an analysis of one-dimensional schelling segregation
DESCRIPTION
An Analysis of One-Dimensional Schelling Segregation. Gautam Kamath , Cornell University Christina Brandt, Stanford University Nicole Immorlica , Northwestern University Robert Kleinberg, Cornell University. Goal: Mathematically model and understand residential segregation. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/1.jpg)
An Analysis of One-Dimensional Schelling Segregation
GAUTAM KAMATH, CORNELL UNIVERSITYCHRISTINA BRANDT, STANFORD
UNIVERSITYNICOLE IMMORLICA, NORTHWESTERN
UNIVERSITYROBERT KLEINBERG, CORNELL UNIVERSITY
![Page 2: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/2.jpg)
- White
- Black
- Hispanic
- Asian
The New Yorker Hotel
In a one-dimensionalnetwork with local neighborhoods,
segregation exhibits only local effects.
map by Eric Fischer
Goal: Mathematically model andunderstand residential segregation
![Page 3: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/3.jpg)
![Page 4: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/4.jpg)
Start with a randomcoloring of the ring
n = size of ring
![Page 5: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/5.jpg)
Happy if at least 50% like-colored
near neighbors.w = window size
![Page 6: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/6.jpg)
At each time step, swapposition of two unhappy
individuals of opposite color
![Page 7: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/7.jpg)
Segregation:run-lengths in
stable configuration
![Page 8: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/8.jpg)
Local dynamics lead to segregation.• Schelling’s experiment: n = 70, w = 4• Average run length: 12
Schelling’s Experimental Result:
![Page 9: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/9.jpg)
The Big Questions
• Is run length a function of n or w? Global vs. local segregation
• If function of w, polynomial vs. exponential?
![Page 10: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/10.jpg)
In the perturbed Schelling model, segregation is global and severe. (stable run length: O(n))
Young’s result:
[Young, 2001]
In the unperturbed Schelling model, segregation is local and modest. (stable run length: O(w2))
Our main result (informal):
[Brandt, Immorlica, Kamath, Kleinberg, 2012]
![Page 11: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/11.jpg)
Working our way up
1. With high probability, process will reach a stable configuration
2. Average run length independent of ring size3. Average run length is modest
![Page 12: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/12.jpg)
Techniques
Defn. A firewall is a sequence of w+1 consecutive individuals of the same type.
Claim: Firewalls are stable with respect to the dynamics.
![Page 13: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/13.jpg)
Convergence
Theorem. For any fixed window size w, as n grows, the probability that the process reaches a stable configuration converges to 1.
Proof Sketch: 1. With high probability, there exists a firewall in
the initial configuration2. Individual has positive probability of joining a
firewall, and can never leave
![Page 14: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/14.jpg)
Working our way up
1. With high probability, process will reach a stable configuration
2. Average run length independent of ring size3. Average run length is modest
![Page 15: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/15.jpg)
An easy bound on run length
Theorem. Ave run length in final state is O(2w).Proof.
• Expect to look O(2w) steps before we find a blue firewall in both directions
• Bounds length of a green firewall containing site• Symmetric for a blue firewall containing site
random site
look for blue firewalllook for blue firewall
![Page 16: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/16.jpg)
Working our way up
1. With high probability, process will reach a stable configuration
2. Average run length independent of ring size3. Average run length is modest
![Page 17: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/17.jpg)
Techniques
1. Define firewall incubators, frequent at initialization.2. Show firewall incubators are likely to become firewalls.
A blue firewall incubator,rich in blue nodes.
A green firewall incubator, rich in green nodes.
![Page 18: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/18.jpg)
Simulation, n = 1000, w = 10
time t = 0
![Page 19: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/19.jpg)
time t = 40
Simulation, n = 1000, w = 10
![Page 20: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/20.jpg)
time t = 80
Simulation, n = 1000, w = 10
![Page 21: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/21.jpg)
time t = 120
Simulation, n = 1000, w = 10
![Page 22: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/22.jpg)
time t = 160
Simulation, n = 1000, w = 10
![Page 23: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/23.jpg)
time t = 260 (final)
Simulation, n = 1000, w = 10
![Page 24: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/24.jpg)
time t = 160
Simulation, n = 1000, w = 10
![Page 25: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/25.jpg)
time t = 120
Simulation, n = 1000, w = 10
![Page 26: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/26.jpg)
time t = 80
Simulation, n = 1000, w = 10
![Page 27: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/27.jpg)
time t = 40
Simulation, n = 1000, w = 10
![Page 28: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/28.jpg)
Simulation, n = 1000, w = 10
time t = 0
![Page 29: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/29.jpg)
time t = 40
Simulation, n = 1000, w = 10
![Page 30: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/30.jpg)
time t = 80
Simulation, n = 1000, w = 10
![Page 31: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/31.jpg)
time t = 120
Simulation, n = 1000, w = 10
![Page 32: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/32.jpg)
time t = 160
Simulation, n = 1000, w = 10
![Page 33: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/33.jpg)
time t = 260 (final)
Simulation, n = 1000, w = 10
![Page 34: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/34.jpg)
Four steps to O(w2)
1. Firewall incubators are common
![Page 35: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/35.jpg)
Firewall Incubators
w sites w sites
Definition. The bias of a site i at time t is the sum of the signs of sites in its neighborhood.
+1 -1 -1 -1-1 +1+1+1 +1 +1 +1 +1
Bias = +3
![Page 36: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/36.jpg)
Firewall Incubators
w+1 sites w+1 sites
defender defenderinternalattacker attacker
w sites w sites
left right
Definition. A firewall incubator is a sequence of 3 biased blocks – left defender of length w+1, internal, right defender of length w+1.At least 2w + 2 sites, where the bias of every site is > w1/2.
![Page 37: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/37.jpg)
Birth of a Firewall Incubator
Lemma. For any 6w consecutive sites, there’s a constant probability that a uniformly random labeling of sites contains a firewall incubator.Proof Sketch. Random walks + central limit theorem
0
5w1/2
-2w1/2
![Page 38: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/38.jpg)
Four steps to O(w2)
1. Firewall incubators are common2. Define an event in which an incubator
deterministically becomes a firewall
![Page 39: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/39.jpg)
Lifecycle of a Firewall Incubatorattacker defender
GOOD swap
![Page 40: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/40.jpg)
Lifecycle of a Firewall Incubatorattacker defender
BAD swap
![Page 41: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/41.jpg)
Lifecycle of a Firewall Incubator
Definition. The transcript is the sign-sequence obtained by associatingeach blue attacker with a +1, each green defender by a -1, and thenlisting signs in reverse-order of when individuals move.
attacker defender
+1 -1 -1+1 +1
5 2 8 6 1swap time:
sign:
transcript: +1, -1, +1, +1, -1
Proposition. If the partial sums of a transcript are non-negative, thenthe firewall incubator becomes a firewall
![Page 42: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/42.jpg)
Four steps to O(w2)
1. Firewall incubators are common2. There is an event in which an incubator
deterministically becomes a firewall3. Assuming swap order is random, this event
happens
![Page 43: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/43.jpg)
Lifecycle of a Firewall Incubator
Ballot Theorem. With probability (A – D)/(A + D), all the partial sums of a random permutation of A +1’s and D -1’s are positive.
Firewall incubator definition implies1. A-D ≥ Ɵ(w0.5) (bias condition of incubator)2. A+D ≤ Ɵ(w) (length of attacker + defender)Each defender survives with probability Ω(1/w0.5)Incubator becomes a firewall with probability Ω(1/w)
![Page 44: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/44.jpg)
Four steps to O(w2)
1. Firewall incubators are common2. There is an event in which an incubator
deterministically becomes a firewall3. Assuming swap order is random, this event
happens4. Swap order is close to random
![Page 45: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/45.jpg)
Lifecycle of a Firewall Incubator
Swaps are random if there is # unhappy green = # unhappy blue.
![Page 46: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/46.jpg)
Wormald’s Technique
•We show numbers are approx. equal using Wormald’s theoremTheorem [Wormald]. Under suitable technical conditions, a discrete-time stochastic process is well approximated by the solution of a continuous-time differential equation•Technically non-trivial due to complications with infinite differential equations•Don’t need to solve diff. eq., only exploit symmetry
![Page 47: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/47.jpg)
Firewall incubators occur every O(w) locations+
Incubator becomes firewall with probability Ω (1/w)=
Firewalls occur every O(w2) locations
![Page 48: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/48.jpg)
An better bound on run length
Theorem. Ave run length in final state is O(w2).Proof.
• Expect to look O(w2) steps before we find a blue firewall in both directions
• Bounds length of a green firewall containing site• Symmetric for a blue firewall containing site
random site
look for blue firewalllook for blue firewall
![Page 49: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/49.jpg)
Summary
•First rigorous analysis of Schelling’s segregation model on one-dimensional ring•Demonstrated that only local, modest segregation occurs–Average run length is independent of n and poly(w)–Subsequent work: Ɵ(w) run length
![Page 50: An Analysis of One-Dimensional Schelling Segregation](https://reader036.vdocuments.us/reader036/viewer/2022070500/568168ac550346895ddf579c/html5/thumbnails/50.jpg)
Open Questions
• Vary parameters (proportion and number of types, tolerance)
• Study segregation on other graphs– 2D grid