The clique complex X(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There are nevertheless higher dimensional analogues of the Erdős-Rényi Theorem.We study here the higher homology groups of X(G(n, p)). For k > 0 we show the following. If p = n α , with α < −1/k or α > −1/(2k + 1), then the kth homology group of X(G(n, p)) is almost always vanishing, and if −1/k < α < −1/(k + 1), then it is almost always nonvanishing.We also give estimates for the expected rank of homology, and exhibit explicit nontrivial classes in the nonvanishing regime. These estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and we cannot rule out the possibility that they are homotopy equivalent to wedges of spheres.
We study the expected topological properties ofČech and Vietoris-Rips complexes built on random points in R d . We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H k 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.
We study Linial-Meshulam random 2 2 -complexes Y ( n , p ) Y(n,p) , which are 2 2 -dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n − 1 / 2 p = n^{-1/2} . This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log n / n p = 2 \log n / n . We use a variant of Gromov’s local-to-global theorem for linear isoperimetric inequalities to show that when p = O ( n − 1 / 2 − ϵ p = O( n^{-1/2 -\epsilon } ), the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2 2 -dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.
There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology groups, and in some cases expectations of the Betti numbers; however, little seems known so far about limiting distributions of random Betti numbers.In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complexes: Erdős-Rényi random clique complexes, random Vietoris-Rips complexes, and randomČech complexes. These results may be of practical interest in topological data analysis.
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