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In both social and neural cases, there are interesting blends of all-or-nothing and smooth ramps of transmission. Any given neuron fires or doesn't, but can fire with different intensity — and it can have a variable excitatory or inhibitory effect on other neurons. Similarly, someone can pass an opinion to someone else (or not), and then there are several other factors that come into play: how strong is the opinion, and how much does the receiver believe it. Then there are things like personality factors that moderate whether the receiver acts on the information and/or tells others about it, which starts the cycle over again.

At a simple level, you need to have people tell more than one person about a cool new idea or product to give it virality. How much more than 1 depends on the other probabilities — but higher values increase spread quickly.

Do you have any information about actual formal models of these phenomena? The more math the better!

And thanks so much for the comment — really awesome.

The random factor in the cascade model exists as part of the model only because we do not know the weight of an individual relationship in terms of its degree of influence. Put another way, if the weight in threshold model is described as a probability [0,1] that an individual will follow the influencing person's lead, then we can aggregate all such influences (not necessarily by simple addition) to describe the threshold and cascade function.

To make that more specific, if three guys at work extol a particular idea, they will each have a variable effect (based on your trust in their opinion — it's not random) on whether you start thinking positively about this idea as well. The aggregation matters: if you think highly of two of them and detest the third, the probability is less than if you trusted the opinions of all three. If however another far more influential person (a well-known blogger or celebrity) also extolled that position, that influence would have a disproportionate effect on your opinion commensurate with your trust in that person's opinion.

The mechanism bears a significant similarity to dendritic summation in neurons and probably other similar phenomena; it's not restricted to social contexts. Variable weight inputs aggregate in terms of their influence or effect, encompassing both what you have called cascade and threshold.

Oh, it's also worth noting that (as in neurons :) ) the *absence* of input can be important too — this is likely part of what drives early adopters. For them, getting high-influence input from a few trusted individuals isn't sufficient to trip the threshold; there must also be a lack of ambient input. That is, if everyone else is already doing it, the early adopter loses interest. The variance in reliance on a few high-trust inputs without ambient input or many more ambient low-trust inputs describes the spectrum from early adopter to laggard in any population.

Celebrities and the like are an interesting phenomenon in social networks. They are always highly connected, moreso on directional SNs like Twitter and MySpace, but the extent to which they are influenced or cause influence are much more ambiguous. Their effect on behavioral and opinion dynamics is probably a research topic all in itself.

And I actually wrote a Ruby script to generate BA graphs just the other day!

One feature you may be missing is preferential attachment. Celebrities like scooble and oprah have drawn in a huge volume of twitter users. Myspace had similar dynamics due to bands in it's early days. The growth of graphs under preferential attachment is formalized and it's characteristics mostly well known:

http://en.wikipedia.org/wiki/Barabási-Albert_model