corresponding class $$\overline{s}_k(t)=\frac{1}{N_k}\sum_{j=1}^N\, \delta_{k,k_j}\, s_{k_j}(t).
  \label{eq:sk}$$ Therein, the sum runs over $N_k$ items due to the action of the $\delta$-function. Since the number of nodes with degree $k$ scales as $N_k \propto N$, application of the limit of large $N$ yields $$\begin{aligned}
  \lim_{N \to \infty} \overline{s}_k(t) = P(1,t|k), 
  \label{eq:thlimes1}\end{aligned}$$ which was introduced in .

Thus, the mean-field $r(t)$ introduced in becomes in this limit $$r(t)=\frac{1}{\langle
    k\rangle}\sum_{k'=k_{\mathrm{min}}}^{k_{\mathrm{max}}}p(k')k'P(1,t|k)
  =\frac{\langle k\,P_{1,k}(t)\rangle}{\langle k\rangle}.
    \label{eq:r}$$ Eventually, by inserting and via into the mean-field description and the following expression is obtained $$\begin{aligned}
\label{eq:MF}
&P(0,t|k)=1-P(1,t|k),\\ \nonumber
&\dot{P}(1,t|k)=\gamma\left[\frac{k}{N}r(t)\right]\left(1-P(1,t|k)\right)-
\\ \nonumber
&\int_0^{\infty}\gamma\left[\frac{k}{N}r(t-t')\right]\left(1-P(1,t-t'|k)\right)w_1(t')\mathrm{d}t'.\end{aligned}$$ This is now a set of coupled non-linear integro-differential equations, since the mean-field $r(t)$ depends on $ P(1.t|k)$ via . Though the structure of the equations looks similar to the previous Master equation it is qualitatively different. As the result of the replacement of the adjacency matrix, the number of equations has reduced drastically compared to the system and . The index $k$ in is only running over the different possible degrees in the network $k\in[k_{\mathrm{min}},k_{\mathrm{max}}]$ whereas in and it runs over all nodes $N$. In addition, the dependence of the activation rate $\gamma$ on the degree $k$ appears uniquely in the argument for all nodes as 