Linear nets and convex polyhedra

It is proved that the set $[G,\varphi]_\Gamma$ of immersed linear networks in $\mathbb R^N$ which are parallel to a given immersed linear network $\Gamma\colon G\to\mathbb R^N$ and have the same boundary $\varphi$ as $\Gamma$, can be configuration space of movable vertices of the graph $G$. Also, the dimension of the space $[G,\varphi]_\Gamma$ is calculated, and the number of faces is estimated. As an application, the space of all local minimal and weighted local minimal networks in $\mathbb R^N$ with fixed topology and boundary is described.