The asymptotic topology of the multineighbor complex of a random graph

We introduce the multineighbor complex of a graph, which is a simplicial complex in which a simplex is a subset of the graph with a sufficient number of mutual neighbors. We investigate the asymptotic homological properties of such complexes for the Erdos-Renyi random graphs and obtain a number of vanishing and nonvanishing results. We use this construction to perform a topological data analysis classification of noisy synthetic point clouds obtaining favorable accuracy as obtained by the standard methods. The case when there is a single neighbor has been studied earlier by Mathew Kahle.