# Counting Cycles with AI: Counting Cycles with AI: Computationally Efficient Equivalent Forms with Applications

arXiv: https://arxiv.org/abs/2505.17964 · HTML: https://mlsanity.com/paper/2505.17964/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 53 references that are in the map.
- First posted on arXiv 23 May 2025.

## Builds on (most cited first, 21 of 21)

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- [HyperTree Proof Search for Neural Theorem Proving](https://mlsanity.com/paper/2205.11491/) (2022)
- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [A Neural Network Solves, Explains, and Generates University Math Problems by Program Synthesis and Few-Shot Learning at Human Level](https://mlsanity.com/paper/2112.15594/) (2021)
- [Agentic AI for Scientific Discovery: A Survey of Progress, Challenges, and Future Directions](https://mlsanity.com/paper/2503.08979/) (2025)
- [Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models](https://mlsanity.com/paper/1807.00891/) (2018)
- [Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers](https://mlsanity.com/paper/2410.08304/) (2024)
- [Testing for high-dimensional geometry in random graphs](https://mlsanity.com/paper/1411.5713/) (2014)
- [Testing network correlation efficiently via counting trees](https://mlsanity.com/paper/2110.11816/) (2021)
- [Optimal hypothesis testing for stochastic block models with growing degrees](https://mlsanity.com/paper/1705.05305/) (2017)
- [Two-sample Hypothesis Testing for Inhomogeneous Random Graphs](https://mlsanity.com/paper/1707.00833/) (2017)
- [Token-Hungry, Yet Precise: DeepSeek R1 Highlights the Need for Multi-Step Reasoning Over Speed in MATH](https://mlsanity.com/paper/2501.18576/) (2025)
- [Tackling small eigen-gaps: Fine-grained eigenvector estimation and inference under heteroscedastic noise](https://mlsanity.com/paper/2001.04620/) (2020)
- [Optimal Adaptivity of Signed-Polygon Statistics for Network Testing](https://mlsanity.com/paper/1904.09532/) (2019)
- [Efficient Algorithms for Attributed Graph Alignment with Vanishing Edge Correlation](https://mlsanity.com/paper/2308.09210/) (2023)
- [Optimal Network Pairwise Comparison](https://mlsanity.com/paper/2408.06987/) (2024)

## Built on by (most cited first, 1 of 1)

- [Solving a Research Problem in Mathematical Statistics with AI Assistance](https://mlsanity.com/paper/2511.18828/) (2025)

## Abstract (arXiv)

Cycle count statistics are fundamental tools in statistics and engineering, with applications in motif counting, channel coding, and statistical inference of network and matrix data. However, how to compute high-order cycle count statistics efficiently is still an open problem. In this paper, we aim to derive Computationally Efficient Equivalent Forms (CEEF) for cycle count statistics of any given order, where we express each cycle count statistic equivalently as a linear combination of finitely many terms. Using the CEEF, we provide a much more efficient way to compute the cycle count statistics. The CEEF problem has no known general solution and requires delicate combinatorial arguments together with extensive calculations. While this task is hard to accomplish by humans alone, it provides an ideal setting in which Artificial Intelligence (AI) can be useful. We solve the problem by combining several theorems we derive with powerful coding skills of modern AI systems. Our results leverage graph-theoretic arguments and yield new formulas for general cases that were previously unknown. We find that, although AI cannot solve the problem independently, it becomes highly effective when guided by humans through theorems we derive as well as a clear derivation strategy, step-by-step instructions, and carefully-written prompts. We consider several statistical applications, including spiked matrix testing, estimation of weak spike eigenvalues, and pairwise network comparison. For each problem, we demonstrate that optimal statistical performance is achieved by using high-order cycle count statistics, and our CEEF formulas make their computation feasible on large-scale data sets.