It depends a bit on the type of network an advertising router is sending an LSA about. I'm a bit lazy so I've taken a screenshot from Routing TCP/IP Vol. 1 which describes some of the differences in LINK ID fields:
So for the broadcast network (transit network) type 2 LSA, the advertising router will be the router-ID of the DR on that specific subnet advertising its interface IP address (LINK ID).
When it comes to type 3, 5 and 7 it would most likely be the specific network address as a LINK ID and the ABR/ASBR's router-id as the ADV ID. For type 4 LSAs the LINK ID would be the router-ID of the ASBR and it would be advertised by the ABR router-id.
Check-out this for elaboration on the type 4 LSA: Where are type 4 LSAs generated? - PacketLife.net