k\frac{\partial \log(\gamma^*_{k})}{\partial r^*}\right\rangle-4\langle k\rangle = 0.                                 
\label{eq:SNEllipseCrit_rbnet}$$ For the $\gamma_k$ given by , results in $$\frac{\sigma_{\text{crit}}}{4 D_\text{crit}}\frac{\langle k^2\rangle}{N\,\langle k\rangle}=1.
\label{eq:scusp_rbnet}$$

Comparing this to the well known result of all-to-all coupled stochastic two-state units [@Kouvaris2010], $$\frac{\sigma_{\text{crit}}}{4 D_\text{crit}}=1,$$ it is visible that it differs only by a scaling factor given by the ratio of the first two moments of the degree (density) distribution. This is a typical network effect in mean-field coupled oscillators [@Restrepo2005; @ArenasKuths08; @Boccaletti06; @Newman03]. The factor can be interpreted as the mean of the degree distribution of the nearest neighbors [@Dorogovtsev02] assuming that the local structure of the network is treelike. Therefore the effective coupling strength $$\sigma_{\rm eff} = \sigma\frac{\langle k^2\rangle}{\langle k\rangle} 
= \sigma\frac{\nu k_1^2 + (1-\nu) k_2^2}{\nu k_1 + (1-\nu) k_2} 
\label{eq:sstar_rbnet}$$ will be introduced. Note that it is evident from that this scaling is only obtained if $\gamma$ depends exponentially on the mean-field $r$.

In case of low noise a more detailed picture with possibly multiple solutions and ordered states occur. These solutions can be discussed solving graphically and plotting the r.h.s. versus the l.h.s as presented in Fig. \[fig:rvsl\] for typical situations.

![image](./rbnet_2ss_d_0-1_s_2_p_0-6){width=".48\textwidth"}![image](./rbnet_3ss_d_0-1_s_2_p_0-37){width=".48\textwidth"}

The l.h.s. of equation is a straight line and unbounded whereas the r.h.s. 