I want to print an image of a network in 3D — what's the best way to convert it? Suppose I have an image like this below that I want to print in 3D. What would be the best format to convert it in?
Unfortunately, with only a .PNG or other file format that is traditionally two dimensional, you are missing some critical information. Unless your objective is to print something "flat with thickness," you can't create a 3D model. If your objective is to print something flat with thickness, even a program such as Inkscape with the Path to OpenSCAD extension will accomplish your goal. 

I believe that Tinkercad will import images and allow you to extrude them to create the thickness aspect. Another program which supports such activity is Fusion360 (free for hobbyists) which would convert your drawing to a sketch. You'd have to add some width to the lines in order to extrude them in any program you select. It will also be necessary to trace the drawing to convert it to a vector file type. Inkscape excels in this task.

Lastly, you can arbitrarily add a third dimension to the diagram by importing it with SolveSpace. It is another free program which allows one to import a DXF file. Note that DXF is a vector file not raster, excluding your existing PNG format. It might be easier to use Inkscape to create the vector file, add width to the lines, then save it in DXF form and load it to SolveSpace. At that point, it becomes extremely challenging due to the complexity of the drawing.

Perhaps a hint or two (image, sketch, drawing) of your final objective?