G(n,p) And G(n,d) Models In Random Networks

The G(n, p) is a probability space consisting of random graphs with n vertexes and any edge with probability p. G(n,d) is a probability space of random regular graphs with n nodes and degree d. First, we introduce the fundamental concepts and probabilistic methods. Second, we deal with the cycles’ problem of G(n,p) and G(n,d) by using classical methods, and gain an asymptotic estimate of p when G(n, p) has cycles of length [3,n/3], and the cycles’problem in random hypergraph. Then, explore the pairing model and switching model of G(n,d), and apply them to prove the asymptotic distribution of 3-length cycles and the independence of the distributions of the cycles of all length. Last, apply the classical methods to d-regular κ-uniform hypergraph and obtain the asymptotic expectation of loops and multiple edges. In the end, there are two unsolved problems of G(n, d) proposed.