In Robert M. Sapolsky's book A Primate's Memoir, Sapolsky suggests that alliances very definitely affect rank among the primates he studies.  At one point, for instance, near the end of the book, he describes his return to a group of baboons with whom he is familiar but whom he has not seen for a while:

You see how fast you can identify the new alpha male. You see what alliances are still intact, what friendships have solidified further, who is feuding with whom (p. 300).

Much earlier in the book, Sapolsky also comments on the relationship between rank and alliances:

. . . I would go out and study the behavior of baboons, see who was doing what with whom – fights, trysts and friendships, alliances and dalliances (p. 15).

Once again, then, Sapolsky suggests that an individual baboon’s rank within the group depends very much on its connections and conflicts with other baboons.

Later still, Sapolsky gives a specific example of the ways alliances affect rank among baboons when he notes of two animals that

To bolster his rank, Benjamin formed a coalition with Joshua, who supported him in the alpha position (p. 170).

Throughout his volume, then, Sapolsky states and shows that alliances and friendships definitely help affect the fluctuating social rankings of the animals he studies – especially among males, whose places in the hierarchy are much more fluid than are the places of females.