I just played a game where I played the Terran Alliance.  I was right at the center of the Universe and I managed to keep all the major civilizations in a friendly disposition towards me.

I researched the techs that increased my diplomacy and 2 levels of miniaturization that allowed me to convert scouts obtained from other races into trade modules that had no engines.  I quickly had trade routes with most of the civilizations.

I build up my army by trading ships from other races.  Whenever a race would start to catch up militarily, I'd buy ships from them.  Having the most powerful army discouraged extortion attempts.

Right up to the end, when I was 10 days away from an influence victory, I allied with all of the major races except with the Korx who were warm towards me.

Then something strange happened:

The Carinoids went and attacked the Krynn who were one of my allies.  This is strange because the AI didn't appear to take into account that the Krynn had a powerful ally even though they themselves were weak. 

So I took out the Carinoids, and went on to have my influence victory.  Asides from that one skirmish.  I didn't engage into battle any of the major races in the game.

The difficulty level was "crippling" with my 9 opponents set at "Genius" level of intelligence.