A Model for the Dynamics of Polarization

All the functions and structures needed are located in polarization.py. Implemented for Python 3.

Dependencies

This project depends on NumPy (tested on 1.19.0) which may be installed from a console using pip:

python3 -m pip install numpy

Optionally, we recommend installing Matplotlib for creating graphs and charts, and Jupyter Notebooks for running simulations interactively.

python3 -m pip install matplotlib notebook

To install all dependencies you can also run:

python3 -m pip install -r requirements.txt

Running Simulations

If you want to try, or recreate, the simulations in Simulations.ipynb for yourself, download this repository, install the corresponding dependencies, and run the jupyter notebook server with access to polarization.py. For example, in the same folder as the polarization.py file is located, run:

jupyter notebook

Select Simulations.ipynb, and make sure to set it as trusted, or just re-run the desired cells.

The Simulation Class

You can run simulations creating an instance of the Simulation class and either its run(max_time=100, smart_stop=True) method or iterating it like any other python iterator. As always, you need to provide the initial belief state, an influence graph, and a belief update function. Optionally you can define an alternative function for computing polarization, other than Esteban-Ray.

By default, Simulation uses the CLASSIC (or regular) update function, but can be changed by passing an update_fn parameter. Also by default, the polarization measure is Esteban-Ray with 5 bins to discretize the belief state into a distribution, you can change the polarization measure by passing a pol_measure function, or just change the num_bins argument.

import polarization as plr

simulation = plr.Simulation(
    plr.build_belief(plr.Belief.UNIFORM),
    plr.build_influence(plr.Influence.CLIQUE),
    plr.make_update_fn(plr.Update.CLASSIC)
)

polarization_history, belief_history, pol = simulation.run()

We may want to plot the evolution of polarization over time:

import matplotlib.pyplot as plt

plt.plot(polarization_history)
plt.show()

Polarization Measure and Update Function for Simulation

For creating an update function to use with the Simulation class:

• make_update_fn(update_type): The resulting function is of type update_fn(belief_vec, inf_graph).

For customizing the polarization measure with the Simulation class:

• make_pol_er_discretized_func(alpha, K, num_bins): The resulting function is of type pol_ER_discretized(belief_state).

For more examples you can look at code in the Jupyter Notebooks of this repository, e.g. Simulations.ipynb for the usage of the polarization module.

Initial Belief Configurations

The definition of Belief.{UNIFORM, MILD, EXTREME, TRIPLE} is as follows:

There is a new function that allows us to generate new initial belief configurations based on a 5 bins Esteban-Ray Polarization measure, evently distributing all agents in clusters between the [0, 1] interval.

Belief	[0, 0.2)	[0.2, 0.4)	[0.4, 0.6)	[0.6, 0.8)	[0.8, 1]
UNIFORM	o	o	o	o	o
MILD		o		o
EXTREME	o				o
TRIPLE	o		o		o

To generate such configurations build_belief is provided.

Alternative Functions for Simulations

Apart from the Simulation class, the polarization module provides:

• build_belief: Function that returns a belief vector based on a Belief scenario type (UNIFORM, MILD, EXTREME, TRIPLE, CONSENSUS), this function produces the new definition of a Belief configuration, as such, it is the recommended way of building initial belief vectors.

• build_influence: Helper function that returns an influence graph based on an Influence scenario type (CLIQUE, GROUP_2_DISCONECTED, GROUP_2_FAINT, INFLUENCERS_2_BALANCED, INFLUENCERS_2_UNBALANCED, CIRCULAR).

• run_until_stable: Legacy helper function that runs a simulation until convergence is achieved, or max_time is reached (defaults to 100), given an initial belief vector, influence graph, and the old definition of an update function (defaults to CLASSIC, see the OldUpdate enum). Deprecated. Produces different output than calling the Simulation's run() method.

• build_old_belief: Legacy helper function that returns a belief vector based on a Belief scenario type (UNIFORM, MILD, EXTREME, TRIPLE, CONSENSUS). Deprecated. Produces different output than using build_belief.

Helper Functions

For creating the initial beliefs:

• build_belief(belief_type: Belief, num_agents, **kwargs)

For creating the influence graph:

• build_influence(inf_type: Influence, num_agents, **kwargs)

Alternatively:

• build_inf_graph_clique(num_agents, belief_value)
• build_inf_graph_2_groups_disconnected(num_agents, belief_value)
• build_inf_graph_2_groups_faint(num_agents, weak_belief_value, strong_belief_value)
• build_inf_graph_2_influencers_balanced(num_agents, influencers_incoming_value, influencers_outgoing_value, others_belief_value)
• build_inf_graph_2_influencers_unbalanced(num_agents, influencers_outgoing_value_first, influencers_outgoing_value_second, influencers_incoming_value_first, influencers_incoming_value_second, others_belief_value)
• build_inf_graph_circular(num_agents, value)

It is recommended to use the general build_influence(*args, **kwargs).

For creating the legacy initial beliefs:

• build_uniform_beliefs(num_agents)
• build_old_mild_beliefs(num_agents, low_pole, high_pole, step)
• build_old_extreme_beliefs(num_agents)
• build_old_triple_beliefs(num_agents)
• build_consensus_beliefs(num_agents, belief)

It is recommended to use the general build_belief(belief_type: Belief, num_agents, **kwargs) for the current configurations.

It is recommended to use the general build_old_belief(belief_type: Belief, num_agents, **kwargs) for the old configurations.