The same value of VertexSize for vertices in different graphs causes different vertices size

Question

I generated some graphs. The smallest vertices in the 1st graph should be the same size as the smallest vertices in the second graph. The smallest vertices have the same size (VertexSize value is the same) in each graph. When I generate those graphs the size of the smallest vertices seems different:

Those 2 graphs in cdf: dropbox link .

The size of graphs is set to 'Large'. How it can be fixed? I need that to do some animation. Thanks!

Answer 1

Use Scaled sizes to set size relative to displayed size:

Row[{CompleteGraph[5, VertexSize -> .5],
  CompleteGraph[10, VertexSize -> .5],
  CompleteGraph[5, VertexSize -> Scaled[.1]],
  CompleteGraph[10, VertexSize -> Scaled[.1]]}]

You likely want to force your graphs to have the same width using ImageSize->w

Answer 2

The reason you get different vertex sizes is that by default the specified vertex size is interpreted in terms of the minimum distance between vertices. This is probably done so that no vertices touch or overlap as long as you have the vertex sizes smaller than 1.

g1 = Graph[{1 -> 2, 2 -> 3}, VertexSize -> 1/3, ImageSize -> {300, 300}];
g2 = Graph[{1 -> 2, 2 -> 3, 3 -> 1}, VertexSize -> 1/3, ImageSize -> {300, 300}];
Row[{g1 // Framed, "  ", g2 // Framed}]

The trick for many of these problems is to use Scaled as mentioned by ssch in his answer. This scales the object in terms of image size. Its use for VertexSize is undocumented on the VertexSize doc page, but follows logically from many of its uses in other Graphics stuff.

However, this means that you now have to find a suitable size yourself, whereas the designed scaling method really makes sense for graphs.

You could keep using scaled sizes in terms of inter-vertex distance by using the following:

sc = Min[EuclideanDistance @@@ Subsets[AbsoluteOptions[g1, VertexCoordinates][[1, 2]], {2}]]/
     Min[EuclideanDistance @@@ Subsets[AbsoluteOptions[g2, VertexCoordinates][[1, 2]], {2}]]

Row[
 {
  g1 // Framed, "  ",
  Graph[{1 -> 2, 2 -> 3, 3 -> 1}, VertexSize -> 1/3 sc, 
    ImageSize -> {300, 300}] // Framed
  }
]