The 1st two could well mean what you have indicated. But I would say "XXX" company has the most number of network in the world. ‘A large number’ is not the same as ‘the biggest’ or ‘the largest’. The 3rd one might well mean greatest in qualify, reputation, service and so forth but not in quantity.
There must be a lot of network companies in the world. Having ‘a large number of network’ is just one way to describe those large companies, including XXX. XXX company has the largest network in the world means not only does it have ‘‘a large number’’ of network, it has the most and it’s the largest of them all. Let say we compare the largest 3 in the world XXX, YYY & ZZZ Co. They all can have a large number of network, but XXX is also the largest.
This is how I interpret your sentence. I could be wrong.