I suppose this can be solved with some trigonometry, see the second point under the Applications section here:
http://en.wikipedia.org/wiki/Law_of_cosines

Note that there might not be a solution at all, say if you have links in a cycle and one of them is longer than the sum total of all others. On the other hand, there might be multiple solutions, e.g. four links of equal length could form a square or rhombus.

If there are multiple faces (closed regions defined by links) in your graph, you will have to solve a system of equations to find a solution.

All in all, if you know these fixed lengths correspond to a shape that actually exists, better use the shape's vertex coordinates to place the nodes, instead of trying to find their positions from the distances.

Stoyan