An interesting experiment to try is to spend one day walking around making eye contact with everyone who passes by, and another walking around not making eye contact with people.
In my experience, my day always ends up being better when I'm making eye contact. Not only do I not bump into people (because I can tell where they are going), generally you get better responses from people, as they see you as being more approachable/honest/etc. People remember you.
On the flip side, if you have a desire to be invisible, walk around not making eye contact. Hardly anyone will remember who you are.
I did this for the first time a few months ago and it was incredible. It made me more confident and especially during conversations where I'm able to hold someones gaze while talking. It's also quite powerful when you do this with a member of the opposite sex with a sly smile on your face. These days whenever I'm out I'll be scanning hungrily for anyone to make eye contact with me. I've had stare down battles with people on the tube, a game of first one to look away loses. Battles can last between 5-10 seconds.
Level of acknowledgement of strangers is a function of population density. In a big city you can't possibly acknowledge everybody you pass, so you acknowledge nobody. There's one exception: people who go for walks early in the morning are often keen to greet one another; partly because you might be the first person they've seen today, and partly because early risers are somewhat of a secret sub-society within the city.
Combine the "small town" thing with the "early morning" thing and you've got trouble, though. In a small town in Maine at six in the morning I encountered a guy who thought it was incredibly rude that I wouldn't stop and discuss (a) the weather and (b) where I was from as I walked past his house. Sorry dude, there's four million people in my home town, can't chat with all of 'em.
In my experience, my day always ends up being better when I'm making eye contact. Not only do I not bump into people (because I can tell where they are going), generally you get better responses from people, as they see you as being more approachable/honest/etc. People remember you.
On the flip side, if you have a desire to be invisible, walk around not making eye contact. Hardly anyone will remember who you are.
That's purely anecdotal, but try it for yourself.