The social internet garners its masses by feeding, often simultaneously, into the most basic human needs. Our needs for social information exchange are fulfilled, we can find and cultivate our own community. Humans are hardwired to be connected. Socially we evolved to be as such because at its base it is life sustaining. We survive because somebody bit the dust eating a mushroom and somebody else learned from it and then taught us.

The other side of the success of Facebook is that it serves to entertain us. Americans watch on average 34 hours of TV a week. Most of it is nearly worthless and not a value add other than to be amused.

But, people in the course of fulfilling these tasks on FB are leaving behind information trails. We know know what they are talking about, who they are talking to, who they just friended. Now we can compile this information and do something with it - back to the purpose.

Let's take a look at a site like hunch.com - a very good use of collecting data about an individual by answering questions (a form of amusement to many) written by other members. They then leverage it to solve some problems that Amazon's rating system fails at - letting me know what user reviews are more like me. Am I more similar to people who give it 1 star reviews or 5 star reviews. I really enjoy the site and I willingly give it information about me so it can help me more.

This data is usually used as you stated to find similarities . In people, for matchmaking sites, here are things purchased by people who...read the same stuff, watch the same movies, etc. It can turn out well, but for what was known by the Netflix challenge teams as the Napoleon Dynamite problem. A movie that based on similarity clusters was almost impossible to predict how a user would rate it.

Now as you ask what are the candidate purposes for which people want the results of software that leverages social graphs.

Companies are much easier than people because unlike our list of uses and goals earlier, companies just want to make money. Facebook for example.. How do they make money? By selling advertising (other sources comprise only about 10%). How do they make more money? By entertaining more people, for a longer time, and charging more for their ads. If you can serve a targeted ad you can charge more, etc. If you know what feed information to show a user, they will stay on longer, you then serve them more ads. And as such, their purpose for using the social data and derivatives thereof will be for those purposes. Similar to amazon, the purpose of the data are to sell you more goods by recommending things you might buy based on predictions made.

So why does Facebook lack dating as part of their core offering? Because while they could be good at it they up to now have just been a platform company. This has changed a bit with the launch of FB messaging and @facebook email. But look deep into the reason why? Gmail makes Google money because they serve you contextual ads based on your content. Why does Facebook hope you use their mail? Ah yes, you get it now...so you spend more time on their site and they have more data about you so they can serve you more ads. With one small feature change you might say the showed a glimpse of who they are becoming.

Before they were a true social media company – a platform where users generated content to share. Now they want you to spend more time on the site, use it as a common login for other sites, send and receive email through it? Why..again the money. So they can do what they do best. Serve you better ads and have a longer time to do it. Every move you see FB make you will see it is now to broaden the platform and increase your dependence on it so you spend more time on it.

Facebook is becoming the new money printing machine by taking exactly the same path Google originally did – by getting people to the information they want faster. How did Google do it? Google page rank – they figured out that rather than word density – the social element of a link to the content was more important. Google is scared now because the content that is being generated inside of Facebook’s closed walls is massive and it is a powerful predictor of what is relevant.

In both Facebook and Google’s case they make money by helping us perform the transactions we desire by using social data.
Social search on the other hand, at least how I see it going, is leveraging all this data to wade through the murky garbage everybody is producing with social media and actually help me find what is relevant.
Relevancy is key because what is relevant to me I look at, I take action with, I buy. This makes other people money usually (if it does not then they usually fail to build or go under).

Again the purpose is user specific and the social data helps you determine relevancy. A matchmaking site would use it to find you a date you liked faster, a consumer retail site would use it to help you find the garlic press you are going to like faster.

Finally - back to the graph. What are your thoughts on using them? Personally I would find them useful for discovering trends in myself, my research etc. They help us explore, sort through data. But they are the end product and for most companies just a form of entertaining you with all the data they have about you.