Matthew Kahle

Random Geometric Complexes

We study the expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝd. We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology Hk is not monotone when k>0.In particular, for every k>0, we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes.

Topology of random geometric complexes: a survey

AbstractIn this expository article, we survey the rapidly emerging area of random geometric simplicial complexes. Random geometric complexes may be viewed as higher-dimensional generalizations of random geometric graphs, where vertices are generated by a random point process, and edges are placed based on proximity. Extending the notion of connected components and cycles in graphs, the main object of study has been the homology of these complexes. We review the results known to date about the probabilistic behavior of the homology (and related structures) generated by these random complexes.

The neighborhood complex of a random graph

For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well-known result of Lovasz that if @?N[G]@? is k-connected, then the chromatic number of G is at least k+3. We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, O(logd), compared to the expected dimension d of the complex itself.

The Threshold for Integer Homology in Random d-Complexes

Let

Random graph products of finite groups are rational duality groups

Y \sim Y_d(n,p)

Topology of random clique complexes

Y∼Yd(n,p) denote the Bernoulli random d-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology

Limit the theorems for Betti numbers of random simplicial complexes

H_{d-1}(Y; \mathbb {Z})