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In the following, the properties of a binary random network [@Lambiotte2007; @Sonnenschein2013; @Carro2016] will be studied. It allows a full sketch of the possible bifurcation scenario appearing in these networks. These ones are randomly connected with two different degrees $k_1$ and $k_2$.

In terms of the degree distribution for a binary network it is supposed that $$p(k)=\nu\delta_{k,k_1}+(1-\nu)\delta_{k,k_2}.
    \label{eq:pbeta_rbnet}$$ where $\nu\in]0,1[$ is the fraction of nodes with degree $k_1$. In this paper $k_1 > k_2$ is set, but due to the symmetry $(k_1, \nu) \longleftrightarrow (k_2, 1-\nu)$ the alternative case is also included.

The set of two master equations for binary networks read: $$\begin{aligned}
  \nonumber &\dot{P}(1,t | k_1) = {\gamma_{k_1}}(t)\;(1-P(1,t|k_1)-\\
  \nonumber &\int_0^\infty {\gamma_{k_1}}(t-t') (1-P(1, t-t'|k_1))w_1(t') \mathrm{d} t',\\
  \nonumber &\dot{P}(1,t | k_2) = {\gamma_{k_2}}(t)\;(1-P(1,t|k_2))-\\
  &\int_0^\infty {\gamma_{k_2}}(t-t') (1-P(1,t-t'|k_2))w_1(t') \mathrm{d}t',
    \label{eq:MFeq_rbnet}\end{aligned}$$ where we have denoted $\gamma_{k_1}(t)=\gamma[k_1 / N r(t)]$ and $\gamma_{k_2}(t)$, respectively. Equations can be brought into the integral form [@phdKouvaris]: $$\begin{aligned}
    \nonumber P&(1,t|k_1) = \\
    \nonumber &\int_0^\infty {\gamma_{k_1}}(t-t') (1-P(1,t-t'|k_1))z_1(t') \mathrm{d} t',\\
    \nonumber P&(1,t|k_2)  = \\
     &\int_0^\infty {\gamma_{k_2}}(t-t') (1-P(1, t-t'|k_2))z_1(t') \mathrm{d}t',
    \label{eq:MFeqInt_rbnet}\end{aligned}$$ supplemented by initial conditions. Therein, $z_1(t)$ is the survival probability of state $1$, $$z_1(t) = 1-\int_0^{t}w_1(t') \mathrm{d} t'.$$ Equations have to be supplemented by initial conditions.

Qualitative discussion of steady states 