Random Deletion in a Scale-Free Random Graph Process

Abstract

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time $t$, with probability $\alpha_1>0$ we add a new vertex $u_t$ and $m$ random edges incident with $u_t$. The neighbours of $u_t$ are chosen with probability proportional to degree. With probability $\alpha-\alpha_1\geq 0$ we add $m$ random edges to existing vertices where the endpoints are chosen with probability proportional to degree. With probability $1-\alpha-\alpha_0$ we delete a random vertex, if there are vertices left to delete. With probability $\alpha_0$ we delete $m$ random edges. Assuming that $\alpha+\alpha_1+\alpha_0>1$ and $\alpha_0$ is sufficently small, we show that for large $k,t$, the expected number of vertices of degree $k$ is approximately $d_kt$ where as $k\to\infty$, $d_k\sim Ck^{-1-\beta}$ where $\beta=\frac{2(\alpha-\alpha_0)}{3\alpha-1-\alpha_1-\alpha_0}$ and $C>0$ is a constant. Note that $\beta$ can take any value greater than 1.