The Study Of Networks Induced By Simplicial Complex Triangulation And Their Relations With Invariants

Simplicial homology group is a topological invariant,and is also a homotopy type invariant.It means that if two topological spaces are homotopically equivalent but not homeomorphic,their simplicial homology groups of the same dimension are isomorphic.Then the two topological spaces cann't be distinguished by homology group.The proposed concept of scale-free and small-world networks makes it natural to think about whether we could embody the homotopy type invariance from three spectacular quantities of networks including the degree distribution,the clustering coefficient and the average distance.As homology group is not complete in distinguishing homeomorphism,this paper will try to search for some new topological invariants.These invariants are not only topological invariants,but also not homotopy type invariants. Then it's hopeful for us to inquire into the problem of topological classification of topological spaces,not being confined to their homotopy classification. The cylinder and the M|¨bius strip are homotopically equivalent but not homeomorphic.This paper will use the results of complex networks induced by the two simplicial complexes to surmise new topological invariants.This paper investigates the above questions based on two simplicial complexes:the cylinder and the Mo|¨bius strip.From simplex triangulation we obtain three kinds of network models and compute their three quantities mentioned above.By comparing the three quantities,the similarity and the difference between the cylinder and the M|¨bius strip can be known.At last,this paper raises a concept about the orientability of networks,which can be a new parameter of ordinary networks.