The Homology Of Alpha Complex And Alpha Hypergraph

The methods of algebraic topology have been popularly used in data analytics in Europe and America.However,its development is still in its infancy in China.In order to analyze topological data with persistent homology,we currently mostly use the persistent homology of the Vietoris-Rips Complex,the Cubical Complex,the Neighborhood Complex,and the Alpha Complex,etc.In this thesis,the Alpha Hypergraph is introduced,and some existing theoretical methods of topological data analysis of the Alpha Complex are generalized at the mathematical level.At the same time,the effect can be tested with some data.First of all,we introduce the Alpha Complex and its persistent homology in this thesis and give some properties of the Alpha Complex.Then the concept of the Alpha Hypergraph is introduced and its topological properties are discussed.The algorithm for the Alpha Hypergraph is given.It can not only draw the Alpha Hypergraph but also output the number of virtual hyperedges.Secondly,the homology correlation and persistent homology correlation between Alpha Hypergraph and Alpha Complex are studied.Finally,a new barcode is obtained by using the persistent homology of the Alpha Complex and combining the features of the Alpha Hypergraph.The barcode can record the total number of virtual hyperedges and the birth time of each virtual hyperedge in the filtering process of the Alpha Hypergraph,which reflects that compared with the filtering process of the Alpha Complex,the Alpha Hypergraph lacks certain information during filtering.