Tipping points and models of collective behavior

A mathematical explanation for why Doja Cat's "Say So" was a TikTok sensation

This week’s newsletter builds off of last week’s installment on Schelling’s model of segregation. If you haven’t read it yet, I’d highly recommend checking it out for background context.

Threshold models of social contagion

Recall that the primary takeaway from Schelling’s model was that system-wide patterns can arise agnostic of individual preferences. Changes in a system can be driven by the dynamics of the system as a whole as much as they can be impacted by the choices of individual agents.

Granovetter proposes a complementary extension to Schelling’s work in his paper “Threshold Models of Collective Behavior”, which shows how individuals’ behavior depends on the number of other individuals already engaging in that same behavior.

I’ll explain the model in greater detail below, but there are two key differences between these two models: the distribution of preferences and the composition of the population.

  • In Schelling’s model, the same threshold preference is shared by different types of agents within a system

  • In Granovetter’s model, different threshold preferences are shared by the same type of agent within a system.

The second-order implication here is this. Schelling shows that individual uniformity does not lead to collective uniformity, while Granovetter shows the converse — that collective uniformity is not indicative of individual uniformity.

How the model works

At the highest level, Granovetter’s threshold model can be understood through the three diagrams below. It can look like an overwhelming hodgepodge of letters and variables at first, but I’ll try my best to break it down in this section.

A. Each agent in the system has a threshold at which they will take action

This diagram shows that at a certain threshold specific to an individual (represented by phi on the x-axis), that individual will adopt a behavior, i.e. take action. The threshold in Granovetter’s model, similar to the threshold in Schelling’s model, represents an agent’s preference for taking action given the behavior of other agents. Outcomes are represented in a binary 0-or-1 fashion, meaning that there are only two states of the world — action and inaction.

B. There is a frequency distribution of thresholds in every system

The diagram shows a system where thresholds are distributed in a normal/Gaussian manner. What this means intuitively is that there are a lot of individuals with middle-of-the-road preferences, but not that many that are incredibly radicalized or many that that will not take any action.

C. Agents take action based on personal thresholds and the actions of other agents

Throwback to high school calculus — diagram C results from integrating the frequency distribution in B. A cumulative distribution function! Intuitively, this reflects a chain reaction in which the participation of each incremental individual causes the threshold of another individual to be surpassed, subsequently causing that individual to participate as well. This is a critical mass model in that after the function in diagram C intersects the y = x line, the entire system “tips” to 1, meaning that everyone in the system partakes in the behavior. This is the point at which the niche becomes the norm.

The nature of the frequency distribution directly affects that point at which the system “tips” and the behavior becomes “contagious”. A simple example — a right-skewed distribution produces a lower threshold for tipping than a left-skewed distribution. This is because the median individual in a population that is right-skewed in distribution has a lower threshold for adopting a behavior than an individual in a normally distributed population, meaning that it’ll take fewer other adopting individuals in the system for the median individual to also then adopt.

Implications

Similar to Schelling’s model of segregation, Granovetter’s model of social contagion is powerful in its simplicity and elegance.

This was meant to serve as a (hopefully) simple introduction to one of my favorite research papers and mental models, but there are some incredibly interesting subsequent studies in the literature worth exploring. Watts’ “A simple model of global cascades on random networks”, which challenges Granovetter’s fundamental assumption of every agent in a system having complete information about the actions of every other agent, is one such example.

Extensions to the model aside, the fundamental thinking behind Granovetter’s work serves as a useful framework for understanding why people join some revolutions but not others, or why some TikTok dances become fads but not others.

Moreover, through the lens of another project I’m working on (Across The Lines podcast) Granovetter’s model is also a humbling reminder of how myopic the individual-centric narratives often found in Western society can be. As much as we’d like to subscribe to an unshakable belief in the power of the individual, threshold models of behavior show that ten times out of ten, complex systems behave in a way that doesn’t necessarily correspond to individual preferences and behaviors. In a poetically circular manner, the individual is both a product of and an input into the crowd.


If you couldn’t already tell, I’m a massive nerd when it comes to behavioral economics, computational social science, decision making, etc. I’m planning to write about more topics similar to this one in future newsletter installments — subscribe below to stay in the loop!