This paper is the second in a series of studies on developing efficient artificial intelligence-based approaches to pathfinding on extremely large graphs (e.g. $10^{70}$ nodes) with a focus on Cayley graphs and mathematical applications. The open-source CayleyPy project is a central component of our research. The present paper proposes a novel combination of a reinforcement learning approach with a more direct diffusion distance approach from the first paper. Our analysis includes benchmarking various choices for the key building blocks of the approach: architectures of the neural network, generators for the random walks and beam search pathfinding. We compared these methods against the classical computer algebra system GAP, demonstrating that they "overcome the GAP" for the considered examples. As a particular mathematical application we examine the Cayley graph of the symmetric group with cyclic shift and transposition generators. We provide strong support for the OEIS-A186783 conjecture that the diameter is equal to n(n-1)/2 by machine learning and mathematical methods. We identify the conjectured longest element and generate its decomposition of the desired length. We prove a diameter lower bound of n(n-1)/2-n/2 and an upper bound of n(n-1)/2+ 3n by presenting the algorithm with given complexity. We also present several conjectures motivated by numerical experiments, including observations on the central limit phenomenon (with growth approximated by a Gumbel distribution), the uniform distribution for the spectrum of the graph, and a numerical study of sorting networks. To stimulate crowdsourcing activity, we create challenges on the Kaggle platform and invite contributions to improve and benchmark approaches on Cayley graph pathfinding and other tasks.

翻译：本文是系列研究的第二篇，旨在开发基于人工智能的高效方法，以在超大规模图（例如包含 $10^{70}$ 个节点）上实现路径发现，重点关注凯莱图及其数学应用。开源项目CayleyPy是我们研究的核心组成部分。本文提出了一种新颖的组合方法，将强化学习与第一篇论文中更直接的扩散距离方法相结合。我们的分析包括对方法关键构成要素的各种选择进行基准测试：神经网络架构、随机游走生成器以及束搜索路径发现。我们将这些方法与经典计算机代数系统GAP进行了比较，结果表明，在所考虑的示例中，它们“超越了GAP”。作为一个特定的数学应用，我们研究了带有循环移位和换位生成元的对称群的凯莱图。我们通过机器学习和数学方法，为OEIS-A186783猜想（即直径等于n(n-1)/2）提供了强有力的支持。我们识别出了猜想的极长元素，并生成了其所需长度的分解。通过给出具有给定复杂度的算法，我们证明了直径的下界为n(n-1)/2-n/2，上界为n(n-1)/2+3n。此外，我们还提出了若干由数值实验引发的猜想，包括对中心极限现象（增长近似于耿贝尔分布）的观察、图谱的均匀分布，以及对排序网络的数值研究。为促进众包活动，我们在Kaggle平台上创建了挑战赛，并邀请各方贡献方案，以改进和基准测试凯莱图路径发现及其他任务的方法。