write a paper with fabio: schelling models with structural holes

In my last mathematical sociology post, I noted that mathematical sociology does not have a core set of results. Mike3550, longtime reader, asks:

Fabio, what areas of sociology do you see as ripe for formal mathematical models and potential candidates for core theorems?

Good question. Most “core” results in mathematical fields prove something about a model that formalizes a key concept in a discipline (e.g. welfare theorems) or it resolves some technical issue that comes up all the time (e.g., Nash equilibria). So what about mathematical sociology? I think theorems that address core sociological insights deserve their own post. So I’ll focus on the second type of “core theorem,” a theorem that addresses a technical issue that comes up in sociology all the time. I’ll describe the problem in detail and make an offer to co-write a paper with anyone who can solve it.

The issue that I think is ripe for a “core theorem” is the Schelling segregation model. It’s an old model and has been studied a million times. And it’s very important in both urban economics and sociology. However almost all studies omit something very, very important. They all assume that the space is a grid – a square or rectangle. Most studies focus on the agents who move around.

Both dimensions of variation are important. Clearly, we want to know if segregation still occurs if people have different utility/cost functions or if we have more than two types of agents (e.g., Black/White/Yellow vs. just Black/White). Most of the literature focuses on this type of model variation.

However, real urban spaces are not nice squares. Water ways, parks and highways chop up space so that the space is no longer a uniform grid. Urban spaces become disconnected (e.g., Manhattan/Harlem vs. the rest of New York) or they have holes in them (e.g., Central Park in Manhattan). Some have funny shapes. A ring, for example – like the cities around the San Francisco Bay.

So let’s set up some terminology so we can clearly state the theorem that I might be looking for. Let start with a square grid of size NxN. Call it G.Let’s put some holes in G. Than means we select some squares in the grid you can’t go into. This may represent an urban spaced fill by water, a big road, or zoning laws that prohibit residences. Let’s call this grid with hole G’.

The theorem I want to know is how the segregation dynamics of G’ differ from G. Here’s my proposal for the measurement. Let’s say you have a measurement of inter-group segregation I. It measures the degree of segregation in a Schelling model with two types of actors after T rounds. And say that I(G,T) = 1 if people only have same group neighbors after T rounds and I(G,T)=0 if nobody has a same group neighbor.

The issue then is the ratio of I(G’,T)/I(G,N). So potential theorem for Schelling models on modified grids is:

• Let G’ be a modification of the NxN grid G where you create structural holes, then I(G’,T)/I(G,T) = F(G’,T).

The research problem is to figure out F explicitly. If that fails, one could explore properties of F. For example, what does G’ look like for F(G’,T)~1? If that is too hard, then one could simulate the process for selected G’.

Many years ago, I took a brilliant IU PhD student with a math background and asked her to do some simulations, to get some intuitions about F(G’,T). However, she was only a first year and had little time to work on the project. Then she decided to make $$$ in the consulting world. My loss. The project stalled.

If you need a path breaking project to work on and you have a taste for either math or computer simulations, I make the following offer – let’s write a paper that solves this problem. Maybe not completely, but we can probably come up with some good results.

This paper will be “open source.” Though the blog, we’ll discuss the paper and toss out analytic strategies for discussion. Anyone who contributes will be added on as a co-author – lit review, coders, theorem provers, etc. I’ll be the factotum. I’ll assemble and revise the paper, and herd it through the journal publication process.

In my view, we need the following:

• Somebody to do a lit review, to make sure this hasn’t been done yet. My knowledge of the Schelling model literature is now out of date.
• Somebody who can code pretty darn quickly and efficiently, so we can get some toy models up and running and then explore the model.
• Some theorem provers.

The final outcome is one or more papers that explore how spatial structures change residential segregation patterns.

This offer is open to anyone in any discipline – sociology, economics, math, cs, engineering, urban studies/planning. I also don’t care about your status. If you are in high school and can code, I’ll work with you. As long as you can accept the “open source” nature of the project and you can work with a team, you’re in. Credit will go to anyone who writes, codes or proves any part of the paper. Just send me an email or start using the comment section of this post.

And of course, if you can do this yourself, go for it. I’ve been waiting nearly fifteen years for the answer to this question.

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