I got a prized early invite to Google+, the search giant’s highly anticipated Facebook competitor, thanks to the magic of Wired’s Steven Levy. But I could not find it anywhere.

Gmail, it turns out, decided the invite was a phishing attack and stuck it in the spam folder — where search doesn’t naturally reach — and slapped a giant red bar on the top, warning me it might not be from the sender the e-mail claimed to be from.

Gmail’s algorithm need not be so protective. Google+ turns out to be a well-designed social network, centered on the concept of Circles — the idea that relationships are more complicated than the binary friend-or-not-friend model that has served Facebook so well.

Instead of asking you to decide whether your mother, dog-walker and old summer camp pal all count as friends and sending them a request to reciprocate, Google+ has a set of circles — with some default ones like Family, Friends and Acquaintances. To start your social network, Google shows you contacts from your e-mail inbox: Gmail by default and options to import from Yahoo and Microsoft — but not, ahem from Facebook. You simply drag a person to the circle you want to put them in. You can put them in as many circles as you like, and it has a nifty — and visually very Apple-like — feature where you can click on a number of people individually and then drag them en masse to a group, an action accompanied by some swooping into a bundle of little cards, held together with a paper clip, before bouncing into the right circle.

Sharing is both simple and complicated in this model; we’ll tackle who sees what and how in a bit.

Like the familiar Facebook, users see a stream of posts from people in their network, and you can filter it by which circle you want to concentrate on — say your book club, or family, or fellow windsurfers. You can easily share photos — and the service naturally integrates closely with Google’s picture-hosting site Picasa.

Google has thrown in a few other features that seemed aimed to attract the always-on Facebook set. “Hangout” is a visual chat room that holds up to 10 people, giving the top part of a browser window to the person talking — and arranging the other participants in smaller boxes below, Brady Bunch style. To start one, you click “Hangout” and send the invite to one or more of your Circles and wait for people to stop by. It might sound odd, but it’s not hard to see it as a way to easily connect families or work teams.

Then there’s Huddle — a group text-chat app that’s currently not very useful since it only works on Android phones that have the Google+ app installed. Users of other phone OSes currently have to rely on an admittedly snappy mobile web version, but sending out an invite to “Huddle” is currently frustrating since desktop users and non-Android phone users get an invitation that doesn’t explain that they technically can’t join. Expect this to change soon so that Huddle will work across a range of devices.

But what might turn out to be Google+’s real killer app isn’t a feature inside Google+.

It’s the new black bar that sits atop every Google property including the Google.com search page and Gmail accounts. The bar replaces the old links to other Google services, putting your name in the far left corner, giving you one-click access to your Google+ profile.

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But the real key is that on the right side, you have a box that turns bright red when you have a notification from Google+. If someone has commented or reshared your post, or added you to a circle, the box lights up. When you click on it, a dropdown box appears — and in the case that someone has added you to a circle, you see their profile pic and with a single click add them to one — or more — of your circles, hide them or outright block them.

When there is action in your stream, the notification bar includes info like “Charlie Bloom commented on your post, Andrea Michaels, Bitt, Charlie Bloom, and 2 others +1?d it, Andrea Michaels reshared it.”

And even easier, if someone comments on a post you have made or comments after you do on someone else’s post, you can reply directly from the notification drop down window, without ever leaving the other Google page you are on.

And there’s of course a share box on the far right side that makes it simple to type an update, add a photo or send a link to any of your circles or publicly to Everyone (the box defaults to remembering which group you last shared with and the option to share with Everyone is dead last.)

Which is a simple way of saying that every time you do a search or check your Gmail, you’ll be just inches away from Google+ — without it being tacked onto Gmail the way that buzz was.

And given how popular Google is — it claimed more than 1 billion unique users last month — that little red notification box and share box is likely to be the siphon that pulls people into and returning to Google+ — mostly without even having to visit the site itself.

Of course, there’s no reason to comment from outside, if what’s inside isn’t interesting. But so far, early reviews by Alpha geeks are very positive, lauding the real-time speed of communication (seemingly borrowed from the now moth-balled Google Wave product) and the camraderie of having a social space that’s somewhat in-between the models of Twitter and Facebook — though Google+ does let you publish to the world, where it will show up on your Google profile.

So far there’s been no glaring privacy issue. The e-mail notifications are a bit aggressive — and could annoy those who choose not to join the network. Posts to Circles can be re-posted by members of the group to other groups with a single click, unless the original poster takes care to find the button to turn off one-click re-sharing after they post an update. That’s fixable, but like anything else on the web, anything you post on Google+ can be easily reshared by anyone familiar with copy-and-paste so the ability to turn off re-sharing is really more of a signal to trusted friends than a technical limitation.

One other important, but so far largely unmentioned feature is that the entire network seems to be using HTTPS. That means that users aren’t vulnerable to having their identities hijacked when using the system over a wireless network. Facebook, by contrast, only uses HTTPS for login by default, though users can turn it on in Account Settings for all of their time on the site via a setting, though this disables certain features.

So what to think of it after just a few days? Well, tech prognostication is always iffy — it’s very hard to guess what will actually capture people’s attention over time, once the gloss rubs off.

But Google+ is smooth, fast and intuitive — a product that feels far more akin to the revolution that was Gmail than to its less successful initiatives like Wave and Knol.

It’s success in the end, however, won’t be about engineering — it will be about psychology and human nature.

Which is to say, Google+’s challenge to usurp Facebook will turn on whether people really will take the time — however little that is — to create and maintain circles. Facebook has successfully bet that while users say they want share more discreetly, they actually prefer to overshare — and that there are tangible benefits that come from letting an acquaintance or business partner see the photos from your weekend putt-putt golf outing.