Crackle: the homology of noise.(English)Zbl 1350.60010

Let $${\mathcal X}_n$$ be a set of $$n$$ points in $${\mathbb R}^d$$ sampled from either a power-law, exponential, or Gaussian probability distribution, i.e., a distribution of the form $$(1+\|x\|^\alpha)^{-1}$$, $$e^{-\|x\|}$$, or $$e^{-\|x\|^2/2}$$. This paper examines the homology of the union $$U$$ of closed balls $$B_\varepsilon(x)$$ of radius $$\varepsilon$$ centered at the points $$x$$ from $${\mathcal X}_n$$. This is done by computing the expectation values, for large $$n$$, of the Betti numbers of the Čech complex formed from these closed balls – a space that is homotopy equivalent to $$U$$. Specifically, let $$\{R_n\}_{n=1}^\infty$$ be an increasing sequence of positive numbers. For each of the three distributions, the authors find the core radius: the largest value of $$R_n$$ for which the probability that $$B_{R_n}(0)$$ is covered by closed balls of the form $$B_1(x)$$, with $$x\in{\mathcal X}_n\cap B_{R_n}(0)$$, approaches unity for large $$n$$. For the Gaussian distribution, the homology outside of the core is shown to be trivial. That is, let $$\beta_{k,n}$$ denote the $$k$$-th Betti number of the Čech complex constructed from the points of $${\mathcal X}_n$$ that lie outside of $$B_{R_n}(0)$$. The authors find a sequence of radial values $$\{R_n\}_{n=1}^\infty$$ such that the expectation values $${\mathbb E}(\beta_{k,n})$$ approach $$0$$ for large $$n$$. On the other hand, for the power-law and exponential distributions, the homology is nontrivial, and exhibits what the authors term crackle. Sequences $$\{R_n\}_{n=1}^\infty$$ are found such that the expectation values $${\mathbb E}(\beta_{k,n})$$ are either trivial, finite, or infinite for large $$n$$. In particular, outside of the core the homology can be organized by annuli of increasing radius, but decreasing homological complexity. For the first annulus outside of the core, the Betti numbers in dimensions less than $$d$$ are infinite, and zero otherwise. For the second annulus, the Betti numbers are infinite in dimensions less than $$d-1$$, finite in dimension $$d-1$$, and trivial otherwise. The pattern repeats until the Betti numbers are trivial in all dimensions.

MSC:

 60D05 Geometric probability and stochastic geometry 55U10 Simplicial sets and complexes in algebraic topology 57Q99 PL-topology 05C80 Random graphs (graph-theoretic aspects) 60F15 Strong limit theorems 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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