Experiment 20250413-BROD¶

Note: The workflow used in this experiment is oddw,br,of,barc (i.e., perform the synchronizations after propagating both of opinions and beliefs). This is not what is intended. The experiment with corrected workflow is performed as [20250414-BROD].

Experiment design¶

Date: 2025-04-11

Designer: Hiro KATAOKA (University of Tsukuba)

Hypotheses: Echo chambers are reinforced if opinions and beliefs evolve together with synchronization

100 agents; 1 runs; 5000 games

Variables¶

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

controlled variables: EPSILONS DELTAS SEEDS

dependent variables: eo eb

Values¶

Measures¶

Before verifying each hypotheses, 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., $D_O^t(C)$

to all of the communities, 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., $M_B^t(C)=0$;
• beliefs have been reinforced, i.e., $D_B^t(C)$

to all of the communities, 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: 5 hours and 30 minutes
• Output: 22 GB

Raw Results¶

Raw results are available at Zenodo:

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¶

There are only one experiment whose $T_O'$ is larger than $4500$:

In this experiment, agents' opinions have been evolved as follows:

Some opinions are oscillating.

Beliefs¶

There are only one experiment whose $T_B'$ is larger than $4500$: the same experiment as above.

Network¶

There are 955 experiments out of 1400 such that $T_N'$ is larger than $4500$:

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 is stable if this is possible (i.e., no agents are stable).

Computing the measures¶

Overall results in a table¶

Initial status¶

Following table shows some statistic of the number of components which satisfy corresponding conditions:

And the number of agents in the corresponding components is:

After the $T$ interactions¶

We show the same tables as before but obtained from the results after the $T$-th interaction.

Pairwise comparison between Eq. (both) and Eq. (coherence)¶

Average agents in echo chamber?¶

Agents in echo chambers?¶

$eo$ and $eb$¶

The following two tables shows the number of experiments grouped by how $eo$ and $eb$ changed between the two experiments and the ratio to all of the experiments (1400).

Following table is the list of experiments such that both of opinion and belief echo chambers have been reinforced.

Especially, the list of experiments with $\varepsilon=0.2$ and $\delta=2$ is:

The initial network is:

On the contrary, the list of experiments such that $\varepsilon=0.2$, $\delta=2$, and $eo$ and $eb$ are reduced is:

Since initial conditions and how network/beliefs evolve is randomized, we compare the mean of $eo$ and $eb$.

The following table is the list of configurations of $\varepsilon$ and $\delta$ such that $eo$ decreases xor $eb$ decreases:

The direction is the same.

When opinions and beliefs stabilize¶

Evolution of the network in plot¶

We see the evolution of the network using some plots:

The plots below are taken from the experiment 0.2-2-722393.

Comparisons by plot¶

Effect of $\varepsilon$ on $eo$¶

Effect of $\delta$ on $eb$¶

Statistical tests¶

We use 0.01 as the significance threshold.

First, we test the whole of the data:

Both has the normality. Thus we perform the paired $t$-test:

Thus we can see that the two serieses of data are different statistically.

Next, we test the pairwise test. We perform:

• paired $t$-test if two series of data have a normality;
• Wilcoxon signed-rank test otherwise.

We could not see clear difference in general.

Conclusion¶

The hypothesis is supported on average.