Thread by @KevinZollman

For week 3 of my Network Epistemology class, we looked at models of "probabilistic pooling." The basic question is what happens when individuals change their mind by moving "in the direction of" the opinions of their friends.

The basic model is that people have a belief that is represented by a number and they change their belief by averaging with others that they interact with.

The model has been reinvented many times and is sometimes interpreted as a normative model for belief change.

Our focus is on an empirical interpretation: as a model of social influence. Each individual is pulled -- for whatever reason -- in the direction of the beliefs of others. This is, of course, a very imperfect model of how people change their minds, but it's a starting point.

The first paper we read is by DeGroot, who defines the basic mathematics of this model. It turns out that there is a mathematical connection between this model and the theory of Markov chains that is really helpful in proving results.

tandfonline.com/doi/abs/10.108…

(Philosophers may be more familiar with this book by Lehrer and Wagner. DeGroot anticipated them, although DeGroot wasn't the first either. It's one of those ideas that many people had.)

link.springer.com/book/10.1007/9…

The important result from DeGroot is that the society will reach *a consensus* under extremely weak conditions. That is, if people are modeled as being pulled toward one another's beliefs, you don't need much social contact in order for there to eventually be agreement.

Consensus is one thing, but that doesn't necessarily mean people are right. We could all agree on the wrong thing. To ask that later question we read this wonderful paper by @ben_golub and Jackson.

aeaweb.org/articles?id=10…

If we assume that each individual's belief is somewhat related to an underlying truth, we might ask: what kind of social networks will lead to better or worse consensus? This is the question that Golub and Jackson answer.

In particular, they consider societies that grow (eventually to infinity) and ask: what ways can a society grow that make it, ultimately, perfect at getting to the truth? (This hearkens back to the discussion of the Condorcet Jury Theorem from our first week.)

They show that every person in the group must have vanishing social influence. That is, no one can maintain disproportionate social influence as the group gets larger.

This is a very democratic ideal, I think, and makes sense given the model.

One concern with DeGroot's model is that it predicts we will eventually reach consensus. Spoiler alert: in reality, we don't always reach consensus! So how can we account for this difference without a radical change in models?

This leads into our third reading from Hegselmann and Krause. They keep the basic structure of the model but add one tweak: you start ignoring people when their opinion is "too far" from your own.

jasss.soc.surrey.ac.uk/5/3/2.html

Already, this small change makes things like perpetual disagreement possible. They also show how it can account for polarization of opinions. Their paper explores many different variations on the model and they show how it can account for many different outcomes.

It's somewhat remarkable that such a simple model is so flexible. And one thing we discussed a lot in the class is what to make of this. We know the model is wrong and misses a lot about how we influence one another. So, can we learn anything from the model?

For those who don't know, the Hegselmann-Krause model has inspired a large literature in philosophy and related fields, and there are many variations that have been explored.

scholar.google.com/scholar?cites=…

Our last paper is one of mine that looked at a similar question to @ben_golub and Jackson, but for both the DeGroot and Hegselmann-Krause model. I focused on finite cases, rather than infinite limits, but I came to very similar conclusions and G&J

journals.sagepub.com/doi/full/10.11…

Broadly: symmetric networks are good because no one person has too much influence. However, in some cases asymmetric networks are better than some symmetric ones in the Hegselmann-Krause model because they provide more opportunities to learn from one another.

So, this might be one limitation of the Golub & Jackson paper. If we think that Hegselmann-Krause is a better model, their conclusions might not always hold true.

The models for this week really focuses a lot on the importance of unequal influence in social networks. This is particularly poignant because... as you all know... some people have much larger twitter followings than others.

If those differences don't track differences in knowledge about the world (no comment), then we mind end up being pushed toward a less reliably consensus. This is probably a serious problem, but is somewhat limited by the simplicity of the models we studied.