How can I create an array or 'node net' of randomly sized shapes, with thin paths between them and dots where each path meets another path?
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It would be fairly straight forward in Illustrator although not automated.
You could draw the paths with the Pen Tool however you want them. Then draw a small circle and place it at the intersections.
Conversely, you could place your dots wherever you want them, then use the Pen Tool to draw lines between the dots.
There's nothing automatic or "magic button" in Illustrator which would create this for you.
You can create it using http://dmesh.thedofl.com/ and import the results into illustrator.