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.

Adverts: From Black Power/Grad Skool Rulz


Written by fabiorojas

March 23, 2012 at 12:02 am

Posted in fabio, sociology

15 Responses

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  1. I did some work some years ago with educational administration students at both Temple U and Widener U with Schelling’s simulation in order to get the students to understand that macrolevel dysfunctions might arise from quite innocent microlevel dispositions.

    I experimented using grids with holes. Holes reduce the “crowdability” of their neighboring squares and individuals with low crowding tolerance and other privileges (wealth) tended to monopolize neighborhoods bordering holes. I had several factors working at once at the microlevel, but these only tended to exacerbate the power of the holes.

    For a quick overview of what I was working with (that does not mention holes, since that was a later serendipitous development) see “RANPOP Simulation Experiment #1 ” at

    Edward G. Rozycki


    Edward G. Rozycki

    March 23, 2012 at 2:07 am

  2. I love this post. By the way, I ran into the following paper by a pair of physicists (see below). They summarize the key components of Schelling-type models (which are actually quite diverse) as consisting of (1) a network, (2) initial conditions (3), a satisfaction function, and (4) transfer probabilities. What I like about your proposal is that (if I understand your post correctly) it alters the edges in the network, in my view the most interesting part of the set up. Here’s the paper:


    Ethan Fosse

    March 23, 2012 at 4:24 am

  3. I’m buried with projects but love the idea and ambition!



    March 23, 2012 at 4:57 am

  4. Fabio, thank you for your extensive reply! In your post, just before the proposed theorem, I believe that the ratio should read: I(G’,T)/I(G,T) [currently it reads as I(G’,T)/I(G,N)].



    March 23, 2012 at 11:39 am

  5. To expand my reply above: I used a grid composed of 4×4 checkerboards and used masking tape to define an “island” within which the squares could be inhabited. A common outcome was that wealthy, low-tolerance people tend to reside on the “shoreline.” (See rules for wealth, given in the instruction sheet cited in my earlier reply.)

    I think the problem could be substantially more generalized by making the “squares” interactive with the characteristics of the population. For Schelling’s original game, squares only had an effect on each other ( and consequently, the populations) depending on their location to other squares, i.e. “crowdability. ”

    Imagine a function SL with two values s = stay, and l = leave. It is determined by some combination of stay-or-leave factors, e.g. C, crowding, W, wealth, E, ethnicity, etc. For example, SL = f(C, W, E) = C@W@E, where @ is some operation.

    Now let each SL-factor be operated on by corresponding factors for each “square,” a, accessibility, c, crowdability, w, expense, e, ethnic uniformity, ….

    If you assigned to each square a 1xn matrix of factors based on the n stay-or-leave characteristics of the population (coded, possibly as a nx1 matrix — forgive my clunky math, it has been ages since I have used it), then an empty Schelling square would have the matrix — as column vector, which I cannot enter here — [k, k, k, ….] constructed so it would have no effect on the SL outcome.

    Holes — whatever we might interpret them to represent — are squares with accessibility, a=0. Since they’re not accessible, they’re not crowdable. But neither do they add to the crowdability to neighboring squares.

    But there’s no reason why accessibility need be either binary. I did a simulation once where one shoreline was subject to “flooding.” Depending on the risk involved, my students would have their atavars avoid that shore for a safer one. The “poorer people” tended to inhabit it.

    Each population characteristic pairs up with a square, residence-characteristic to create a surface across the playing board that can be constructed to influence the stay-move outcomes of the population. You could depict each surface in a step-wise manner, or smooth them out so that the were continuous.

    I think an important consideration here is that there is lots of empirical data against which hypotheses using such a model could be tested, e.g. neighbor parkland increases expense of property; or, wealth will not move out of an ethnic neighborhood if additional space can be inhabited — baronial effect.

    Just some thoughts.


    Edward G. Rozycki


    Edward G. Rozycki

    March 23, 2012 at 6:03 pm

  6. Interesting challenge.
    I haven’t read the following thoroughly yet, but it seems that some people did look at potential spatial irregularities. see here:

    I also quickly built an R code that enables simulating the schelling model for any spatial pattern (by defining explicit “neighbourhood” for each potential location. I added a function that creates such neighbourhood maps for 2 dimensional grids with any number of holes, and a sample session that run 2 such models. can’t attach it here – so emailed to you.



    March 24, 2012 at 1:17 am

  7. I remember you bringing this up over lunch some time in 2007. Sounded fun then, sounds fun now.



    March 24, 2012 at 7:26 pm

  8. […] even goes on to call for a sort of open-science collaboration to develop a Schelling model of segregation which considers the influence of topography.  I think […]


  9. Fabio, what makes you suspect that segregation would be different on a grid with holes in the first place?



    March 27, 2012 at 7:22 am

  10. Rense: In math, things behave differently when there are holes. Mathematicians call no hole “simply connected.” Holes means that things get split up, which might have an impact on segregation dynamics. That’s vague. which is why we need proofs.



    March 27, 2012 at 5:13 pm

  11. I suspect holes, per se, will not matter at all to segregation dynamics — unless you give them properties which can interact with population characteristics. In Schelling’s original model, it was crowding tolerance and increasing population that effected segregative movement. The contribution of the squares of the checkboard was their four-sidedness and location, which permitted crowding to interact with the population’s crowding tolerance.

    People live under bridges and roadways, in sewers and boats and doorways, on mountain slopes, inside volcanos and in trailers parked off the sides of highways. What are squares supposed to mean?

    By the way, I have had this simulation played with multiple-story “apartment buildings” of height n < lowest crowding tolerance of the individual inhabitants. Each apartment resident was considered to be crowded by only those residents above and below them. So an apartment building could house 5 residents of crowding tolerance 6 on the single building square even though 5 squares surrounding the building square were filled. Result: higher crowding tolerance in a population permitted housing them in higher apartment buildings. Every see an post-WWII government housing project?


    Edward G. Rozycki

    March 27, 2012 at 6:46 pm

  12. Fabio, I would imagine that the presence of structural holes is less important than the shape and/or structure of the holes. A random assortment of holes will simply mean that the evaluations of neighborhoods will be based on fewer respondents; however, linear holes or amoebas might influence dynamics much more strongly — especially in the presence of rules regarding how agents can move.

    I think it possible that patterns can emerge even if the structural holes are “inert” rather than responsive to other agents in the model. Although I think that what Edward writes is correct, I think that patterns can emerge even in the absence of patches directly influencing agents.



    March 27, 2012 at 10:31 pm

  13. To everyone: Good comments, I’ll try to get a follow up post. But to answer mike3550 and others, the importance of holes is the research question. I could be wrong, but only simulation and proofs are the answer. Talk can’t definitely answer how topology affects segregation dynamics.



    March 28, 2012 at 4:01 am

  14. I agree that simulation is a way to test the research question. My guess was from running Schelling-type simulations and seeing the results, so it was more of a hypothesis about the structure of the holes rather than the number themselves.

    Your last response also leads to some confusion on my part: is the research question whether *any* structural holes cause changes in outcomes, the *frequency* of structural holes cause changes in outcomes, or the *relations between* structural holes cause changes in outcomes? It seems the question is certainly not the latter given your last response, but I could read your post saying either of the first two (and one solution is a subset of the other).



    March 29, 2012 at 12:28 pm

  15. […] Rojas at Orgtheory launches an “open source”  paper project, trying to figure out how “holes” in a grid would influence segregation in […]


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