The topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since a single parameter usually governs the randomness in these models. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex. In particular, we introduce a dynamic variant of this model and look at how its topology evolves. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. Special cases of this setup include the dynamic versions of the clique complex and the Linial–Meshulam complex. Our key result concerns the regime where the face-count of a particular dimension dominates. We show that the Betti number corresponding to this dimension and the Euler characteristic satisfy a functional strong law of large numbers and a functional central limit theorem. Surprisingly, in the latter result, the limiting process depends only upon the dynamics in the smallest non-trivial dimension.

现有随机简单复形的拓扑研究是不平凡的，并导致了几项开创性的工作。但是，由于单个参数通常控制这些模型中的随机性，因此此类研究的适用性受到限制。考虑到这一点，我们在这里集中于最近提出的多参数随机简单复形的拓扑。特别是，我们介绍了该模型的动态变体，并研究了其拓扑结构是如何演变的。在这种动态设置中，单纯形的时间演化由具有更新结构的平稳过程以及可能的非马尔可夫过程确定。这种设置的特殊情况包括集团动态和Linial-Meshulam复合的动态版本。我们的主要结果与特定维度的面额占主导地位的制度有关。我们表明，与此维对应的贝蒂数和欧拉特征满足大数的函数强定律和函数的中心极限定理。出乎意料的是，在后一结果中，限制过程仅取决于最小非平凡尺寸上的动力学。