Reduction algorithm for random abstract simplicial complexes

Laurent Decreusefond (1, 2) , Philippe Martins (1, 2) , Anaïs Vergne (1)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 Télécom ParisTech

In this paper, we present a reduction algorithm for abstract simpli- cial complexe that we apply to Vietoris-Rips complexes based on random point processes. Our algorithm aims at reducing the number of simplices of an abstract simplicial complex without modifying its topology, i.e. its Betti numbers. The vertices are removed in a optimal order for the ab- stract simplicial complex implementation complexity. Some mathematical properties of the reduction algorithm and its resulting abstract simplicial complex are derived. Then the complexity of the algorithm is investigated: we are reduced to compute the behavior of the size of the largest simplex in the abstract simplicial complex, which is known in graph theory as the clique number. We find its asymptotic behavior when the number of vertices goes to infinity depending on the percolation regime for the underlying random geometric graph on the torus.

Domaines: Probabilités [math.PR]

Laurent Decreusefond, Philippe Martins, Anaïs Vergne. Reduction algorithm for random abstract simplicial complexes. 2013. ⟨hal-00864303v2⟩