Betti number signatures of homogeneous Poisson point processes

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: beta(0) is the number of connected components and beta(k) effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centered spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha shapes, a construction from computational geometry that deserves to be more widely known in the physics community.