Weight-painting mesh containing only vertices I created a mesh that has only vertices and converted it to a particle system representing the stars of the Hipparcos catalog. 

I used image textures to control the colors and brightnesses of the stars, with each pixel of the textures encoding the color of each star by vertex index. I accessed these indices using a Particle Info node in the material for the dupli object (a simple sphere). This seems to have worked.

I want also to control the sizes of the individual stars, but for that I can't use nodes (or at least it doesn't look like it). All the posts I came across suggest that I should either weight-paint the mesh or generate an image texture and assign it to the particle system. I did the latter, partly because I don't know if I can weight-paint a mesh that has no faces or edges.

But it seems that I need to mind the mapping between the texture and the mesh. I suspect this is the reason I haven't succeeded at controlling particle size with image textures. I don't know how to go about doing the mapping, because it's just vertices, so I'm now considering weight-painting as an option.

So my question is... can I weight-paint a mesh that has only vertices and use those weights to control the sizes of 119,617 dupli objects in my particle system?

I know the challenge is the same—it's just vertices and how do I assign each a weight or a spot in an image texture to control particle size... I guess I'm just putting one foot in the weight-painting option in case it turns out to be doable.

Many thanks in advance for helping if you can!
Yes you can. Weights are all stored per vertex and have nothing to do with edges or faces.

That said, without faces to shade, you will have trouble visualising what the weights are, so with 119,617 points, this is going to be very difficult to manage without just having a script set the correct values automatically.