\\
    & \hspace{3.7cm} i \in \{ 1, \dots, N \}.
    \label{eq:dtP2}\end{aligned}$$

This is a set of $N$ coupled linear integro-differential equations for the $P_i(1,t)$. The complexity is given by the chosen adjacency matrix $A_{i,j}$ which links the almost $N$ equations by the the signal function, see Eq. . By applying a heterogeneous mean-field approximation as described in Section \[sec:HMF\], the structure of this set of equations is much simpler and the number of differential equations in the set reduces significantly. As consequence of this approximation, the set will become analytically treatable for special cases in the stationary limit.

But before describing this approximation, an appropriate notation for the values describing the dynamical behavior on a complex network will be introduced.

Master equation on complex networks {#sec:ME}
-----------------------------------

To include the network topology into the description, it is assumed that the complex network structure can be described as a random network. Central value in this description are the degrees $k_i$ of the $N$ nodes. For a given adjacency matrix $A_{i,j}$ with values $0,1$ the degree $k_i$ of the $i$th node is the number of existing links to other nodes of the network. It becomes $$\begin{aligned}
    \label{eq:degree}
            k_i = \sum_{j=1}^{N} A_{i,j}.\end{aligned}$$

Let 