On the degree distribution of a growing network model

In this note we make some specific observations on the distribution of the degree of a given vertex in certain model of randomly growing networks. The rule for network growth is the following. Starting with an initial graph of minimum degree at least $k$, new vertices are added one by one. Each new vertex $v$ first chooses a random vertex $w$ to join to, where the probability of choosing $w$ is proportional to its degree. Then $k$ edges are added from $v$ to randomly chosen neighbours of $w$.