After much trial and a lot of error, I finally think that I have gotten Influence Strategy down pat.

Influence, like much in the game, is exponential, not linier. The best analogy I can give is it works like gravity for each of your planets and bases. Each of your worlds generates a certain amount of Influence Points (ip's). This influence spreads out around each of your planets like a gravity well, powerful close in, but drops off rapidly the further you get away. This means that twice the distance = 1/4th the influence. If two of your influence spheres overlap, they add together. If your and a rivals spheres overlap, they subtract from each other. This leads to the wierd shapes you see on the influence maps.

Now, there are three factors that determine how much influence a planet has. The first is population, the second is influence improvements, and the third is your civilization wide influence bonus. The civ-wide bonus becomes very important in influence conquests as it is applied equally to all of your planets and influence starbases.
On starbases, influence works completely differently. Each starbase has a set amount of influence that is applied evenly over its set radius. This is a very small amount and is easily drown out by any nearby planets. An Influence Starbase starts out like this, but quickly changes as soon as you add an influence module. Its Influence becomes the Civ-wide influence bonus multiplied by 1 + the starbases influence bonus. This allows an Influence starbase to project in its radius nearly five times (4.97) your civ-wide influence bonus. This leads us to how to use influence to conquer the galaxy!

Influence Starbases are for for one purpose only! To influence rival civs worlds to defect to you. (Yes, you can fortify them and put sensors on them too.) An Influence starbase that doesn't have any rival worlds within its radius is completely useless! They have no effect upon your worlds at all. Influence Starbases go wherever there are rival worlds within your civ's influence or are right at the edge. Note that more influence does not necessarily mean faster defection. Once your influence is 4 or more times their own civs influence (Which is noted in the planet small display just after morale,) it has a chance of defecting. There is no more chance at 4 times than at 20 times as far as I can tell. It also does not matter how long it been at this status either.

This leads to a discussion of how to dominate through influence. If the main thing you like in a game is blowing things up, this is not the strategy for you. If you are a micro-manager who loves to tune each world into a well oiled machine this is the ultimate strategy for you. This is a patient strategy that is much like gardening. You must plant the proper seeds, tend your crops carefully, and wait for your fruit to ripen on their own. The idea is to create a group of strong balanced worlds that are a beacon that lead others to join in your ways. It is the way of passive aggression that absorbs friend and foe alike.

Your homeworld is your template that each world should be modeled after. In a standard gane it is a PQ 11 that will eventually upgrade to a PQ14. The first thing that should be built is a factory. The first factory on every world should be bought as soon as you can afford it because it allows everything else you do to happen faster. The suggested build order is:

Factory (bought)
Starport
Economic
2nd Factory
Research
Morale
3rd Factory(midgame once you have gotten over the first economic hump)
Influence
Farm
After this repeat until you run out of room:
Factory
Research
Economic
Influence

Notes on building

Until your economy stabilized out mid game you need to modify the list a bit. The early game will not support more than 2 Factories and a Research on your planets without running out of money.
Early in the game, it is useful to build up to 3 additional economics improvements once I have built the initial set, some of which I will replace after my economy stabilizes. I always build the appropriate buildings on bonus tiles even if this means deviating from the initial pattern.
The exceptions to this are:
Farm bonuses, which I never use because they kill you later in the game.
Multiple morale bonuses.
I build a morale on the first one and economic building on any additional morale bonus tiles. This works out the best when you get to the Stock exchange improvement.
When you get to the repeating section of improvement buildings, keep the total number of buildings equal. This means skip the Factory Buildings the first couple of times because you should have alread built several of them.
Also factor bonus tiles in counting how many there are. I.e. 100% bonus = 2 buildings, 300% = 4 buildings, and 700% = 8 buildings.

Research:
As a general rule, research whatever will be finished the quickest. Priority shoud go to Propultion, anything that gives you a civ-wide bonus. Military technologies should only be researched when they are quicker than anything else to research.The reason military is last is because everyone has to research military technologies before they can attack you, so you are safe for quite some time.
AI civs usually concentrait on one tech branch to the exclusion of all others. The well rounded approach gives you a better platform that will catch and pass their concentraited approach in every area by midgame. This also leaves you plenty of outdated techs to trade for anything useful that they get before you, or to sell when cash gets low.
One last note: I always give away all planetary upgrade tech (like soil enhancement) and trade techs as soon as I get them. The reason for this is that this actually benifits you for them to have these. The AI always gives priority to building planetary upgrades, which actually slows down their empire building, and gives you a better planet when it defects.
Trade serves a threefold purpose when given away. It makes them friendlier for the gift, it doubles you trade income on average, and it also tends to pacify them so that they do not attack you.

Colony Rush:

The colony rush will determine the initial size and stength of you civ for the first half of the game.
(read my post on The First turn here in the strategies section)
(PS. if anyone could show me how to link to another post I would appreciate it)

I use as aggressive policy as I can because this has the best payoff later in the game.

Each planet builds ships in this order:
Defensive ship (when it becomes available)
Survey ship (until I get 6-8 of them)
Colony ship (as soon as the planet's population hits 1.5 billion)
Constructor (as soon as I run out of colonizable worlds)

I always upgrade my colony ships as fast as I can, with the most engines of the best type I have as often as possible. (I do the same with constructors and suryey ships built on a freighter hull as well)
Defensive ships are built with no engines or life support and as many weapons and defenses as I can cram onto it. Each world builds one defensive ship, which stays in orbit of that world permanently. These are upgraded with new versions with newer tech as appropriate. New defence ships are designed as soon as medium and large hulls become available. New ships should be built for every world when this happens. As the larger defensive ships go online the smaller ships are converted to standard warships and sent to rally points where they can react to any aggression.

Every time one of my survey ships finds a resource, I switch a colony ship under construction to a constructor and go claim it as quickly as possible. This gives you a big advantage later in the game.

Survey ships are used to expore as much as fast as possible. A trick to use for finding anomalies is to move your survey ship first, then hit the 'a' key, which will show you where the closest anomily is so that you can head for it when you are ready, and then hit the 'a' key again. Getting the Eyes of the Universe Galactic Wonder is absolutely essential! After you get this Wonder, your survey ships become 10 times as effeciant. Your defensive ships also act as sentenials, their sensor radius making sneak attacks very difficult.

Colony ships are sent to the highest PQ planet you have discovered. Try to get everything pq10 or above before getting anything smaller. Other civs will come in and fill in the lower pq worlds that you don't get to: this is part of the plan. If there is a choice between two worlds that one is in a system you havn't colonized and one is in a system you already have a world in, go for the one you don't have a presence in.

Expansion after the colony rush:

After all the worlds are colonised it is time to concentrait on starbases.

Every world that has built its appropriate defensive ship should be grinding out constructors. Every cluster of three or more worlds should have a economy starbase covering them (I have gotten as many as 10 worlds under a single starbase.) Economic and Rescource starbases should be maxed out with every available module as soon as they become available.

Influence Starbases should be placed wherever there are more than one rival world within you area of influence. They can also be placed where there are rival worlds just outside of your zone of influence. Warning: Trying to get more than 3-4 worlds at a time is much less effective due to their overlapping influence zones. Once all rival worlds within their influence circle have defected, influence starbases can be scrapped, as they serve no further purpose. Influence Starbases should only be given influence modules.

Military Starbases are really not useful under this strategy and I never bother to build them.

The best way to I have found to get rival worlds to defect is the domino effect. This is where your civ's overall influence level really comes into play. Every world gets your civ's influence bonus added to its own influence no matter how small or new that world is. Each world that defects puts pressure on all nearby worlds and makes them more likely more likely to elegible for defection. If you put this kind of pressure on all your borders, about 5% of all the eligable worlds defect each turn. I doesn't seem like much, but it adds up steadily allowing you to optimise each world that defects in an organized and efficient manner. Each world that defects adds to you strength and brings you closer to eventual domination.

Alliances and war:
Alliances in this strategy are counterproductive. You tend to be on good terms with just about everyone, and alliances just drag you into wars you don't want. If a civ appeals to you for help, give them an outdated tech and they are happy.

Wars are to be avoided as much as possible with few exceptions. There always seems to be one minor race that snaps up several rescource starbases. When you are in the starbase building phase and you have available constructors nearby, get everything in position destroy all their Rescource Starbases in one turn and claim them for your own.
Offer them peace that same turn after taking their rescources or the next and they will probably accept, if not take their homeworld. This can also be done to any civ that just fails to thrive and only has a few worlds left.

When other civs go to war with each other, check out their Rescource Starbases every turn as they are a targeting priority. Snap up any unclaimed rescources as soon as they become available, even if they are nowhere near your civ.

Civs seldom become unfriendly, but if they do, give them an outdated tech each turn until they become neutral.

When a civ drops down to three worlds the chance of defection drops to about 1%. You can usually wipe them out in a single turn, but I seldom bother; they come to me eventually.

There are more tips and tricks for this strategy, but this covers most of it. This strategy works best in large+ galaxies with scattered, abundant stars and abundant worlds. Average difficulty is assumed, but can be reasonably adjusted for harder settings.

Scincerely,
Scintor