# From Stochastic to Stable: Rank Stability and Structural Sufficiency in AI Visibility Measurement

arXiv: https://arxiv.org/abs/2607.10341 · HTML: https://mlsanity.com/paper/2607.10341/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 19 references that are in the map.
- First posted on arXiv 11 Jul 2026.

## Builds on (most cited first, 6 of 6)

- [Enabling Large Language Models to Generate Text with Citations](https://mlsanity.com/paper/2305.14627/) (2023)
- [Auditing large language models: a three-layered approach](https://mlsanity.com/paper/2302.08500/) (2023)
- [A Weighted Correlation Index for Rankings with Ties](https://mlsanity.com/paper/1404.3325/) (2014)
- [Generative AI Search Engines as Arbiters of Public Knowledge: An Audit of Bias and Authority](https://mlsanity.com/paper/2405.14034/) (2024)
- [Source Coverage and Citation Bias in LLM-based vs. Traditional Search Engines](https://mlsanity.com/paper/2512.09483/) (2025)
- [Quantifying Uncertainty in AI Visibility: A Statistical Framework for Generative Search Measurement](https://mlsanity.com/paper/2603.08924/) (2026)

## Abstract (arXiv)

AI visibility measurement is comparative: practitioners want to know which domains generative search engines cite most often and whether observed differences are large enough to support decisions. Yet the industry lacks a principled way to determine whether enough data has been collected. Collection budgets vary widely across studies and platforms, and conclusions are often drawn from rankings whose stability and precision are unknown. We introduce a sequential convergence framework based on two complementary criteria: rank stability evaluates whether the rank-correlation trajectory has reached a structural plateau, while structural sufficiency evaluates whether the spread of citation shares among established domains -- those whose confidence intervals exclude zero -- exceeds the uncertainty of those estimates. Together, these criteria distinguish rankings that have merely stabilized from those sufficiently resolved to support inference. Both are derived from regularities in the observed citation distribution, including its rank structure, uncertainty profile, and the boundary between observed and established domains. The framework retains a small number of structural constants but requires no externally specified query count, correlation target, or confidence-interval width target; stopping is driven by observed measurement uncertainty and remains robust across a range of sufficiency thresholds. Applied across 30 platform-topic combinations spanning Gemini, SearchGPT, and Perplexity, the framework adapts to platform- and topic-specific citation distributions. Results show that no fixed collection budget can be justified across contexts and that convergence can instead be evaluated from the structure of the observed distribution. The framework provides a practical basis for determining when AI visibility measurements are ready to support comparative analysis.