Conditional Computation and Automatic Sharding](https://mlsanity.com/paper/2006.16668/) (2020)
- [gpt-oss-120b & gpt-oss-20b Model Card](https://mlsanity.com/paper/2508.10925/) (2025)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [DeepSeekMoE: Towards Ultimate Expert Specialization in Mixture-of-Experts Language Models](https://mlsanity.com/paper/2401.06066/) (2024)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [ST-MoE: Designing Stable and Transferable Sparse Expert Models](https://mlsanity.com/paper/2202.08906/) (2022)
- [The Vendi Score: A Diversity Evaluation Metric for Machine Learning](https://mlsanity.com/paper/2210.02410/) (2022)
- [RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank](https://mlsanity.com/paper/2210.02885/) (2022)
- [Let's Sample Step by Step: Adaptive-Consistency for Efficient Reasoning and Coding with LLMs](https://mlsanity.com/paper/2305.11860/) (2023)
- [Escape Sky-high Cost: Early-stopping Self-Consistency for Multi-step Reasoning](https://mlsanity.com/paper/2401.10480/) (2024)
- [Fast Inference of Mixture-of-Experts Language Models with Offloading](https://mlsanity.com/paper/2312.17238/) (2023)
- [Landscape of Thoughts: Visualizing the Reasoning Process of Large Language Models](https://mlsanity.com/paper/2503.22165/) (2025)
- [LLM Reasoning as Trajectories: Step-Specific Representation Geometry and Correctness Signals](https://mlsanity.com/paper/2604.05655/) (2026)
- [Mapping the Minds of LLMs: A Graph-Based Analysis of Reasoning LLM](https://mlsanity.com/paper/2505.13890/) (2025)

## Abstract (arXiv)

Test-time scaling produces cohorts of reasoning rollouts, yet there is no standard label-free account of how their internal computation reorganizes as inference unfolds. We introduce routing effective rank deff, the entropy-effective dimensionality of a cross-rollout graph built from MoE expert-routing similarity. Across ten MoE configurations and five math/science benchmarks, deff exhibits a reproducible low-high-low trajectory, with a prominent interior maximum in 98.5% of 3,105 model-question cohorts: routing similarity is concentrated early, maximally differentiated at intermediate budgets, and reconcentrated later, and the timing of this maximum varies systematically with architecture and reasoning effort. An exact decomposition separates cohort-wide common-mode mass from residual spectral dimensionality: common-mode reallocation accounts for about two thirds of the trajectory, while the residual spectrum contributes about one quarter and retains substantial variation beyond the common mode. The decomposition further localizes behavior: among non-unanimous cohorts, increases in common-mode concentration strongly predict same-answer recoverability, and higher reasoning effort delays the maximum by 2.59 octaves (doublings of the token budget) and consistently expands the high-rank period across all four tested architectures, locating the effort effect in timing and duration rather than peak amplitude. Correctness comparisons separate structural monitoring from answer selection, positioning routing effective rank as a decomposable, label-free diagnostic of cohort organization - a principled spectral lens on how MoE reasoning cohorts differentiate and reconcentrate over inference time.