If you export a Figma file, (containing a vector network) as an SVG, you will see how Figma implemented this feature. The SVG will contain all the individual lines in the network, and there will also be little overlapping generated fillets to implement all the joins. I am pretty sure that method isn't just how they do SVG exports either. It's very likely that this how they do the rendering in the app as well.
If you know a little trigonometry, it shouldn't be all that hard to reproduce their method yourself. Where it might get complicated is if you need to support rounded corners etc. But if you don't need that, then the task of implementing the fillet generation doesn't seem an impossible task.
I think there are probably also other ways to implement vector networks. For example by careful application of clipping paths. Clipping each enclosed path against all other enclosed paths, may get you close to a solution.
Paul