ImediaConnection has an article on Word of Mouth by Joseph Carrabis  called Yes, You Can Predict Viral Marketing that has a breakdown on the parts of a viral campaign, some of which you can put web metrics measurements against.

Starting with what you can control, according to Joseph Carrabils (my comments are in blue):

1. How many individuals does the campaign need to start with (seed)?

You can start a campaign with sending to each individual a url tagged that identifies that individual as a seed person (each seed in individually tagged).

2. How fast will the campaign spread (propagation factor)?

I disagree with Joseph - how can you know this for sure beforehand?  When observing this through web analytics you could notice how quickly traffic to tagged urls increases based on who the person (seed is) and over a period of time might be able to predict how quickly viral campaigns can spread - but not the first couple of times.

3.How will the campaign spread (vectors)?

You might be able to track the campaign spreading via Blogpulse conversation tracker and tools like that (if it’s being released via a blog).  I suspect that a custom high end tool would be needed to fully track the spread of a viral message or viral campaign.  I don’t think the regular Web Analytics are really designed for this (though I could see people using it that way - with the proper tracking set up beforehand).

4.How large a group is required to sustain the propagation (viral burden)?

Over time, running several campaigns, a metric could be developed for this that says something like - you have to have and "X" sized group to determine this much propergation of your viral message.

5.What is the campaign’s goal (maintenance factor)?

Joseph refers to his when he writes ""How will we know when this campaign is successful?" Often people will answer nebulously with something like "Sales will double" or "We get a million downloads," neither of which is good enough.

6.How large a group is required to sustain the campaign once the goal is achieved (threshold point)?

I think you’d need to have data for at least a half dozen campaigns to come up with a number.

7.At what point is the campaign too successful (saturation point)?

When no matter what you do, you can’t get more of the viral message to spread (you’ve expended all your effort or all the client wants to pay for) I would think your at the point where your campaign is saturated (or you are, at least) and it’s time to stop.

Joseph Carrabis ends by bringing up guidelines that can be used for any campaign, including SEO/SEM and Web Analytics>

1. Clearly define the end goal. Write it down as a paragraph and draw a diagram (sales chart, geographic or demographic penetration, et cetera) that demonstrates the goal as recognizable numbers.
2. Clearly define the current situation. Write it down as a paragraph and draw a similar diagram that demonstrates the current state as recognizable numbers.
3. Begin "stepping back" from the goal. Use as many steps as necessary and ignore any time-management requirements at present. The purpose is to clearly "look back" from the goal to the current state and document each milestone or benchmark that existed between the goal and current state, why each was important, how they occurred, how they were passed or exceeded and what was required to ensure each step’s success.