  We consider the following activation process in undirected graphs: a vertex
is active either if it belongs to a set of initially activated vertices or if
at some point it has at least $r$ active neighbors. A \emph{contagious set} is
a set whose activation results with the entire graph being active. Given a
graph $G$, let $m(G,r)$ be the minimal size of a contagious set.
  We study this process on the binomial random graph $G:=G(n,p)$ with $p: =
\frac{d}{n}$ and $1 \ll d \ll \left(\frac{n \log \log n}{\log^2
n}\right)^{\frac{r-1}{r}}$. Assuming $r > 1$ to be a constant that does not
depend on $n$, we prove that $$m(G,r) =
\Theta\left(\frac{n}{d^{\frac{r}{r-1}}\log d}\right),$$ with high probability.
We also show that the threshold probability for $m(G,r)=r$ to hold is
$p^*=\Theta\left(\frac{1}{(n \log^{r-1} n)^{1/r}}\right)$.
