I dunno.
The way I see Advent, they can (err.."should be able to") expand extremely quickly to unoccupied planets with Shield Regen on the mothership taking out tons of local militia then lock down what they have by turtling and spreading culture. The Vasari can go ahead and make attacks on other player's planets quite quickly with their Assailant at Lv1, and the TEC can survive either way with their economic advantage.
The problem is that in a small map situation you'll end up getting Embargoed/LRM rushed so quickly that if you build those two civ stations to cap a bunch of planets you will get destroyed by a military-focused player. You're then forced to play to your disadvantages: Advent suffer from pretty weak military at the beginning of the game. In a large map, it really doesn't matter how you expand.
The problem, as I see it, is twofold:
1. The early game strongly penalizes a "boom" player. With credits being so advantageous and LRMs being so far down the tech tree, the TEC can afford to boom, but other players are pretty much screwed. Vasari is pushed into rushing, and Advent are pushed into turtling: while the Vas can easily rush, if the Advent player doesn't get lucky and have a lot of resources before they get in fleet conflicts, there is no way for them to defend what they have. You have the choice of two tactical or two civilian research stations at your home planet before you have to pay for logistics. That's fine - but it's slanted heavily toward tactical before the game even begins.
2. Culture is too easy to mitigate. Only a couple times have I had to put real effort into stopping an enemy from dropping a planet via culture, most of the time it can be stopped by simply adding a culture station close. "Stacking" culture stations tends to be less advantageous for the stacker as the defender: the defender, of course, can always warp in and take out those stations instead of defending with their own culture. Advent is clearly pushed to use culture, but unfortunately the tree seems relatively useless.
The solution to #1 is to either slow down the game or make rushes more expensive while penalizing early-game mass fleets. A reduction in supply, a increase in upkeep, an increase in LRM prices would all help. The Vasari should be the ones that can rush effectively and focus on taking other people's planets rather than unoccupied ones - right now, the TEC can do it much too effectively as well.
The solution to #2 is to simply buff culture in general, or at least give much more of an advantage to the Advent's culture research tree. Someone with the tree maxed out should be dangerous with only one or two stations, not so easily countered.
In short, the Advent are screwed on defense, they're screwed on culture, and they're screwed on offense. It's still doable (I tend to play Advent anyway), but it seems to be inherently starting you farther behind the other players. Which is a shame - they're my favorite race!