First off, I cannot praise highly enough the author, and his whole site is full useful information. This is all from this page dealing with Alpha Street Behavior in a self defense context. I'll attempt to extract some of the most 'TRP relevant' insights in quote form for discussion.

What is Alpha

Being an alpha has to do with one's involvement in a group. More specifically, it's about the helping develop and maintaining a group dynamic, hierarchy and the functionality of the group. And this not just for your own benefit.

Social Standing

Being an aggressive, obnoxious, selfish prick does NOT put you in control of a situation. Nor does it improve your standing as an alpha in your community. In fact, it weakens it.

Power

Power is granted to you by the group. You don't have power unless other people give it to you. Here's the catch, the group gives you power on the condition that you look out for their needs.

Trust

Trust is a fundamental issue of being an alpha. People do not give power to someone they do not trust to look after their needs.

Alphas have extra resources

A layman's definition of stress is: The belief that you don't have the resources necessary to deal with a problem.

We thought about calling this section "Alphas are calm in the face of adversity" but the simple fact is, the reason they are calm is that they know they can handle it. They have the resources.

Faking it

Basically Caesar (Milan)'s summation of an insecure alpha is a beta dog that is thrust into a leadership position. This dog lacks alpha attributes. Most of all it doesn't have the calm assurance of a true alpha. As such it is unstable and overly aggressive.

Needs of the group

Realize humans are social animals. We need other humans in order to exist. The nature of the social hierarchy is two-fold. One is so the greatest number of people can 'get by' in order for the group/species to survive (it organizes and protects the group). Two is it allows individuals within that group a place of security to obtain what that individual needs in order to survive.