at most 2 cooperating neighbours.

Taking these ideas into account, and assuming that cluster expansion or death is not influenced by the presence of cooperators outside the cluster, we can try to predict what happens when cooperators are placed at random in a network. In the case of a lattice network, the probability of an initial set of cooperators taking over the network is simply the probability that there is at least one square cluster of cooperators: $P(p)=1-(1-p^4)^N$. For a random regular network the probability is $P(p)=1-(1-f(p,k,N))^N$, where $f(p,k,N)$ is the probability that a given node is the center of a 3-cluster and that it expands during at least two steps:

$$\begin{aligned}
f(p,k,N)  &=& (1-(1-k/N)^{4(k-2)(k-2)})(1-(1-k/N)^{\frac{k(k-1)}{2}}) \nonumber \\
&& (1-(1-p)^k-kp(1-p)^{k-1})\end{aligned}$$

where the first term in the product is the probability that a given node has at least two cooperating neighbors, the second is the probability that the 3-cluster expands in the first step, and the third is the probability that it continues expanding in the second step. We assume that after the second step the expansion goes on until all the lattice is occupied by cooperators. Fig. \[k8t\] shows that these functions overestimate the fraction of populations that are able to 