20bits by Jesse Farmer

Why and how do people adopt new behaviors? Why do they start using new products? Did you sign up for Facebook because all of your friends were on it, or because a specific friend recommended it to you? Or do you refuse to sign up at all?

In this article I’m going to outline two models that describe how new behaviors, ideas, and messages propagate through social networks.

The Threshold Model

The first model is called the Threshold Model.¹ It says that people adopt a new behavior bceause a sufficiently large proportion of their friends have adopted that behavior. Early adopters have a very low threshold, say 5% or 10%, while late adopters would have a much higher threshold. Every person, however, has their own individual threshold.

For example, my girlfriend’s stated reason for signing up for Twitter was that “all my friends were using it.” And during the 2008 US Presidential election, some Obama supporters would adopt Hussein as their middle name.² When I saw that lots of my friends were doing it I was certainly tempted to do the same.

The underlying psychological principle is one of “missing out” or “when in Rome.” The key variable here is the initial distribution of thresholds across a social network, which describes in totality the final extent of the behavior.

It’s worth noting that this model says nothing about how people initially adopt behavior. That is, it says nothing about innovators, only about the spread of innovation through a social network.

The Cascade Model

The second model is called the Cascade or Word-of-Mouth Model³, and is the method of “viral growth” that most social application developers are familiar with. It says that every person has a chance of adopting a new behavior whenever one of their neighbors adopts it.

This model describes phenomena like product recommendations or user-to-user notifications on Facebook. The probability that a person adopts the new behavior is the conversion rate for the notification.⁴

This probability is both a function of the sender and the recipient, so more influential people are more likely to convince you to adopt a behavior (or purchase a product, or install an application).

Practical Implications

Both of these models describe facets of real-world interaction on social networks. My take is that the cascade model is more accurate at the beginning of a social network’s life, where behavior is spreading through sparse areas, connected by influencers. Later on, after a critical density has settled in, people start adopting the behavior because everyone else is adopting it and there’s a social cost to not doing the same.

We see this pattern in services like Facebook and MySpace, both of which got their start by harvesting emails and spreading through word-of-mouth (and spam) across a social network.⁵ Eventually each network reached a point where a sufficient number of people were familiar with the product and new users adopted it not because their friends recommended it (the cascade model), but because there was a social expectation that they do (the threshold model).

Also, with respect to analytics and viral growth, the threshold model is more difficult to track. In the cascade model we record who sent what to whom and which messages they responded to. It’s clear who gets credit for a user’s conversion. In the threshold model you have to track passive exposures, and there’s no clear causal relationship.

If ten of my friends are doing something and I decide to start doing the same thing, who gets credit? Most analytics packages will show this behavior as a direct visit, with no connection to other users’ behavior, even though there is a viral process underlying it.

In short, the threshold model requires a certain level of behavioral density, while the cascade model doesn’t. However, we see both models expressed in how people actually adopt new behaviors in social contexts.

Formalisms

In the threshold model every person u has a threshold

and each of their neighbors v is weighted according to

If

then the person u adopts the behavior.

The set of thresholds, weights, and initial adopters completely determines the extent of the behavior in the social network.

In the cascade model, for every person u and neighbor v there is a random variable

which describes the likelihood of u adopting the behavior if v has adopted it.

Takeaways

I’ll try to boil all this down into a few, practical takeaways.

1. The Threshold and Cascade Models describe two mechanisms of behavior adoption in social networks.
2. The Threshold Model says that people do something if enough of their friends are doing it.
3. The Cascade Model says that people have a chance of doing something if one of their friends is doing it.
4. Both models correspond to different real-life adoption patterns.
5. The typical “viral loop” involves the cascade model, but most successful social networks rely on the mechanics of the threshold model in the long run, i.e., density is important for long-term success.
6. The cascade model is a good tool for analyzing acquisition scenarios, but the threshold model is probably more helpful for understanding retention and engagement — it at least implies that density is a key factor in social network growth, a metric that’s not often discussed publicly.

Agree? Disagree? Leave a comment, send me an email, or follow me on Twitter!

1. See Threshold Models of Collective Behavior (1978) by the famous sociologist Mark Granovetter. [↩]
2. See Obama Supporters Adopting Middle Name “Hussein” As Their Own [↩]
3. See Talk of the Network: A Complex Systems Look at the Underlying Process of Word-of-Mouth (2001) by Goldenburg, Libari, and Muller. [↩]
4. More accurately, we’d model the “probability” as a random variable whose mean was the conversion rate. [↩]
5. See Stealing MySpace: The Battle to Control the Most Popular Website in America for details about the MySpace team’s background in direct marketing. The ConnectU vs. Facebook court documents, which you can find via Google, paint a similar story for Facebook’s early years. [↩]