playing against a defector). To understand the origin, and quantify the limits, of these steps, we must consider the necessary conditions for the propagation of the cooperating behavior. For a cooperator to have a chance to turn a defecting neighbour into a cooperating one, its payoff should be at least larger than that of the defecting neighbour. This leads to the condition $n_{CC}+(k-n_{CC})(1-t) > n_{DC} t$, where $n_{CC}$ is the number of cooperator neighbors of the cooperator and $n_{DC}$ is the number of cooperator neighbors of the defector. The condition on $t$ can be written as $t>k/(k-n)$ where $n=n_{CC}-n_{DC}$. Note that, as $n_{CC} \leq k-1$ and $n_{DC}
\geq 1$, $n$ is a natural number that must satisfy $1 \leq n \leq
k-2$. This gives a maximum of $k-1$ possible steps. Note however that in some networks the range of possible values for $n$ is smaller, and therefore the number of steps of $\rho_c$ is at most $k-2$. In general, for networks with the same number of $k$ the number of possible steps will be smaller for the networks with smaller clustering coefficients. As an example, consider the two extreme cases of a tree and a lattice network with $k=8$: whereas 