Simplets, constituting elementary units within simplicial complexes (SCs), serve as foundational elements for the structural analysis of SCs. Previous efforts have focused on the exact count or approximation of simplet count rather than their frequencies, with the latter being more practical in large-scale SCs. This paper enables simplet frequency analysis of SCs by introducing the Simplet Frequency Distribution (SFD) vector. In addition, we present a bound on the sample complexity required for accurately approximating the SFD vector by any uniform sampling-based algorithm. We also present a simple algorithm for this purpose and justify the theoretical bounds with experiments on some random simplicial complexes.

翻译：简单子结构作为单纯复形中的基本单元，是单纯复形结构分析的基础要素。已有研究主要聚焦于简单子结构的精确计数或近似计数，而非其频率分布——后者在大规模单纯复形中更具实用价值。本文通过引入简单子结构频率分布向量，实现了对单纯复形的简单子结构频率分析。此外，我们给出了任意基于均匀采样算法精确逼近简单子结构频率分布向量所需的样本复杂度上界，同时提出了一种实现该目标的简易算法，并通过在若干随机单纯复形上的实验验证了理论界值的有效性。