Communications and Diplomacy
The Network is a very old AI, with lots of knowledge of different languages and plenty of diplomatic experience, hard-earned. It makes contact with many different species, and acts as an interpreter, arbitrator, and judge between them.
Because the Network is often the first "species" that interstellar species make first contact with, it is a favorable trading partner with many species. It buys raw materials from other nations and refines and processes them. It sells "castoffs," FTL drives and computers that don't work as well as it needs.
In modern computer manufacturing, the difference between a 4GHz and 2GHz chip is how well they did on the quality control checks. In the same way, much of the galactic FTL drive market is dominated by the Network, which sells the FTL drives that aren't good enough for it to everyone else. The galactic market doesn't have to create the huge infrastructure needed for FTL drives, and the Network can get huge amounts of raw resources, far more than it actually needs.
An alliance, and a threat
One of the first things the Network tells any race it meets is about the species that it's programmed to kill. It paints this species as horrendous, aggressive, deceitful, and xenocidial, hating and wanting to kill all aliens. This species did wipe out the Network's creators, after all*.
It then suggests that in the event of this species showing up, they should band together in an alliance and fight it together. It's not going to make a formal alliance before that happens - it doesn't want to waste its resources fighting in the organics' wars, but it will as soon as the species shows up.
*This might not actually be true in-story, but the Network will certainly claim this was the case. After all, if they've wiped out one race already, what's another?