Random Deletion in a Scale-Free Random Graph Process

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time t, with probability α1 > 0 we add a new vertex ut and m random edges incident with ut. The neighbours of ut are chosen with probability proportional to degree. With probability α − α1 ≥ 0 we add m random edges to existing vertices where the endpoints are chosen with probability proportional to degree. With probability 1 − α − α0 we delete a random vertex, if there are vertices left to delete. and with probability α0 we delete m random edges. Assuming that α + α1 + α0 > 1 and α0 is sufficently small, we show that for large k, t, the expected number of vertices of degree k is approximately dkt where as k → ∞, dk ∼ Ck −1−β where β = 2(α−α0) 3α−1−α1−α0 and C > 0 is a constant. Note that β can take any value greater than 1.