Topology of random geometric complexes: a survey <s> Stationary point processes <s> Cech and Vietoris-Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either iid observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction, or repulsion, we find phenomena quantitatively different from those observed in the iid and Poisson cases. From the point of view of topological data analysis, our results seriously impact on considerations of model (non) robustness for statistical inference. <s> BIB001 </s> Topology of random geometric complexes: a survey <s> Stationary point processes <s> We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality. <s> BIB002
The results we presented so far in this survey describe the behavior of geometric complexes constructed from either the binomial process X n or the Poisson process P n . Both models exhibit a strong level of independency which plays a significant role in the proofs. For the binomial process X n the number of points is fixed, while the locations of the points are independent. For the Poisson process P n the amount of points in different regions are independent, and given the number of points in a region their locations are independent. Recent work by Yogeshwaran and Adler BIB001 extends some of the results presented in this survey to a more general class of spatial point processes allowing certain attractive and repulsive point processes, as well as stationary determinantal processes. In this section we wish to briefly review their results. A general point process in R d can be thought of as a random measure Φ(·) = i δ Xi (·) where δ x is the Dirac delta measure concentrated at x. In that case, for every subset A ⊂ R d , Φ(A) is a random variables counting the number of points lying inside A. The distribution of a random point process Φ can be characterized by its factorial moment measure functions α (k) defined as follows -  which depends only on the volumes of the sets and therefore invariant to translations. Note that if Φ is a stationary point process, and C(Φ, r) is the correspondingČech complex, then depending on r either E {β k (C(Φ, r))} = 0 or E {β k (C(Φ, r))} = ∞ (since the process is supported in an infinite domain). Therefore, it does not make sense to try to analyze β k (C(Φ, r) ). Instead, we can define and try to study r) ). Note that if Φ is a homogeneous Poisson process with rate µ = 1, and P n is the Poisson process we used previously supported on the unit cube, then C(Φ n , r) is a scaled version of C(P n , n −1/d r), and so β Φ k (n, r) = β k (n, r). Therefore, we can view the results in BIB001 as an extension of the models described earlier in this survey. Similarly to the study of the binomial and the Poisson processes we described before, the limiting behavior of β Φ k (n, r) splits into three main regime. Due to the different scaling, the term that controlling the limiting behavior is r rather than nr d . The sparse (or the subcritical) regime is when r → 0. In this case they show that there exists a sequence of functions f k such that either f k ≡ 1 or lim r→0 f k (r) = 0 (depending on the distribution of Φ), and then and the exact limiting constant is given by a formula similar in spirit to C i in Theorem 4.1. They also provide equivalent limits for the distribution as in Theorems 4.4-4.5. The critical (thermodynamic) regime is when r → λ ∈ (0, ∞). In this case, they show that E β Φ k (n, r) = Θ(n), and provide a limit for the Euler characteristic similarly to Corollary 5.2. Finally, in the super critical regime (r → ∞) they discuss the connectivity regime which is when r d = Θ(log n). Similarly to Theorem 4.12 they show that there exists In addition to the Betti numbers of theČech complex, they also provide equivalent results to the Vietoris-Rips complexes R(Φ n , r) and for the critical point counts N i for the distance function d Φn . In BIB002 these theorems are extended in some cases, to laws of large numbers and central limit theorems.