What is the most important factor that affects social relationships in primates?
Kinship!
What's most important in affecting kinship patterns; what determines who lives with relatives and who doesn't?
Dispersal!

Whether or not you live with kin depends on which sex disperses. The classic story with old world monkeys is that they're like most mammals in that it's the males who disperse and the females who remain in their natal group. You get a female bonded social organization. There are several matrilines in a group. (Matrilines are females who are all related through their maternal lines.) The males who are there are immigrants or are juveniles who haven't emigrated yet. The females form the social core of the group and they have a stable dominance hierarchy. (The male dominance hierarchies are usually really volatile.)

Let's compare gelada baboons and hamadryas baboons:

In hamadryas baboons, females disperse and so they have weak bonds with each other. Most of the strong social bonds in the group are between the male and his females. If the male disappears, the whole group will usually disintegrate.

So while their social organization is pretty much the same, because the dispersal pattern is different, their kinship and social stuff are different.

A quick review on dispersal:

So in most cases it's advantageous for a male to move on. Sometimes they stay in a group only a short time and sometimes they stay a long time. When females reach sexual maturity, the thing that will affect whether or not inbreeding will happen is whether their dad is likely to still be in the group. In some species, males tend to move on before their offspring reach sexual maturity. In these species, there's no benefit for females to disperse, so they don't. In other cases, the males remain in the group long enough for their offspring to grow all the way up. In these groups, if the females stayed, they'd likely end up inbreeding- so they disperse.

Vervets and Gorillas- two example species

Vervets

Aggression

In vervets, the most common forms of aggression are supplants; one individual walks towards the other and makes it leave where it was sitting. These are pretty low-key. To get an idea of how often these kinds of things happen, here's some data from Dorothy Cheney:
She watched 75 individuals for 225 hours total. She saw about 13 aggressive interactions per hour. Unrelated individuals were primarily involved.

What are the contexts of aggression? 11% were over food or water; about 20% were over access to preferred social partners; for the rest there was no obvious resource being fought over- most were probably just one asserting dominance over the other.

Cooperation

One of the most conspicuous forms of cooperation is grooming, mostly between kin, within matrilines. Also, high-ranking females received more grooming than low-ranking females; You may remember seeing data before about how likely a female was to go to a non-relative when they called for help depended on how recently they'd been groomed by that monkey. So, if you're grooming to ensure future agonistic support, then you'd rather cultivate a high ranking individual who could help you out more.
You also find alliances, mostly between kin. When a fight breaks out and someone comes to aid one of the contestants and forms a coalition with them, the recipient of the aid is usually the higher ranking female of the two fighting- this data was based on 666 interactions where two females were having a conflict and one came and helped. 89% of the time, the recipient of aid was the higher ranking female. So, usually they like the support the winner, not the underdog.

Gorillas

Aggression

Recall that they have female dispersal- so the females are for the most part unrelated to each other. This is from data collected by David Watts. Most aggressions take the form of lunges, chases, and displays; a lot of nonphysical contacts. Also, they engage in shoving and hitting as well. Again, most aggression is low-level, not all out fights. In gorillas, most of it is just vocal threats- they have two vocalizations they use in agonistic encounters, called 'screams' and 'pig grunts'.
There were .26 displacements per hour (n=971 h) So only one displacement every 4 hours or so.
There were .9 harassments per hour (n=586 h) (harassment=low-level threats, not necessarily over a specific resource or anything.)
Again, unrelated individuals were the primary participants.

Another factor that turns out to be important in with gorillas is how long you've been in the group;

Frequency of harassment	Resident	Immigrant
Give Harassment	225	50
Receive Harassment	50	225

So the residents are giving a lot of harassment but don't receive very much, while immigrants take a lot and don't dish it out too much. Older females just couldn't care less about the new ones coming in; they're just more food competition.

As far as contexts of aggression, there is some competition over feeding even though gorillas don't have a lot of food competition. Some intolerance of proximity, too; they don't like someone to come too close. Protection of infants, or course, and lastly unclear reasons which is termed harassments.

Another determinant of aggression is where you happen to be; feeding was 15%
most of other was when they were resting- if more than 2m from silverback, 74% but resting less than 2m from silverback was only 4% of harassment.Other was 7%.
So, the silverback male often intercedes to end aggressive encounters between females. It's not in his interest for them to fight since they'd use up energy fighting that they could use to raise his kids. So they're unlikely to be harassed when sitting near the male.

Cooperation

One conspicuous form is grooming but you don't see a lot in gorillas relative to other old world moneys because they're not related to each other for the most part. There are some close relative but they're the exception.
Another form is alliances- they're very structured by relatedness. David Watts saw 59 interventions by a third who formed coalition with one of two fighters. In 56 of those 59 they were going to help kin:
25 mom <-->daughter (r=.5)
7 full sisters (r=.5)
5 half sisters (r=.25)
1 otherwise related. (r=Very slight)

What proximate factors affect patterns of female competition and cooperation?

• Kinship: Kin engage in serious aggression less often than non-kin. Also, a substantial portion of cooperative acts are performed between kin.

• Rank: High ranking females are more attractive social partners.

• Reproductive state: Lactating females are attractive social partners for both immature and adult females- they receive a lot more grooming than when they're not lactating. This is probably mostly because the others are trying to get close to the newborn so they're being nice to the mom.
  Also, sexually receptive females receive more grooming and give less, and are also more likely to receive agonistic support from males against other females.

• Immigration status: Resident females harass and attack immigrant females in both gorillas and chimps, for example.

Evolutionary effects of female competition and cooperation

• Better access to food
  
• Better feeding efficiency since they get interrupted less
  
• Receive more grooming
  
• Get harassed less
  
• More likely to receive agonistic support

How this translates to evolutionary effects:

Robin Dunbar studied Gelada baboons and compared rank to matriline size. Found that a large matriline meant higher rank. Higher rank means less harassment, which means less energy wasted fighting. Thus, higher rank increases the number of offspring that an individual has.

Richard Wrangham studied vervets. It was a severe drought. Vervets always sleep in trees at night, and some of the trees are closer to the rare water than others. Because it was such a drought, it became important to survival how close your tree was to the water, and he found that the higher-ranking individuals got the trees closer to the water. During this drought, more lower-ranking vervets died than higher-ranking; Three out of four higher ranking survived, while only one out of four lower-ranking survived.

Fertility has been linked to higher rank in vervets, macaques, and mustached tamarins, to name a few. In all of these, high-ranking females gave birth more than low-ranking females.