Schelling's segregation model

In this introductory example we demonstrate Agents.jl's architecture and features through building the following definition of Schelling's segregation model:

  • Agents belong to one of two groups (0 or 1).
  • The agents live in a two-dimensional Chebyshev grid (8 neighbors per position).
  • If an agent is in the same group with at least three neighbors, then it is happy.
  • If an agent is unhappy, it keeps moving to new locations until it is happy.

Schelling's model shows that even small preferences of agents to have neighbors belonging to the same group (e.g. preferring that at least 30% of neighbors to be in the same group) could lead to total segregation of neighborhoods.

This model is also available as Models.schelling.

Defining the agent type

using Agents
using StatsBase: mean

mutable struct SchellingAgent <: AbstractAgent
    id::Int # The identifier number of the agent
    pos::Dims{2} # The x, y location of the agent on a 2D grid
    mood::Bool # whether the agent is happy in its position. (true = happy)
    group::Int # The group of the agent,  determines mood as it interacts with neighbors
end

Notice that the position of this Agent type is a Dims{2}, equivalent to NTuple{2,Int}, because we will use a 2-dimensional GridSpace.

We added two more fields for this model, namely a mood field which will store true for a happy agent and false for an unhappy one, and an group field which stores 0 or 1 representing two groups.

Notice also that we could have taken advantage of the macro @agent (and in fact, this is recommended), and defined the same agent as:

@agent SchellingAgent GridAgent{2} begin
    mood::Bool
    group::Int
end

Creating a space

For this example, we will be using a Chebyshev 2D grid, e.g.

space = GridSpace((10, 10), periodic = false)
GridSpace with size (10, 10), metric=chebyshev and periodic=false

Creating an ABM

To make our model we follow the instructions of AgentBasedModel. We also want to include a property min_to_be_happy in our model, and so we have:

properties = Dict(:min_to_be_happy => 3)
schelling = ABM(SchellingAgent, space; properties)
AgentBasedModel with 0 agents of type SchellingAgent
 space: GridSpace with size (10, 10), metric=chebyshev and periodic=false
 scheduler: fastest
 properties: Dict(:min_to_be_happy => 3)

Here we used the default scheduler (which is also the fastest one) to create the model. We could instead try to activate the agents according to their property :group, so that all agents of group 1 act first. We would then use the scheduler property_activation like so:

schelling2 = ABM(
    SchellingAgent,
    space;
    properties = properties,
    scheduler = property_activation(:group),
)
AgentBasedModel with 0 agents of type SchellingAgent
 space: GridSpace with size (10, 10), metric=chebyshev and periodic=false
 scheduler: by_property
 properties: Dict(:min_to_be_happy => 3)

Notice that property_activation accepts an argument and returns a function, which is why we didn't just give property_activation to scheduler.

Creating the ABM through a function

Here we put the model instantiation in a function so that it will be easy to recreate the model and change its parameters.

In addition, inside this function, we populate the model with some agents. We also change the scheduler to random_activation. Because the function is defined based on keywords, it will be of further use in paramscan below.

using Random # for reproducibility
function initialize(; numagents = 320, griddims = (20, 20), min_to_be_happy = 3, seed = 125)
    space = GridSpace(griddims, periodic = false)
    properties = Dict(:min_to_be_happy => min_to_be_happy)
    rng = Random.MersenneTwister(seed)
    model = ABM(
        SchellingAgent, space;
        properties, rng, scheduler = random_activation
    )

    # populate the model with agents, adding equal amount of the two types of agents
    # at random positions in the model
    for n in 1:numagents
        agent = SchellingAgent(n, (1, 1), false, n < numagents / 2 ? 1 : 2)
        add_agent_single!(agent, model)
    end
    return model
end

Notice that the position that an agent is initialized does not matter in this example. This is because we use add_agent_single!, which places the agent in a random, empty location on the grid, thus updating its position.

Defining a step function

Finally, we define a step function to determine what happens to an agent when activated.

function agent_step!(agent, model)
    agent.mood == true && return # do nothing if already happy
    minhappy = model.min_to_be_happy
    neighbor_positions = nearby_positions(agent, model)
    count_neighbors_same_group = 0
    # For each neighbor, get group and compare to current agent's group
    # and increment count_neighbors_same_group as appropriately.
    for neighbor in nearby_agents(agent, model)
        if agent.group == neighbor.group
            count_neighbors_same_group += 1
        end
    end
    # After counting the neighbors, decide whether or not to move the agent.
    # If count_neighbors_same_group is at least the min_to_be_happy, set the
    # mood to true. Otherwise, move the agent to a random position.
    if count_neighbors_same_group ≥ minhappy
        agent.mood = true
    else
        move_agent_single!(agent, model)
    end
    return
end

For the purpose of this implementation of Schelling's segregation model, we only need an agent step function.

When defining agent_step!, we used some of the built-in functions of Agents.jl, such as nearby_positions that returns the neighboring position on which the agent resides, ids_in_position that returns the IDs of the agents on a given position, and move_agent_single! which moves agents to random empty position on the grid. A full list of built-in functions and their explanations are available in the API page.

Stepping the model

Let's initialize the model with 370 agents on a 20 by 20 grid.

model = initialize()
AgentBasedModel with 320 agents of type SchellingAgent
 space: GridSpace with size (20, 20), metric=chebyshev and periodic=false
 scheduler: random_activation
 properties: Dict(:min_to_be_happy => 3)

We can advance the model one step

step!(model, agent_step!)

Or for three steps

step!(model, agent_step!, 3)

Running the model and collecting data

We can use the run! function with keywords to run the model for multiple steps and collect values of our desired fields from every agent and put these data in a DataFrame object. We define vector of Symbols for the agent fields that we want to collect as data

adata = [:pos, :mood, :group]

model = initialize()
data, _ = run!(model, agent_step!, 5; adata)
data[1:10, :] # print only a few rows

10 rows × 5 columns

stepidposmoodgroup
Int64Int64Tuple…BoolInt64
101(4, 16)01
202(9, 19)01
303(18, 14)01
404(12, 11)01
505(1, 1)01
606(19, 1)01
707(11, 11)01
808(5, 3)01
909(3, 7)01
10010(10, 14)01

We could also use functions in adata, for example we can define

x(agent) = agent.pos[1]
model = initialize()
adata = [x, :mood, :group]
data, _ = run!(model, agent_step!, 5; adata)
data[1:10, :]

10 rows × 5 columns

stepidxmoodgroup
Int64Int64Int64BoolInt64
101401
202901
3031801
4041201
505101
6061901
7071101
808501
909301
100101001

With the above adata vector, we collected all agent's data. We can instead collect aggregated data for the agents. For example, let's only get the number of happy individuals, and the maximum of the "x" (not very interesting, but anyway!)

model = initialize();
adata = [(:mood, sum), (x, maximum)]
data, _ = run!(model, agent_step!, 5; adata)
data

6 rows × 3 columns

stepsum_moodmaximum_x
Int64Int64Int64
10020
2120920
3227420
4329620
5431320
6531420

Other examples in the documentation are more realistic, with a much more meaningful collected data. Don't forget to use the function aggname to access the columns of the resulting dataframe by name.

Visualizing the data

We can use the abm_plot function to plot the distribution of agents on a 2D grid at every generation, via the InteractiveDynamics.jl package and the Makie.jl plotting ecosystem.

Let's color the two groups orange and blue and make one a square and the other a circle.

using InteractiveDynamics
import CairoMakie # choosing a plotting backend

groupcolor(a) = a.group == 1 ? :blue : :orange
groupmarker(a) = a.group == 1 ? :circle : :rect
figure, _ = abm_plot(model; ac = groupcolor, am = groupmarker, as = 10)
figure # returning the figure displays it

Animating the evolution

The function abm_video can be used to save an animation of the ABM into a video. You could of course also explicitly use abm_plot in a record loop for finer control over additional plot elements.

model = initialize();
abm_video(
    "schelling.mp4", model, agent_step!;
    ac = groupcolor, am = groupmarker, as = 10,
    framerate = 4, frames = 20,
    title = "Schelling's segregation model"
)

Launching the interactive application

Given the definitions we have already created for a normally plotting or animating the ABM it is almost trivial to launch an interactive application for it, through the function abm_data_exploration.

We define a dictionary that maps some model-level parameters to a range of potential values, so that we can interactively change them.

parange = Dict(:min_to_be_happy => 0:8)
Dict{Symbol,UnitRange{Int64}} with 1 entry:
  :min_to_be_happy => 0:8

We also define the data we want to collect and interactively explore, and also some labels for them, for shorter names (since the defaults can get large)

adata = [(:mood, sum), (x, mean)]
alabels = ["happy", "avg. x"]

model = initialize(; numagents = 300) # fresh model, noone happy
AgentBasedModel with 300 agents of type SchellingAgent
 space: GridSpace with size (20, 20), metric=chebyshev and periodic=false
 scheduler: random_activation
 properties: Dict(:min_to_be_happy => 3)
figure, adf, mdf = abm_data_exploration(
    model, agent_step!, dummystep, parange;
    ac = groupcolor, am = groupmarker, as = 10,
    adata, alabels
)

Replicates and parallel computing

We can run replicates of a simulation and collect all of them in a single DataFrame. To that end, we only need to specify replicates in the run! function:

model = initialize(numagents = 370, griddims = (20, 20), min_to_be_happy = 3)
data, _ = run!(model, agent_step!, 5; adata = adata, replicates = 3)
data[(end - 10):end, :]

11 rows × 4 columns

stepsum_moodmean_xreplicate
Int64Int64Float64Int64
1127710.41622
2232410.60812
3334810.61082
4436410.56492
5536610.58652
60010.49463
7127710.41623
8232410.60813
9334810.61083
10436410.56493
11536610.58653

It is possible to run the replicates in parallel. For that, we should start julia with julia -p n where is the number of processing cores. Alternatively, we can define the number of cores from within a Julia session:

using Distributed
addprocs(4)

For distributed computing to work, all definitions must be preceded with @everywhere, e.g.

@everywhere using Agents
@everywhere mutable struct SchellingAgent ...

Then we can tell the run! function to run replicates in parallel:

data, _ = run!(model, agent_step!, 2, adata=adata,
               replicates=5, parallel=true)

Scanning parameter ranges

We often are interested in the effect of different parameters on the behavior of an agent-based model. Agents.jl provides the function paramscan to automatically explore the effect of different parameter values.

We have already defined our model initialization function as initialize. We now also define a processing function, that returns the percentage of happy agents:

happyperc(moods) = count(x -> x == true, moods) / length(moods)
adata = [(:mood, happyperc)]

parameters = Dict(
    :min_to_be_happy => collect(2:5), # expanded
    :numagents => [200, 300],         # expanded
    :griddims => (20, 20),            # not Vector = not expanded
)

data, _ = paramscan(parameters, initialize; adata = adata, n = 3, agent_step! = agent_step!)
data

32 rows × 4 columns

stephappyperc_moodmin_to_be_happynumagents
Int64Float64Int64Int64
100.02200
210.6152200
320.862200
430.922200
500.03200
610.293200
720.5753200
830.7253200
900.04200
1010.074200
1120.1754200
1230.2554200
1300.05200
1410.0155200
1520.0255200
1630.0555200
1700.02300
1810.8433332300
1920.972300
2030.992300
2100.03300
2210.6033333300
2320.8266673300
2430.9233333300
2500.04300
2610.3666674300
2720.6033334300
2830.724300
2900.05300
3010.1566675300
3120.3533335300
3230.445300

paramscan also allows running replicates per parameter setting:

data, _ = paramscan(
    parameters,
    initialize;
    adata = adata,
    n = 3,
    agent_step! = agent_step!,
    replicates = 3,
)

data[(end - 10):end, :]

11 rows × 5 columns

stephappyperc_moodreplicatemin_to_be_happynumagents
Int64Float64Int64Int64Int64
110.1715300
220.29333315300
330.41333315300
400.025300
510.1725300
620.29333325300
730.41333325300
800.035300
910.1735300
1020.29333335300
1130.41333335300

We can combine all replicates with an aggregating function, such as mean, using the groupby and combine functions from the DataFrames package:

using DataFrames
using Statistics: mean
gd = groupby(data,[:step, :min_to_be_happy, :numagents])
data_mean = combine(gd,[:happyperc_mood,:replicate] .=> mean)

out = select(data_mean, Not(:replicate_mean))

32 rows × 4 columns

stepmin_to_be_happynumagentshappyperc_mood_mean
Int64Int64Int64Float64
1022000.0
2122000.61
3222000.855
4322000.915
5032000.0
6132000.275
7232000.515
8332000.685
9042000.0
10142000.065
11242000.165
12342000.29
13052000.0
14152000.015
15252000.04
16352000.055
17023000.0
18123000.843333
19223000.97
20323000.99
21033000.0
22133000.596667
23233000.796667
24333000.873333
25043000.0
26143000.38
27243000.62
28343000.733333
29053000.0
30153000.17
31253000.293333
32353000.413333

Note that the second argument takes the column names on which to split the data, i.e., it denotes which columns should not be aggregated. It should include the :step column and any parameter that changes among simulations. But it should not include the :replicate column. So in principle what we are doing here is simply averaging our result across the replicates.