中文摘要：

固定α0∈[0,1）及β∈[0,1/2）.该文引入如下随机图过程（Gt）t≥1：设在时刻1及2已存在图G1=G2,其中G1的顶点为v1,v2且它们之间有2条边相连.当t≥3时,Gt定义如下：（i）G（t-1）中任意顶点v不活跃的概率为α0.顶点不活跃意味着其不能与t时刻新增加的顶点相连.此概率独立于自己以及其他顶点t-1之前的状态;（ii）以概率1-β增加一个新顶点vt.在G（t-1）中以概率dw（t-1）/∑vdv（t-1）选一顶点w,其中dw（t-1）表w在G（t-1）中的度.若w是活跃的则在vt与w之间连1条边,否则在vt上加个环;（iii）以概率β在G（t-1）中删去一顶点u,其中u被选中的概率为（1-du（t-1）/∑vdv（t-1））/（n（t-1）-1）.此处,n（t-1）是G（t-1）的顶点个数.令Nk（t）表Gt中度为k的顶点个数.该文证明了Gt度分布的期望在2β/1-α0=1附近存在一相变：当2β/1-α0〉1时,Nk（t）/t的期望是呈指数衰减的;当2β/1-α0〈1时,Nk（t）/t的期望是呈幂律衰减的.

英文摘要：

The following random graph process Gt is introduced. Assuming that at the timestep 1 and 2 there has been a graph G1 = G2 consisting of vertices Vl, v2 and 2 edges between them. At time-step t ≥ 3, Gt is defined as follows： （i） each vertex v ∈ Gt-1 is inactive with probability a0, independently its and other vertices＇ statuses before t - 1. The inactiveness of v means it can not being connected by more edges; （ii） with probability 1 -β a new vertex vt is added along with one edge connected to it. Then a vertex wi is chosen with probability proportional to its degree. If wi is active then connect vt to wi. Otherwise, the edge of vt is connected to itself; （iii） with probability β a vertex is deleted preferentially from the network. It is proved that there is a phase transition on expected degree distribution when 2β/（1 - α0） is in the vicinity of 1. In the supercritical regime （2β/（1 - α0） 〉 1）, its expected fraction of vertices with degree k decays exponentially. While in the subcritical regime （2β/（1 - α0） 〈 1）, the expected fraction of vertices with degree k decreases asymptotically as a power law.