Algebraic and Metric Properties of Random Geometric Graphs and Complexes
Description Phase 2
The study of geometric graphs and simplicial complexes and their properties is a highly active area of research. The question, how a generic graph looks like, i.e. about the average behaviour of algebraic and combinatorial properties of the graph and some graph-generated simplicial complex, is important in many applications and of high intrinsic mathematical interest. Particular focus in research is on their f-vector and Betti numbers. To answer these questions on the generic behaviour, random graphs were investigated. Classical models are the Erdös-Renyi random graph and the geometric random graph in Euclidean space. The geometric random graph and the simplicial complex generated by the graph are in the focus of this application. In the last years several papers have appeared, who link stochastic geometry, stochastic analysis and and topological data analysis to random graphs and random simplicial complexes. This is due to the fact that by the nerve lemma the topology of the Boolean model in stochastic geometry and the socalled Cech complex are very similar. This allows us to translate questions about algebraic and metric properties of random simplicial complexes into geometric questions. In this proposal we consider a Poisson point processes to generate vertices of the geometric random graph. The edges of this graph are defined as those pairs of vertices where their distance is bounded by some fixed parameter. The Vietoris-Rips complex is the clique complex of the Gilbert graph, and a similar construction yields the Cech complex. We are interested in the limiting structure of both random complexes as the intensity tends to infinite and the distance converges to zero. In particular, we would like to analyse the metric and algebraic behaviour of the random geometric graph and these random simplicial complexes, its dependence on the underlying space (Euclidean, spherical, hyperbolic), and to generalize this investigations to related random graphs. In a first step we investigate the f-vector and the volume-power functionals, then the Euler characteristic and more general the h-vector of these random simplicial complexes. This leads to the question of determining the dimension of the Vietoris-Rips and Cech complexes. The algebraic Betti numbers – in contrast to their topological counterparts – have not been investigated at all in the literature using these kind of approaches and models. One of the main goals is to estimate the probability, that in the connected regime the random simplicial complexes have the Cohen-Macaulay property.