the $2(k-2)$ non-cooperating neighbors of the nodes that share the link (see Fig. \[figer\]A). But there is now a non vanishing probability that there is also a link joining the nodes of the cluster ’surface’ which would lead to an increase of its size of $2(k-2)$. And, in general, at any step of its growth it could keep growing with a probability roughly proportional to $(k/N) N_s 2(k-2)$, where $N_s$ is the size of the cluster surface. Considering that most of the nodes of the clusters lie in its surface, implies that once the cluster has reached a size of order $N/(2k(k-2))$ it will keep growing until it spans the whole lattice. Thus, if we consider the evolution of all possible 3-clusters, the distribution of final cluster sizes should be non-vanishing only for sizes $O(N)$ and for sizes smaller than $O(N/(2k(k-2)))$.

In the case of 4-clusters, it is easy to see that the situation is qualitatively the same for regular random networks. On the other hand, for lattice networks the picture is completely different: square 4-clusters expand until they occupy the whole lattice because every cooperator is connected to 2 or 3 others whereas non cooperating neighbours can only have 