History estimation in random recursive trees: Pointwise approach via iterated Jordan centralities

2026-06-24 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

Johannes Bäumler, Simon Briend, and Joost Jorritsma, in their arXiv submission from June 23, 2026, investigate the challenge of estimating vertex arrival times in uniform random recursive trees based solely on their unlabeled structure. Their research adopts a pointwise approach, focusing on the distribution of relative estimation error and deriving tail bounds that are consistent across both vertex and tree sizes. For estimates derived from Jordan centrality, the probability of overestimating arrival time by a factor S decreases at a rate of 1/S, while underestimation by a factor 1/S decays exponentially in S. The authors introduce a refined centrality measure that improves the overestimation tail to (log S)/S^2, though this comes with a heavier lower tail of order 1/S^2. This work highlights a previously unobserved tradeoff between upper- and lower-tail performance in arrival-time estimation, yet the refined measure still achieves the optimal risk order across its parameter range.

Key takeaway

For AI Scientists or statisticians developing algorithms for graph history reconstruction, you should recognize the inherent tradeoff between overestimation and underestimation error tails when estimating vertex arrival times in random recursive trees. While Jordan centrality offers a baseline, exploring refined centrality measures, even those with heavier lower tails like the one presented, can achieve optimal overall risk. Your choice of centrality measure should align with the specific costs associated with over- or underestimation in your application.

Key insights

Estimating vertex arrival times in random recursive trees involves a tradeoff between overestimation and underestimation tail performance.

Principles

• Jordan centrality yields 1/S overestimation decay and exponential underestimation decay.
• Refined centrality can improve overestimation to (log S)/S^2 at cost of 1/S^2 underestimation.
• Optimal risk order can be maintained despite tail performance tradeoffs.

Method

The study employs a pointwise approach to analyze the distribution of relative estimation error, deriving uniform tail bounds for vertex arrival times in random recursive trees using centrality measures.

Topics

• Random Recursive Trees
• Jordan Centrality
• Arrival Time Estimation
• Tail Bounds
• Graph Theory
• Probability Theory

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.