graph = (Uncompress@*FromCharacterCode@*
    Flatten@*(ImageData[#1, "Byte"] &)@*Import)[
  "https://ooo.0o0.ooo/2017/02/27/58b427c17047e.png"]

weightGraph=Graph[graph,
 EdgeWeight ->
  Thread[EdgeList[graph] ->
    EuclideanDistance @@@ (EdgeList[graph] /.
       Thread[VertexList[graph] -> GraphEmbedding[graph]])]]