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Spatial Model of Segregation
Leonid E. Zhukov
School of Data Analysis and Artificial IntelligenceDepartment of Computer Science
National Research University Higher School of Economics
Structural Analysis and Visualization of Networks
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Spatial model of segregation
”Dynamic Models of Segregation”, Thomas Schelling, 1971
Micro-motives and macro-behavior
Personal preferences lead to collective actions
Global patterns of spatial segregation from homophily at a local level
Segregated race, ethnicity, native language, income
Cities are strongly racially segregated. Are people that racists?
Agent based modeling: agents, rules (dynamics), aggregation
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Segregation
Integrated pattern Segregated pattern
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Racial segregation
New York Washington Chicago
Seattle Los Angeles Miami
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Bay area high school graduates
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2012 US Presidential Elections Map
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Schelling’s model of segregation
Population consists of 2 types of agents
Agent reside in the cells of the grid (2-dimensional geography of acity), 8 neighbors
Some cells contain agents, some unpopulated
Every agent wants to have at least some fraction of agents(threshold) of his type as neighbor (satisfied agent)
On every round every unsatisfied agent moves to a satisfactory emptycell.
Continues until everyone is satisfied or can’t move
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Spatial segregation
satisfied agent unsatisfied agent
preference threshold λ = 3/7
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Spatial segregation
T. Schelling, 1971
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Spatial segregation
vacancy 5%, tolerance λ = 0.5
L. Gauvin et.al. 2009
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Spatial segregation
L. Gauvin et.al. 2009
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Model
N - nodes, θ - fraction of occupied by A and B
nA + nB = θ · N
Proportion of ”unlike” nearest neighbors, ki = #NN
Pi =
{#nB/ki , if i ∈ A#nA/ki , if i ∈ B
Utility function, λ - sensitivity (tolerance threshold) level
ui =
{1, if Pi ≤ λ0, if Pi > λ
Every node moves to maximize its utility
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Spatial segregation
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Algorithm
time steps 1..T
At every time step randomly select an agent, compute utility
If utility is u = 0 move to an empty location to maximize utility
Movements: 1) random location 2) nearest available location
Repeat until either all utilities are maximized∑
i ui = θNor reaches ”frozen” state, no place to move, then
∑i ui < θN
Total utility of society
U =∑i
ui
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Measuring segregation
Schilling’s solid mixing index
M =1
nA + nB
∑i
Pi
Freeman’s segregation index
F = 1− e∗
E (e∗)
e∗ = eAB(eAB+eAA+eBB) - observed proportion of between group ties,
E (e∗) = 2nAnB(nA+nB)(nA+nB−1) - expected proportion for random ties
Assortative mixing
Q =1
2m
∑ij
(Aij −kikj2m
)δ(ci , cj)
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Spatial segregation on networks
Fixed degree k = 10 neighboring graphs: regular, random, scale-free,fractal
Arnaud Banos, 2010
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Spatial segregation on networks
λ = 0.5, θ = 0.8
Banos, 2010
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Spatial segregation on networks
Banos, 2010
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Spatial segregation on networks
ν = 10% of random ”noise” added for decision to avoid freezes
Banos, 2010
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Spatial segregation on networks
λ = 0.3, θ = 0.8 Sensitivity to initial conditions
Banos, 2010
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Spatial segregation on networks
ν = 0.1, θ = 0.8, λ = 0..0.5
Banos, 2010
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Mathematical description
Agents of two types A, B
σi = ±1 if site i is occupied, σi = 0 if empty
System state vector: σ = (σ1...σN)
Adjacency matrix Aij
Fraction of vacant sites θ = 1/N∑
i (1− |σi |)Proportion of ”unlike” neighbours
Pi =
∑j Aij(|σiσj | − σiσj)∑
j Aij |σiσj |
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Summary
Spatial segregation is taking place even though no individual agent isactively seeking it (minor preferences, high tolerance)
Network structure does affect segregation
Fixed characteristics (race) can become correlated with mutable(location)
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References
Dynamic Models of Segregation, Thomas C. Schelling, 1971
Segregation in Social Networks, Linton Freeman, 1978
Gauvin L, Vannimenus J, Nadal JP. Phase diagram of a Schellingsegregation model. The European Physical Journal B, 70:293-304,2009
Arnaud Banos. Network effects in Schelling’s model of segregation:new evidences from agent-based simulations. 2010
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