How to scale my grid graph?

Question

I am learning to draw graphs for my thesis and came up with the following ugly result:

\documentclass[tikz]{standalone}
\usetikzlibrary{cd,graphs,graphs.standard}
\begin{document}
  \begin{tikzpicture}
  \graph [simple, nodes={empty nodes, fill, circle, inner sep=1pt, draw}] {
    subgraph I_n [n=4, grid placement, radius=2cm];
   3[label=left:$v_3$]--["$K_1$"]1[label=left:$v_1$]--["$K_2$"]4[label=right:$v_4$]--["$K_4$"']2[label=right:$v_2$];
  };
  \end{tikzpicture}
\end{document}

The label "K_2" is positioned inappropriately, but I guess it will be better if the graph is scaled such that the edges are longer. Neither scale=2, nor radius=2cm (see How to scale a tikz graph) works. The latter option does work if I replace grid placement by clockwise, but I prefer the positioning of the nodes as presented in the image.

How can I solve this?

Note: Maybe the graphs library is a bit of overkill for this case, but I am trying to find some kind of generic way for drawing graphs, since I will have to draw complex examples later on.

Answer 1

By default, x and y are each units of 10mm. However, you can change this for a picture or a scope. For example,

\documentclass[border=10pt,multi,tikz]{standalone}
\usetikzlibrary{cd,graphs,graphs.standard}
\begin{document}
  \begin{tikzpicture}[x=20mm, y=20mm]
  \graph [simple, nodes={empty nodes, fill, circle, inner sep=1pt, draw}] {
    subgraph I_n [n=4, grid placement];
   3[label=left:$v_3$]--["$K_1$"]1[label=left:$v_1$]--["$K_2$"]4[label=right:$v_4$]--["$K_4$"']2[label=right:$v_2$];
  };
  \end{tikzpicture}
\end{document}