Agent-Based Modeling

Agent-based models (ABMs) simulate collections of autonomous agents following simple local rules. Global patterns — segregation, flocking, traffic jams — emerge without central control.

Schelling's Segregation Model

Thomas Schelling (1971) showed that mild individual preferences for same-type neighbors produce strong global segregation. Even agents who are happy with 50% same-type neighbors will end up in nearly homogeneous clusters.

1D Model

We use a 1D ring (periodic boundary conditions):

• Grid values: 0 = empty, 1 = type A, 2 = type B
• An agent at position $i$ looks at neighbors within distance neighborhood
• The agent is happy if at least fraction $f$ of its non-empty neighbors are the same type
• Satisfaction: fraction of agents who are happy

Segregation Index

The segregation index measures how same-type pairs dominate adjacent slots:

$\text{SI} = \frac{\text{same-type adjacent pairs}}{\text{total adjacent non-empty pairs}}$

For a periodic ring of $N$ sites, there are $N$ adjacent pairs $(i, (i+1) \% N)$. Only non-empty pairs count.

• Perfectly alternating [1,2,1,2,...]: $\text{SI} = 0$
• Perfectly segregated [1,1,...,2,2,...]: $\text{SI} \to 1$ (minus boundary pairs)

Implementation

def segregation_index(grid):
    # Count adjacent same-type pairs / total adjacent non-empty pairs
    # Periodic boundary conditions
    ...

def count_happy(grid, f=0.5, neighborhood=1):
    # Count agents where same-type fraction of neighbors >= f
    # Skip empty sites (value 0)
    # Periodic boundary conditions
    ...

def schelling_satisfaction(grid, f=0.5, neighborhood=1):
    # Return fraction of non-empty agents that are happy
    ...

def interface_density(grid):
    # Fraction of adjacent non-empty pairs that are DIFFERENT type
    # = 1 - segregation_index(grid)
    ...