It is difficult to exaggerate the importance of eye contact when it comes to social dominance and psycho-sexual rank. A significant amount of human non-verbal communication takes place via the eyes; one can readily perceive intelligence or the lack thereof by nothing more than a one-second exchange of eye contact.
Deltas, Gammas and Omegas habitually avoid eye contact, especially with the opposite sex. Alphas, on the other hand, tend to turn it into a dominance game. Therefore, to communicate high status to the opposite sex, it is important for men to not look away when a woman happens to notice them looking at her. This doesn't mean one should leer or stare, (and by all means, do not smile), all that is necessary is to hold eye contact for about one second before deliberately blinking, then looking away. It is important to blink first, before looking away, as this sends the clear message that one is consciously deciding to break eye contact, not instinctively or reactively showing embarrassment at being "caught". The slower one blinks, the more deliberation it conveys.
The technique is extremely effective in same-sex dominance challenges too. That is the situation where I first learned to utilize it in a conscious manner. When one is being challenged in some way, either verbally or physically, it is a powerful dominance move to remain silent and expressionless, lock eyes for two or three seconds, then deliberately blink and turn away. This conveys contempt and is usually received as a message that one is not afraid of the other individual and his challenge does not even merit one's attention. An audible sniff and slight curl of the lip can serve as an exclamation point if one wishes to actively assert dominance rather than simply reject the challenge. Warning: such an action does run the slight risk of sparking an immediate verbal or physical attack.
On the other hand, the small minority of men who are naturally dominant may need to deliberately avoid eye contact on occasion in order to evade accidental dominance clashes. Usually, a simple acknowledgement nod of the head, (be it upward or downward) is sufficient to defuse any potential clash between strangers. For example, I was at a big theme park last year and there was a tall, socially dominant man running one of the rides - in his robes, he looked like an aristocratic monk straight out of the Middle Ages - who was clearly enjoying his role directing traffic and making flirtatious small talk with the many mothers passing through. What I found interesting as I observed him was that he was constantly scanning the crowd as he engaged the customers; when he saw me watching him from our position about 20 rows back, he froze for a second, then nodded in impassive acknowledgement. He also made a point of addressing me, rather than Spacebunny, when we reached the front of the line. Obviously, he was an alpha (at the theme park, at least), who mistook me for a fellow alpha, but that happens on a regular basis to sigmas.
Human beings are predators and predators always watch their prey. This is why women correctly find men who watch them without expression to be either frightening or arousing depending upon the man's psycho-sexual status. Eye contact is the Force Recon of social dominance, so its conscious management can be useful.