Estudio de la Homología Persistente de una filtración de Complejos de Clique

Abstract

In this paper the basic notions of homotopy for topological spaces and simplicial homology are presented, as well as the relationship between them. Later, the definition of clique complex is given and its relationship with the power of a graph is presented to introduce the concept of persistent homology for a graph filtration. Afterwards, the concept of beat-point is introduced and its effect on the homotopy of the complex is proven. Finally, transformations which preserve the persistent homology of a clique complex filtration, while reducing the points of the graph powers and thus the calculations of its homology, are defined.