An $(n,k,r)$-network is a triple $N=(G,in,out)$ where $G=(V,E)$ is a graph and $in,out$ are integral functions defined on $V$ called input and output functions, such that for any $v \inV$, $in(v)+out(v)+ deg(v)\leq2r$ with $deg(v)$ the degree of $v$ in the graph $G$. The total number of inputs is $in(V)=\sum_v\inVin(v)=n$, and the total number of outputs is $out(V)=\sum_v\inVout(v)=n+k$. An $(n,k,r)$-network is valid, if for any faulty output function $out'$ (that is such that $out'(v) \leqout(v)$ for any $v \inV$, and $out'(V) = n$), there are $n$ edge-disjoint paths in $G$ such that each vertex $v\inV$ is the initial vertex of $in(v)$ paths and the terminal vertex of $out'(v)$ paths. We investigate the design problem of determining the minimum number of vertices in a valid $(n,k,r)$-network and of constructing minimum $(n,k,r)$-networks, or at least valid $(n,k,r)$-networks with a number of vertices close to the optimal value. We first show $\frac3n+k2r-2+ \frac3r^2k \leq\calN(n,k,r)\leq\left\lceil\frack+22r-2\right\rceil\fracn2$. We prove a better upper bound when $r\geqk/2$: $\calN(n,k,r) \leq\fracr-2+k/2r^2-2r+k/2 n + O(1)$. Finally, we give the exact value of $\calN(n,k,r)$ when $k\leq6$ and exhibit the corresponding networks.