Segregation in Social Networks

1
Segregation in Social Networks

• Giorgio Fagiolo
• University of Verona, Italy
• https//mail.sssup.it/fagiolo
• Marco Valente
• University of LAquila, Italy
• Nick Vriend
• Queen Mary, University of London, U.K.

WEHIA 2005 10th Annual Workshop on Economics with
Heterogeneous Interacting Agents Colchester, June
2005
2
Background Literature (1/2)

• Schelling Model of Neighborhood Segregation
  (1978)
• Two types of agents placed on a lattice
• Agents care about the composition of their
  neighborhood
• Agents are unhappy only if more than 50 of
  neighbors is of different type
• If unhappy, they move to a satisfactory (empty)
  position

• Micro-Motives vs. Macro-Behaviors
• Despite no agent strictly prefers segregation, a
  strongly segregated society always emerges
• Very simple model explaining self-organized,
  emergent macro property out of repeated
  interactions among individual agents
• Empirical implications segregation emerging in
  racial, social and natural contexts

3
Background Literature (2/2)

• Robustness of Schellings Results
• Pancs and Vriend (2003)
• Checking Schelling Results vs. Simplifying
  Assumptions
• Utilities Schellings agents are equally happy
  to live in a completely integrated city or in a
  completely segregated city of like agents
• Lattice dimension and metrics (1D vs. 2D),
  percentage of empty nodes, segregation measures,
  inertia, sequential vs. simultaneous moves

• Pancs and Vriend (2003) Main Results
• Schelling Segregation Model very robust to a
  whole range of variations in its setup
• Even in all individuals strictly prefer to live
  in a fully integrated city, segregation will
  occur in the aggregate!

4
Motivations (1/2)

• What happens when agents are not spatially
  located?
• Schelling (1978) and Pancs and Vriend (2003)
• Agents spatially located on lattices (proxy of
  geographic space)
• They care about their local neighbors (proxy of
  actual neighborhoods)

5
Motivations (2/2)

• From lattices to generic networks
• Studying Schelling Model with agents located on
  generic networks

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Motivations (2/2)

• From lattices to generic networks
• Studying Schelling Model with agents located on
  generic networks

• Theoretical motivation
• Topological structure of neighborhoods is regular
  and homogeneous
• Lattice only as a special type of network
• Results can also be affected by lattice
  homogeneity properties

• Empirical motivation
• People also interact through networks of friends,
  relatives, and colleagues, and through virtual
  communities on the internet
• Segregation may not necessarily occur at the
  spatial (neighborhood) level only
• People might be socially segregated despite they
  are spatially integrated

7
The Model (1/2)

• Locations and Players
• M nodes
• ? percentage of empty nodes
• N(1-?)M agents
• Agents can be of two types -1,1

• Networks
• Non-directed graph with M ? M (symmetric)
  socio-matrix Wwhk
• Interaction Group of node k V(k) h
  whkwkh1

• Utilities
• Standard Schellings type of utility
• Agents want at most 50 unlike agents in their
  interaction groups
• Happy agents get u1
• Unhappy agents get u0

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The Model (2/2)

• Initial conditions
• Time t0 Choose
• A particular network instance Wwhk
• A random allocation of empty nodes and N
  agents/types across nodes

• Dynamics Locally-constrained moves plus inertia
• At any t Draw at random one agent
• If the agent is
• Happy (u1) then
• Nothing changes
• Unhappy (u0) then
• Searches for empty nodes within his interaction
  group
• Computes utility if he moves there
• Moves to a node randomly-chosen among all empty
  nodes providing u1

9
Network Classes

• 2-D Von Neumann lattice without boundaries
  (torus)
• Degree of each location D(r) 2r(r1), r
  interaction radius

• 2-D Moore lattice without boundaries (torus)
• Degree of each location D(r) 4r(r1), r
  interaction radius

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Network Classes

• 2-D Von Neumann lattice without boundaries
  (torus)
• Degree of each location D(r) 2r(r1), r
  interaction radius

• 2-D Moore lattice without boundaries (torus)
• Degree of each location D(r) 4r(r1), r
  interaction radius

• D-Regular Graphs
• Each location has the same degree D

• Random Graphs
• Each link is in place with probability p
  (independently of other links)

• Small-World Graphs
• Based on a 2-D VN lattice with rewiring
  probability ?

• Scale-Free Graphs
• Based on M0 initial nodes each holding 4 links

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Segregation Measures

• Freemans Network Segregation Index (FSI)
• Based on Freeman (1972, 1978)
• Rationale If a given attribute (type) does not
  matter for social relationships (links), then
  relationships (links) should be distributed
  randomly with respect to that attribute

• Formally
• (Cross-Group Links Expected by Chance) -
  (Actual Cross-Group Links)
• --------------------------------------------------
  -------------------------------------------------
• (Cross-Group Links Expected by Chance)

• Additional Segregation Indices
• As in Pancs and Vriend (2003)

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Simulation Strategy

• Monte-Carlo Exercises
• Fix percentage of empty nodes ?
• Fix network class and parameters, e.g. (average)
  degree
• Run the model and collect (steady-states or
  long-run) statistics (FSI)
• Repeat K1000 times and study Monte-Carlo
  distributions
• Monte-Carlo average of FSI

• Basic questions
• Do segregation levels differ among different
  classes of social networks?
• Comparing different networks with the same
  (average) degree and of empty
  nodes
• What happens to segregation in each given network
  class when parameters change (degree, of empty
  nodes, graph-specific parameters)?

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Some Preliminary Results (1/3)

• Monte-Carlo Average of FSI vs. Networks
• Percentage of empty nodes ?0.3
• Comparing D4 vs. D8

D4
D8
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Some Preliminary Results (2/3)

• How Large are FSI Values?
• Comparing Monte-Carlo Distributions vs. Random
  Network-Types Allocations
• System always attains segregation levels much
  larger than random averages
• Long-run FSI (kernel-smoothed) densities always
  right of random ones

MOORE (8)
REGULAR (8)
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Some Preliminary Results (3/3)

• How Do Resulting Networks Look Like?
• Plotting steady-state network by arranging nodes
  on a virtual circle
• REGULAR(8) graph segregation patterns emerge
  also visually
• Type-x agents close to each other, with links to
  other type-x agents

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so far

• Schelling results are quite robust to non-lattice
  networks
• A society where agents do not live on spatially
  homogeneous environments (lattices) is still
  characterized by high levels of social
  segregation

• Segregation levels do not dramatically change
  across structurally-different classes of networks
• If any, segregation is weaker in social networks
  where a few agents hold very crowded interaction
  groups, while the majority holds small
  interaction groups (scale-free)

• What happens across different parameter setups?
• Varying the average degree (D) of the network
• Varying the percentage of empty (available) nodes
  (?)
• Varying network-specific parameters (rewiring
  probability, etc.)

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Monte-Carlo Analysis (1/4)

• Average FSI vs. (Average) Degree Regular vs.
  Random Nets

Small of Empty Nodes
Large of Empty Nodes

• Small D Similar segregation levels across
  regular vs. random networks
• Large D Lattices tend to reach slightly higher
  segregation levels
• Segregation tend to decrease with D in all
  network setups (for large ?)

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Monte-Carlo Analysis (2/4)

• Average FSI vs. (Average) Degree Small Worlds

Small of Empty Nodes
Large of Empty Nodes

• Small-Worlds behave similarly to lattices and
  random graphs
• Small ? Lattice Large ? Random Graph
• Weaker segregation as degree increases

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Monte-Carlo Analysis (3/4)

• Average FSI vs. (Average) Degree Scale-Free

Small of Empty Nodes
Large of Empty Nodes

• Segregation is slightly weaker in scale-free
  networks
• Due to presence of hubs?

20
Monte-Carlo Analysis (4/4)

• Average FSI vs. Empty Nodes within each network
  class

Regular Graphs

• Segregation levels are increasing in the of
  empty nodes for any given (average) degree

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Extensions

• Dropping inertia
• Agents always randomize among all available empty
  nodes providing highest utility

• Dropping local search (no moving costs)
• Agents care about their local interaction group
  V(i)
• When they search around, they can do it globally
• Check all empty nodes in the network and possibly
  move there

• Introducing endogenous networks
• Agents placed in a social network
• Agents can choose where to move in social
  networks
• But they can also endogenously add/delete social
  links

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Extensions (1/3)

• Dropping inertia
• Agents always randomize among all available empty
  nodes providing highest utility

2D-VN
2D-MOORE
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Extensions (1/3)

• Dropping inertia
• Agents always randomize among all available empty
  nodes providing highest utility

REGULAR
RANDOM
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Extensions (1/3)

• Dropping inertia
• Agents always randomize among all available empty
  nodes providing highest utility

SMALL-WORLDS
SCALE-FREE
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Extensions (2/3)

• Dropping local search (no moving costs)
• Agents care about their local interaction group
  V(i)
• When they search around, they can do it globally
• Check all empty nodes in the network and possibly
  move there

2D-VN
2D-MOORE
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Extensions (2/3)

• Dropping local search (no moving costs)
• Agents care about their local interaction group
  V(i)
• When they search around, they can do it globally
• Check all empty nodes in the network and possibly
  move there

REGULAR
RANDOM
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Extensions (2/3)

• Dropping local search (no moving costs)
• Agents care about their local interaction group
  V(i)
• When they search around, they can do it globally
• Check all empty nodes in the network and possibly
  move there

SMALL-WORLDS
SCALE-FREE
28
Extensions (3/3)

• Introducing endogenous networks
• Agents placed in a social network
• Agents can choose where to move in social
  networks
• But they can also endogenously add/delete social
  links

29
Conclusions

• Schelling Model Revisited
• Pancs and Vriend (2003) Lattice-based
  segregation model
• Schellings main result about segregation
  emergence holds even when agents have a strict
  preference for complete integration

• This Paper Segregation in Social Networks
• Study what happens when agents are located on the
  nodes of generic networks instead of homogeneous
  spatial structures (lattices)
• Exploring different classes of alternative
  network structures (regular, random,
  small-worlds, scale-free)

• Main Results
• High levels of segregation still occur no matter
  network structures
• Segregation decreases with average degree
  (network density) and with of occupied nodes
  (network crowdedness)
• Scale-free networks display less segregation,
  lattices display more
• Removing inertia/local moves implies higher
  segregation levels