Combinatorial Invariants of Multidimensional Topological Network Data

Modern applications of algebraic topology to point cloud data analysis have motivated active investigation of combinatorial clique complexes -- high-dimensional extensions of combinatorial graphs. We show that meaningful invariants of such spaces are reflected in the combinatorial properties of an associated family of linear matroids, and discuss matroid-theoretic approaches to several problems in computational topology. Our results allow us to derive estimates of the summary statistics of related constructs for random point cloud data, which we discuss for several sampling distributions in $\mathbb{R}^2$ and $\mathbb{R}^3$.