There is no relationship.

Google could care less if you build a million good backlinks
on day one, or before.

If your site does not have 10 visits before building 3 lousy backlinks,
in one week, then you are already behind.

Unless your website is a blogspot, uses GA, or adsense, then
how on earth does google keep track of your traffic? Since the
majority of websites do not fill google in....

And since the majority of traffic comes from things other than
counted backlinks....why would there be a connection?

Paul

It doesn't matter.

Counting backlinks is pointless. You could have a single backlink that's stronger than 1,000 junk links.

Also, Google doesn't crawl all the backlink pages at the same time so, it doesn't matter how fast you build links. Look at your server logs for your own domains, Google doesn't crawl the entire site on each visit.

There are actually a lot of factors google consider, not just the visits nor the backlinks. But if you just think about how a powerful relevant quality backlink compared to a bunch of irrelevant backlinks, right? See the point?

I've literally have seen case studies where people spam 10's of thousands of backlinks to a new site and they have achieved medium- high success ranking their keywords depending on how competitive it is.

There is no connection between how many backlinks you have and how your new site is in determining whether it will get a "penalty" or sandboxed if these things actually exist. The bigger determining factor in the case studies i've seen is consistency of the how many backlinks your site is receiving day to day.

No.
Let's say you have an interesting and beneficial content from your site and people share that story because it could help a lot of users then Google will consider it.