Experiment 20250412-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 reinforced if opinions and beliefs evolve together but without synchronization

100 agents; 1 runs; 5000 games

Variables¶

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

controled variables: EPSILONS DELTAS SEEDS

dependent variables: eo 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 $t$.

eo (Opinion echo chamber)¶

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

• they are segregated, i.e., $L^t(C)\leq 0.5$;
• opinions are homogeneous, i.e., $M_O^t(C)\leq 10^{-4}$;
• opinions have been reinforced, i.e., $D_O^t(C)$,

where

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

$$ M_O^t(C) = \max_{a,a'\in C}|O_a^t-O_{a'}^t| $$

and

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

Here, $[t_C,T]$ is the maximal time window such that $\forall t\in[t_C,T]$, $C\in\mathcal S^t$ and $T$ is the number of iterations.

More formally, $eo$ is defined as:

$$ eo^t=|\{C\in\mathcal S^t;L^t(C)\leq 0.5\land M_O^t(C)\leq 10^{-4}\land D_O^t(C)\}| $$

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., $M_B^t(C)=0$;
• opinions have been reinforced, i.e., $D_B^t(C)$,

where

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

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:

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

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: 2 hours and 9 minutes
• Output: 22 GB

Raw Results¶

Raw results are available at:

Analysis¶

Before analysis: stability check¶

Before we analyze the results, we test whether opinions/beliefs/networks have been stable. We compute the minimum time $T_O',T_B',T_N'$ such that:

• $\forall t\in [T_O',T]$ and $\forall a\in A$, $|O_a^t-O_a^T|\leq 10^{-4}$ where $T=$ 5000;
• $\forall t\in [T_B',T]$ and $\forall a\in A$, $B_a^t=B_a^T$;
• $\forall t\in [T_N',T]$ and $\forall a\in A$, $N_a^t=N_a^T$,

and show some statistic of them.

Opinions¶

In some experiments, opinions have not been stable even after the 4999th interactions. The table below is the list of experiments such that $T'>4500$:

The two plots below shows the evolution of opinions in the two experiments with $T_O'=4999$:

Thus, opinions have been stable while there are a few exceptions.

Beliefs¶

In all experiments, beliefs have been stable as $T_B'<4500$ for all experiments.

Network¶

In 984 experiments out of 1400, $T_N'>4500$:

This is because for some agents,

$$ |N_a^T|\geq |\{a'\in A;|O_a^T-O_{a'}^T|\leq 10^{-4}\land B_a^T=B_{a'}^T\}| $$

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.

Thus, the network has been stable if possible.

Computing measures¶

We compute the measures $eo$ and $eb$.