Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: (1) the vector of critical simplex counts attained by a lexicographical Morse matching, (2) the vector of simplex counts in the link of a fixed simplex, and (3) the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein's method.
We study the persistent homology of an Erdős--Rényi random clique complex filtration on $n$ vertices. Here, each edge $e$ appears at a time $p_e \in [0,1]$ chosen uniform randomly in the interval, and the \emph{persistence} of a cycle $σ$ is defined as $p_2 / p_1$, where $p_1$ and $p_2$ are the birth and death times of the cycle respectively. We show that for fixed $k \ge 1$, with high probability the maximal persistence of a $k$-cycle is of order roughly $n^{1/k(k+1)}$. These results are in sharp contrast with the random geometric setting where earlier work by Bobrowski, Kahle, and Skraba sho…
Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most $δ$. Summing powers of the volume of all $k$-dimensional faces defines the volume-power functionals of these random simplicial complexes. The asymptotic behavior of the volume-power functionals of the Vietoris-Rips complex is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. Univariate and multivariate central limit theorems are…
We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group $\mathbb{Z}/2\mathbb{Z}$. We establish results about the typical asymptotic topology of these complexes. As a consequence we give bounds for the dimension $d$ such that $\mathbb{Z}/2\mathbb{Z}$-equivariant maps from the double cover to $\mathbb{R}^d$ have zeros with high probability, thus establishing a random Borsuk--Ulam theorem. We apply this to derive a structural re…
We consider the random clique complex process - the process of clique complexes induced by the complete graph with i.i.d. Uniform edge weights. We investigate the evolution of the Betti numbers of the clique complex process in the critical window and in particular, show a process-level convergence of the Betti numbers to a Poisson process. Our proof technique gives easily an hitting time result i.e, with high probability, the $k$th cohomology becomes trivial when there are no more isolated $k$-faces. Our results imply that the thresholds for vanishing of cohomology of the clique complex proces…
In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds generalize the result of Aharoni et al. (2005) \cite{ABM} which is valid only for clique complexes. As an application, we investigate the thresholds for vanishing of cohomology of the neighborhood complex of the Erdös-Rényi random graph. We improve the upper bound derived in Kahle (2007) \cite{kahle} by a lo…
The Linial-Meshulam complex model is a natural higher-dimensional analog of the Erdős-Rényi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate scaling of the underlying parameter. The present paper aims to extend that result to more-general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial-Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their…
We study the homological properties of random simplicial complexes. In particular, we obtain the asymptotic behavior of lifetime sums for a class of increasing random simplicial complexes; this result is a higher-dimensional counterpart of Frieze's $ζ(3)$-limit theorem for the Erdős-Rényi graph process. The main results include solutions to questions posed in an earlier study by Hiraoka and Shirai about the Linial-Meshulam complex process and the random clique complex process. One of the key elements of the arguments is a new upper bound on the Betti numbers of general simplicial complexes in …
Central limit theorems for the Euler characteristic in the Random Connection Model for higher-dimensional simplicial complexes
Institute of Stochastics（随机研究所）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Institute of Theoretical Physics（理论物理研究所）；Friedrich Alexander University Erlangen-Nuremberg（弗里德里希·亚历山大埃尔朗根-纽伦堡大学）
Semicircle law for multi-parameter random simplicial complexes
Indian Institute of Science Education and Research Bhopal（印度科学教育研究所博帕尔分校）；New York University Abu Dhabi（纽约大学阿布扎比分校）；Indian Institute of Technology Bombay（孟买印度理工学院）
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 the randomness there is usually governed by a single parameter. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex and, more importantly, of its dynamic analogue that we introduce here. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. The dynamic versions of the …
