Several Problems On Invariants And Random Graphs

In this dissertation,the content is divided into the following two aspects.Firstly,the hypersurface problem in invariants is discussed.In the process of studying the invariant of finite group and its subgroups,the T-functors is proved a powerful tool.Therefore,this paper explores the operation of T-functors on hypersurfaces.We proved the result that the T-functor can only decrease the embedding dimension in the category of unstable algebras over the Steenrod algebra.And then we obtain the conclusion that the T-functor preserves the hypersurfaces in the category of unstable algebras.In other words,if the invariant of a finite group is a hypersurface,then so are the invariant of its stabilizer subgroups.But not vice versa,by constructing counter-examples,we illustrate that if the invariants of the stabilizer subgroups or Sylow p-subgroups are hypersurfaces,the invariant of the group itself is not necessarily a hypersurface.Secondly,we also investigate some parameters of the stochastic block model,which have been proved to be one of the most successful tool for detecting community structure in various networks.The dense stochastic block model is a simple undirected random graph,assuming that every node’s block membership is random taken from a finite set independently according to a fixed law.To capture the asymptotic behavior of the stochastic block graphs when the number of vertices tends to infinity,the definition of the empirical pair measure for any dense stochastic block graph is put forward,exploring the number of edges connecting any given pair of blocks.Then the empirical block measure is provided,computing the number of vertices with given block membership.Our concern here is the large deviation principle for these crucial empirical measures in the corresponding weak topology.So we described the metric space corresponding the weak topology,deduced the large deviation principle for the empirical pair measure under the empirical block measure.Moreover,we obtained the joint distribution is obtained through the mix approach,proved the lower semi-continuity and the exponentially tightness.At last,the numerical simulation of these theoretical results is given.