The maximum degree of the rth power of a sparse random graph

Let G^r_n,p denote the rth power of the random graph G_n,p, where p=c/n for a positive constant c. We prove that w.h.p. the maximum degree Δ(G^r_n,p)∼log n/log_(r+1)n. Here log_(k)n indicates the repeated application of the log-function k times. So, for example, log_(3)n=logloglog n.