Do you want the simple answer or the complicated one? *G*
The simple one is that there are no "ranks" in my home. Even within the "pack" (group) of dogs at the house (all 14), there is no clear hierarchy between the dogs. There is one dog that co-exists with, but doesn't interact at all with the other dogs, and they don't interact with her. There is one dog that will play with this other dog, but who will also play with the Schnauzers. Even within a house of intact animals, where breeding rights are at stake (in their eyes), there is no real "alpha" in either male or female, or between. There are certainly roles, but the roles are super dynamic and there is give and take within all of the relationships. There is never a "This dog is _____" or "This dog is ______".
Now, that being said, if I used the same rules that some people do to apply positions, I could likely do it, and people who looked at the dogs would agree within that paradigm, as I do understand how people are doing it and how they assign ranks. But in reality they are false positions that don't reflect at all the dynamic relations the animals have, with each other and with humans.
As it is, people themselves can't agree with what a "dominant" or "alpha" animal is (and by people I mean people here as well as professionals, as well as researchers!). Some people believe that this is the most calm, laid back animal, who other animals defer to because they want to. Other people say that their dogs "show dominance" to the other dogs, in effect "making" the others inferior by "being" superior. Yet other people discuss how their "dominant" dog is pushy, tells other dogs what to do, etc. All this information conflicts with each other, which makes for impossible discussion.
In the end I observe the behaviours themselves, with no preconception of rank, just the behaviours, and in doing so the fluidity of interactions between animals are amazing, so amazing that none of the relationships can be put down to one status position or rank. They all bring unique aspects to every relation (dog-dog and as a whole social unit), each dog has various roles, each role of which is as important as the next, and can't be quantified into alpha/beta/omega terminology. To give a dog a rank would be to say that "this dog" is "this way" as though you can list out the variables easily. To me it's not that one is dominant over another, or that one is submissive to another, but that with every interaction the strengths and weaknesses of particular dogs come together to form this cohesive unit that makes both live in mutual harmony, and makes the two stronger than either one on their own. Each dog expresses themselves, their likes, dislikes, and certainly control, to another dog, and the other dog takes that and uses it. How it uses it depends on its own, I suppose you would call it personality and wants, likes and dislikes, and control issues. Dogs as a whole tend to strive for peaceful interactions, and will modify their own behaviour (despite apparent "rank";) to try to attain that peace.
I just like to think that the role of the domestic dog is more dynamic than that of the pack wolf, that the hierarchies we may see in wolves (and people have even argued about those! If we can't agree on that, how can we agree on our own pets!) do not match up with what we see in our every day lives with dogs. When I sit down and look at every encounter as it is, and watch the communication between them, it becomes stunningly clear to me that it's not a linear relationship by any means, and that it's impossible to pick out one clear "leader" over another. And in fact, it's not that leadership doesn't occur, but that different dogs within the home lead in different ways, so that each is dependent upon the "leadership" skills of another, for different tasks or roles, to function as a cohesive group. And then there are dogs who don't lead in any way, but are happy just to go along for the ride. And then there are dogs that don't lead nor follow, but just totally do their own thing despite what other dogs in the home are doing.
Others don't see that, and that's perfectly cool. But I'll continue expressing how I see things (and how others see things too) on the off chance that other people may set aside the ideas of linear hierarchy, even for a little bit, to simply explore the possibilities of something more complex, and something much more dynamic.
That was meant to be simple....and honestly I would be happy to discuss it at more length if it interests you, it's really not something you can wrap up in one paragraph as it just involves sooooooooooo many variables and discussions *G* And adding to that that I don't even have a firm grasp on it, because I'm always observing and always watching, and always learning, and the more I do that the more I'm convinced that the relationships and interactions between dogs are not something we can ever really quantify or qualify, because they are so dynamic.