Experiment 20250411-BROD¶

Note. It has been found that the code implementing rewiring was not taking into account the current opinions and beliefs but those at the beginning of the workflow instead. Although this does not impact the main results of the notebook, the data is not reliable anymore. These experiments have been corrected (see [20260209-BROD] and [20260217-BROD]).

Experiment design¶

Date: 2025-04-11

Designer: Hiro KATAOKA (University of Tsukuba)

Hypotheses: Echo chambers are observed if beliefs evolve independently of opinions

100 agents; 1 runs; 5000 games

Variables¶

fixed variables: EPSILON WORKFLOW TOPIC ATOM PACTIVE PREWRITE NBITERATIONS NBAGENTS NBRUNS ALPHA MU REWRITE VALUE PREHOC

controled variables: DELTAS SEEDS

dependent variables: eb

Values¶

Measures¶

Before testing the hypothesis, we define the measures needed to check them. Let $A$ be the set of agents. Let $\mathcal S^t$ be the set of strongly connected components of the network of agents at time $t$.

eb (Belief echo chamber)¶

This measure counts the number of communities (i.e., strongly connected components) such that:

• they are segregated, i.e., $L^t(C)\leq 0.5$;
• beliefs are homogeneous, i.e., $M_B^t(C)=0$ (or equivalently, $\forall a,a'\in C$, $B_a^t=B_{a'}^t$);
• opinions have been reinforced, i.e., $D_B^t(C)$,

where

$$ L^t(C)=\frac{|\{(a,a')\in N^t;a\in C\land a'\notin C\}|}{|\{(a,a')\in N^t;a\in C\}|}, $$

$$ M_B^t(C) = \max_{a,a'\in C}d_B(B_a^t,B_{a'}^t) $$

and

$$ D_B^t(C)=\forall s\in [t_C,T),M_B^s(C)\geq M_B^{s+1}(C). $$

Here, $d_B$ is the Hamming distance over the models of two beliefs:

$$ d_B(B,B')=|\mathcal M(B)\setminus\mathcal M(B')|+|\mathcal M(B')\setminus\mathcal M(B)|. $$

More formally, $eb$ is defined as:

$$ eb^t=|\{C\in\mathcal S^t;L^t(C)\leq 0.5\land M_B^t(C)=0\land D_B^t(C)\}| $$

Experiment¶

Date: 2025-04-11

Performer: Hiro KATAOKA (University of Tsukuba)

The whole experiment, from scratch, can be executed through:

Parameter file: params.sh

Executed command (script.sh):

Hardware: AMD EPYC 7302P (16) @ 3.000GHz, Memory 128GB

OS: Ubuntu 22.04.5 LTS x86_64

Nim version: 2.2.0

Simulator version: 8244359cd4a790a216e5c162fe0faa9f6ffc0754

Duration and Output¶

• Duration: 11 minutes and 30 seconds
• Output: 2.2 GB

Raw Results¶

Raw results are available at Zenodo:

Analysis¶

Before analysis: stabilization check¶

Before we analyze the results, we test whether everything are stable.

Beliefs¶

The following table shows some statistic of the distribution of the minimal time $T_B'$ and $T_N'$ such that:

• $\forall t\in [T_B',T]$ and $\forall a\in A$, $B_a^t=B_a^T$;
• $\forall t\in [T_N',T]$ and $\forall a\in A$, $N_a^t=N_a^T$.

It shows that beliefs are stable after $T$ interactions.

However, the network never stabilizes in some cases. In fact, in 78 experiments out of 140, $T_N'$ is larger than $4500$:

This is because for some agents,

$$ |N_a^T|\geq |\{a'\in A; B_a^T=B_{a'}^T\}| $$

and thus $a$ has to find other concordant agent $a'$ who is not in $N_a$ while this is impossible. We call such agents as unstable agents.

We test whether the two sets of the experiments are the same:

• the experiments whose $T'$ is larger than $4500$;
• the experiments such that there are at least one unstable agents.

This means that unstable agents prevent the network from stabilizing. Thus, the network has been stable after $T$ iterations if this is possible.

Calculating the measures¶

As checked before, everything has been stable if this is possible. Now we compute the measures.

Following boxplot shows the distribution of $eb$:

Clearly, in some experiments we could not observe belief echo chambers. The list of such experiments is as follows:

The following two plots are the plots corresponding to the two experiments listed above.

We can observe that beliefs are not homogeneous.

Overall results in a table¶

Some plots¶

All plots¶

We show the final networks of all experiments.

Conclusion¶

We can observe belief echo chambers, while there are a few exceptions.