On the topology of random complexes built over stationary point processes

D. Yogeshwaran, Robert J. Adler

Published 2012-11-01, updated 2014-09-22Version 2

There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a random point process in $\mR^d$, and the edges and faces are determined according to some deterministic rule, typically leading to 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. Our proofs rely on analysis of subgraph and component counts of stationary point processes, which are of independent interest in stochastic geometry.