Topology of random clique complexes

In a seminal paper, Erdős and Rényi identified a sharp threshold for connectivity of the random graph G ( n , p ) . In particular, they showed that if p ≫ log n / n then G ( n , p ) is almost always connected, and if p ≪ log n / n then G ( n , p ) is almost always disconnected, as n → ∞ . ique 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. dy 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 / ( 2 k + 1 ) , then the k th 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. o 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 a spheres.