This is a question best answered with real data by someone like Stormy Shippy, based on how much Twitter/Facebook/LinkedIn/Google+ graphs predict Quora graphs. I think Google+ is likely to be the most closely correlated.

There is no real definition of "social network." The "social graph" is just the abstract name for the who-knows-who construct that nobody can really see. Every social network technology is basically an aspect view of the social graph.

Quora... I think you'll find two aspect-patterns mixed up. There will be the who-likes-who pattern that's similar to existing networks, but there will also be the social-filter aspect (who is using whom to basically filter the stream).

In a way, this question misses the point of how society works. Every edge on a social graph is a relationship that can be mined in infinite ways, limited only by your imagination. You don't even need two people to know each other. They may be related without knowing it via some common factor, like last name say, which is of predictive value. Here are some "social networks" that have not yet been built. Call 'em "social graph aspect views" if you want to remind yourself about what they are:

1. Who hates whom
2. Who is related to whom via less than n blood links
3. Who has ever been within 100 yards of each other more than k times (this one is kinda intriguing for location-based services... if two people keep haunting the same places, there may be reason to connect them; at least that's what Foursquare may be betting)
4. Who has taught whom
5. Who outranks whom (a partial order graph that would basically be the world's org chart)
6. Who imitates whom in fashion (this is a hugely valuable one if someone can figure out a way to surface it)
7. Who listens to whom in various types of purchase decision-making
8. Who shares a religion with whom
9. Who shares a last name with whom
10. Who shares a birthday or star sign with whom

You could go on all day. People who complain about the YASN effect ("yet another social network") don't get it. The social graph has potentially infinite depth. Each time you surface an aspect that has commercial value, you've got yourself a stake in the game. The mistake most people make is not in building YASNs, but in building YASNs that are neither valuable enough, nor fundamental enough that it isn't within the elastic scope of an incumbent. Facebook for example, is straining its brand elasticity lately, trying to cover Q&A and the more newsy/small-group-permissions style Google+. I think they are at their aspect limit.

Think of it as the equivalent of land. Some people think of land ownership as an atomic thing, but really, land is a potentially infinite set of aspect uses that we generally understand in terms of "rights"

1. Live-on rights
2. Fly-over rights
3. Right of passage
4. Eminent domain
5. Drill-for-oil rights
6. Drill-for-gas rights
7. Mine-for-gold rights
8. Drill-tunnel-under rights
9. View rights (an interesting non-local right, since your neighbor might build something that will block your view)
10. Wind-farming rights
11. Agricultural rights
12. Riparian rights
13. Air pollution rights
14. Build-on rights
15. Heavy-metal contamination rights
16. Sound/light pollution rights

A way to think of both cases is in terms of modeling. You start with finite dimensional models of some rich real-world phenomenon, and keep adding dimensions that define it a little better each time.