You should read it as "progress or sophistication". ie. "Ahead of others in progress OR ahead of others in sophistication." Feel free to ask if that's unclear :-)

You can advance through a process, such as a computerized menu. Or you can be advanced, "He is taking a class called 'Advanced Math.'" or "The child's language skill were advanced for his age." Am I in the ballpark?

The noun does not define the adjective. "Ahead of others in progress or sophistication" defines the word "advanced". I'm not really sure what you're asking.

"Tired" is a state of mind or body. Tired is an adjective. "state" is a noun, "mind" is a noun, "body" is a noun. All three nouns explain what "tired" mean. "advanced" is "ahead of others in progress or sophistication". You seem to have problems understanding why "sophistication" being a noun can define an adjective. But "progress" is also a noun, and you don't seem to have any problem understanding that it defines the adjective. Could you be more specific as to what you do not understand?

*Tangoya - Sorry about that!