动态团复合体中拓扑的演化
On the evolution of topology in dynamic clique complexes
摘要：
考虑Erd{ \H o}s-R{\' e}nyi图的一个时变类似图，并研究了它的团复形的拓扑变化.图的动态是平稳的，由边决定，边作为连续时间马尔可夫链独立地演化。我们的主要结果是，当边包含概率为$p = n^α$（$n$为顶点数，$α\in (-1/k, -1/(k + 1)),$为顶点数）时，这些动态团复形的规范化$k-$的贝蒂数的过程弱收敛于Ornstein-Uhlenbeck过程为$n \to \infty.$
英文摘要：
We consider a time varying analogue of the Erd{\H o}s-R{\' e}nyi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous time Markov chains. Our main result is that when the edge inclusion probability is of the form $p = n^α$, where $n$ is the number of vertices and $α\in (-1/k, -1/(k + 1)),$ then the process of the normalized $k-$th Betti number of these dynamic clique complexes converges weakly to the Ornstein-Uhlenbeck process as $n \to \infty.$
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