other nodes will be considered now. Replacing therein the number of the linked node $j$ by the specific degree $k_j$ of this node, it becomes $$\begin{aligned}
    \label{eq:signal_degree}
         f_i(t) =  \frac{1}{N}\, \sum_{j=1}^{N}\,A_{i,j}s_{k_j}(t).\end{aligned}$$ Sorting different degrees gives $$\begin{aligned}
    \label{eq:signal_degree1}
         f_i(t) =  \sum_{k=k_{min}}^{k_{max}}\, \frac{N_k}{N} \, \frac{1}{N_k}\,\sum_{j=1}^{N}\,\delta_{k,k_j}\, A_{i,j}\,s_{k_j}(t).\end{aligned}$$ Every node receives its input depending on the specific linked environment. Due to the disorder contained in the adjacency matrix $A_
{i,j}$, the summation is still in general different for distinct nodes. The reduced description for given degrees requires an additional approximation.

Heterogeneous mean-field approximation {#sec:HMF}
--------------------------------------

As proposed in [@Ichinomiya2004; @Sonnenschein2012; @Carro2016], the complex network with the adjacency matrix $A_{i,j}$ and with the given degree distribution $p(k)$ will be replaced by a fully connected network with weighted edges. The latter with adjacency matrix $\tilde{A}_{i,j}$ shall possess the same distribution of the degrees at the nodes. In detail, it is required that the degree-values of all nodes in the new network shall coincide with the corresponding ones in the original network. Accordingly, the following shall hold $$\begin{aligned}
    \label{eq:degree1}
    \sum_{j=1}^{N} A_{i,j}\, = \,k_i\,= \sum_{j=1}^{N} \tilde{A}_{i,j}.
    \quad i=1,\ldots,N.\end{aligned}$$ Whereas the sum at the l.h.s. is running over values $1$ and $0$ of $A_{i,j}$, at the r.h.s. the sum 