The condition is similar to the equilibrium condition on the plane, only this time the sum of the forces at each vertex must not be zero but collinear with the radius-vector of the vertex:
$$
\sum_j w_{ij}(p_i-p_j) = c_i p_i
$$

In other words, the graph made by adding a vertex at the origin and joining it with all other vertices must have an equilibrium stress.

This should be contained in Lovasz' article "Steinitz representations of polyhedra and the Colin de Verdière number".