On the phase transition property of one random graph model

Abstract

In this paper, we introduce one model named random connection model RG(n, α, β) defined as follows: the vertex set is Z n 2 , and two vertices u, v ∈ Z n 2 are adjacent with probability α n−H(u,v)β H(u,v) , in which H(u, v) is the Hamming distance between u and v, and α, β ∈ (0, 1) are some fixed constants. This model can be regarded as the discrete counterpart of the random connection model which is given by Penrose (1991) via the Euclidean distance. We obtain some phase transition properties of RG(n, α, β) with the help of some asymptotic results by the First and Second moment arguments; and some possible generalizations of the results mentioned in this paper are discussed in the last section. Keywords: Random connection model, Isolated vertices, Phase transition property, Hamming distance, the Giant Component