# HK (Hegselmann and Krause) opinion dynamics model

This example showcases

• How to do synchronous updating of Agent properties (also know as Synchronous update schedule). In a Synchronous update schedule changes made to an agent are not seen by other agents until the next step, see also Wilensky 2015, p.286).
• How to terminate the system evolution on demand according to a boolean function.
• How to terminate the system evolution according to what happened on the previous step.

## Model overview

This is an implementation of a simple version of the Hegselmann and Krause (2002) model. It is a model of opinion formation with the question: which parameters' values lead to consensus, polarization or fragmentation? It models interacting groups of agents (as opposed to interacting pairs, typical in the literature) in which it is assumed that if an agent disagrees too much with the opinion of a source of influence, the source can no longer influence the agent's opinion. There is then a "bound of confidence". The model shows that the systemic configuration is heavily dependent on this parameter's value.

The model has the following components:

• A set of n Agents with opinions xᵢ in the range [0,1] as attribute
• A parameter ϵ called "bound" in (0, 0.3]
• The update rule: at each step every agent adopts the mean of the opinions which are within the confidence bound ( |xᵢ - xⱼ| ≤ ϵ).

## Core structures

We start by defining the Agent type and initializing the model. The Agent type has two fields so that we can implement the synchronous update.

using Agents
using Statistics: mean

mutable struct HKAgent <: AbstractAgent
id::Int
old_opinion::Float64
new_opinion::Float64
previous_opinon::Float64
end

There is a reason the agent has three fields that are "the same". The old_opinion is used for the synchronous agent update, since we require access to a property's value at the start of the step and the end of the step. The previous_opinion is the opinion of the agent in the previous step, as the model termination requires access to a property's value at the end of the previous step, and the end of the current step.

We could, alternatively, make the three opinions a single field with vector value.

function hk_model(; numagents = 100, ϵ = 0.2)
model = ABM(HKAgent, scheduler = fastest, properties = Dict(:ϵ => ϵ))
for i in 1:numagents
o = rand()
end
return model
end

model = hk_model()
AgentBasedModel with 100 agents of type HKAgent
no space
scheduler: fastest
properties: Dict(:ϵ => 0.2)

Add some helper functions for the update rule. As there is a filter in the rule we implement it outside the agent_step! method. Notice that the filter is applied to the :old_opinion field.

function boundfilter(agent, model)
filter(
j -> abs(agent.old_opinion - j) < model.ϵ,
[a.old_opinion for a in allagents(model)],
)
end

Now we implement the agent_step!

function agent_step!(agent, model)
agent.previous_opinon = agent.old_opinion
agent.new_opinion = mean(boundfilter(agent, model))
end

and model_step!

function model_step!(model)
for a in allagents(model)
a.old_opinion = a.new_opinion
end
end

From this implementation we see that to implement synchronous scheduling we define an Agent type with old and new fields for attributes that are changed via the synchronous update. In agent_step! we use the old field then, after updating all the agents new fields, we use the model_step! to update the model for the next iteration.

## Running the model

The parameter of interest is now :new_opinion, so we assign it to variable adata and pass it to the run! method to be collected in a DataFrame.

In addition, we want to run the model only until all agents have converged to an opinion. From the documentation of step! one can see that instead of specifying the amount of steps we can specify a function instead.

function terminate(model, s)
if any(
!isapprox(a.previous_opinon, a.new_opinion; rtol = 1e-12) for a in allagents(model)
)
return false
else
return true
end
end

step!(model, agent_step!, model_step!, terminate)
model[1]
Main.ex-HK.HKAgent(1, 0.7055807897280046, 0.7055807897280046, 0.705580789728005)

Alright, let's wrap everything in a function and do some data collection using run!.

function model_run(; kwargs...)
model = hk_model(; kwargs...)
agent_data, _ = run!(model, agent_step!, model_step!, terminate; adata = [:new_opinion])
return agent_data
end

data = model_run(numagents = 100)
data[(end - 19):end, :]

20 rows × 3 columns

stepidnew_opinion
Int64Int64Float64
16810.278468
26820.278468
36830.700346
46840.278468
56850.700346
66860.278468
76870.278468
86880.700346
96890.700346
106900.700346
116910.278468
126920.278468
136930.278468
146940.700346
156950.278468
166960.278468
176970.278468
186980.700346
196990.278468
2061000.700346

Notice that here we didn't speciy when to collect data, so this is done at every step. Instead, we could collect data only at the final step, by re-using the same function for the when argument:

model = hk_model()
agent_data, _ = run!(
model,
agent_step!,
model_step!,
terminate;
when = terminate,
)
agent_data

100 rows × 3 columns

stepidnew_opinion
Int64Int64Float64
1510.74935
2520.269865
3530.269865
4540.269865
5550.269865
6560.269865
7570.74935
8580.74935
9590.74935
105100.269865
115110.269865
125120.269865
135130.269865
145140.269865
155150.74935
165160.269865
175170.269865
185180.269865
195190.74935
205200.269865
215210.269865
225220.74935
235230.269865
245240.74935
255250.269865
265260.269865
275270.269865
285280.74935
295290.74935
305300.74935
315310.74935
325320.269865
335330.269865
345340.269865
355350.269865
365360.269865
375370.269865
385380.269865
395390.269865
405400.74935
415410.74935
425420.74935
435430.74935
445440.74935
455450.269865
465460.269865
475470.269865
485480.269865
495490.269865
505500.74935
515510.74935
525520.269865
535530.269865
545540.74935
555550.269865
565560.269865
575570.269865
585580.74935
595590.269865
605600.269865
615610.269865
625620.74935
635630.74935
645640.74935
655650.74935
665660.269865
675670.269865
685680.74935
695690.74935
705700.269865
715710.74935
725720.74935
735730.74935
745740.74935
755750.269865
765760.74935
775770.74935
785780.74935
795790.269865
805800.74935
815810.269865
825820.269865
835830.269865
845840.269865
855850.269865
865860.269865
875870.74935
885880.74935
895890.74935
905900.269865
915910.269865
925920.74935
935930.74935
945940.74935
955950.269865
965960.269865
975970.74935
985980.269865
995990.269865
10051000.269865

Finally we run three scenarios, collect the data and plot it.

using Plots

plotsim(data, ϵ) = plot(
data.step,
data.new_opinion,
leg = false,
group = data.id,
title = "epsilon = \$(ϵ)",
)

plt001, plt015, plt03 =
map(e -> (model_run(ϵ = e), e) |> t -> plotsim(t[1], t[2]), [0.05, 0.15, 0.3])

plot(plt001, plt015, plt03, layout = (3, 1))