Counting Cycles with Deepseek

TL;DR

The paper tackles the problem of deriving a Computationally Efficient Equivalent Form (CEEF) for the cycle count statistics $C_m$. It introduces a humAI approach that combines human guidance with the reasoning capabilities of an AI tool (DeepSeek-R1) to decompose $C_m$ into a finite linear combination of Full-Sum terms via a merging process and then convert those FS terms to SEA or IFS forms through a pruning algorithm on labeled multi-graphs, aided by Möbius inversion. The authors prove two main theorems: a merging-based expansion of $C_m$ with coefficients given by a Möbius-derived expression, and a pruning procedure that terminates in SEA or IFS forms, enabling compact, computable formulas for general $m$. They implement the pipeline with AI, validate the results against known small-$m$ formulas and brute-force tests, and demonstrate practical benefits in high-order cycle statistics for low-rank matrix detection. The work shows that AI can serve as a powerful research assistant when paired with structured strategies, yielding scalable tools for network analysis and matrix testing.

Abstract

Despite recent progress, AI still struggles on advanced mathematics. We consider a difficult open problem: How to derive a Computationally Efficient Equivalent Form (CEEF) for the cycle count statistic? The CEEF problem does not have known general solutions, and requires delicate combinatorics and tedious calculations. Such a task is hard to accomplish by humans but is an ideal example where AI can be very helpful. We solve the problem by combining a novel approach we propose and the powerful coding skills of AI. Our results use delicate graph theory and contain new formulas for general cases that have not been discovered before. We find that, while AI is unable to solve the problem all by itself, it is able to solve it if we provide it with a clear strategy, a step-by-step guidance and carefully written prompts. For simplicity, we focus our study on DeepSeek-R1 but we also investigate other AI approaches.

Figures (4)

• Figure 1: An LMG and its associated full sum (blue vectors: node labels, red matrices: edge labels).
• Figure 2: The pruning process and corresponding updating rule. Left: Type I. Right: Type II.
• Figure 3: An illustration of the pruning process.
• Figure 4: The pipeline of our humAI approach for counting cycles.

Theorems & Definitions (10)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Theorem 2.4
• Definition 2.5: Labeled Multi-Graph (LMG)
• Definition 2.6: Full sum for an LMG
• Definition 2.7
• Lemma 2.8
• Definition 2.9: Default LMG
• Theorem 2.10