(RSN) as shown in Fig. \[fig:ModelOverview\](b) composed of $L \times L$ nodes arranged on a triangular lattice whose nearest neighbours are connected by the same element we introduced for the multi-spring model ($L=50$ for all RSN simulations). Now, upon applying an external load, stress concentration is possible due to the topological restrictions imposed by the network structure.

Following the experimentally found guidelines for making tough double networks [@Gong2010] we consider materials in which the sacrificial network is stiff and weak, while the matrix is soft and strong. This means that the elastic constant of sacrificial springs $\mu_S$ is always higher than the one of the matrix springs $\mu_M$. We therefore vary the ratio of the stiffnesses such that $0<\mu_M/\mu_S<1$. To reflect the asymmetry in network strengths, we fix $\lambda_M=4.0$ and vary $\lambda_S$ such that $\lambda_S<\lambda_M$. In all cases, we keep the parameters of all matrix springs the same. To implement disorder at the bond-level, the failure thresholds $\lambda_S$ are picked from a Gaussian distribution as in the MS model. Nodes at the bottom and top are fixed in the $y$-direction, but they can slide along the $x$-direction. Along the $x$-axis periodic boundary conditions are implemented.

We characterize the mechanical response 