Dr. Gordon Neufeld and Todd Sarner talk about being "the Alpha" so much in their presentation of strong parenting leadership. It's a "Papa Bear" or "Mama Goose" presence, and it flows with/from the source of my unapologetic being. My presence can be "big" or powerful simply from that confident energy.
From my own experience, the more certain we are of how this helps (aka what the research shows through Attachment Theory etc.), the stronger our Alpha sensibility. It's a deep, inner knowing of who we are as parents, and it's a kind of faith really. We are the best answer to their needs. We are the core answer to their difficulties.
Being the strong Mama is directing the energy and attention in a very visual, guided way when they are struggling to follow/behave. It doesn't mean I seek to control their every action or get into a battle with them to assert that control. In fact, if I'm even considering such a squaring off, I've already lost. It's not becoming the constant family entertainer--they need to embrace some boredom for deeper, creative play (and embrace some problems to work out solutions). That's a balance I'm playing with these days.
As a quieter person, I sometimes wonder how to be "Alpha" enough. Personally, I'm learning to reduce my thinking out loud as I reason through my decisions. It's better to think for a moment and then come out with a clear, concise decision. Here, I take Kim John Payne's advice about over-talking to heart as I consider reducing children's stress and their mental stages. (Aside from Simplicity Parenting, he has a great course called Soul of Discipline: 0-9 years and will have a book out in 2015 I think.) The Alpha concept was introduced to me through Todd's Transformative Parenting course and is based on Gordon Neufeld's ideas (as Todd was on his Neufeld Institute faculty back in the day). It was a superb course.
Some questions arise of a different nature from some thinkers regarding control. I think it is healthy, as Alfie Kohn does, to wonder why I have a need to control their behavior always (even if I'm sensitive to how I play my part). It's not a bad idea to reflect and look at where we control for control's sake (and how that relates to our own childhood issues).
When does a parent need to be that guiding star for sure? Most parents I know see the need to be the greater gravitational pull when it deals with safety, the basics of getting along in society and also in character points. (While I might not insist on every point in every moment I could, it is important to help children with what seem like little things so they build attention and habits.) Here it's not control: it's taking care.
In attachment, one person is the Alpha (even if it changes they way we take turns taking care of each other, as in marriage), like it or not. If it's not me, it's my child. When our children are in that roll, there are all kinds of troubles.
Here's an intro for a class I might take from the Neufeld Institute: