Experiment 20260610-BROD¶

Experiment design¶

Date: 2026-06-10

Designer: Hiro KATAOKA (University of Tsukuba)

Hypotheses: The results obtained in [20260210-BROD] are preserved if different oriented rewirings are applied: More strict rewiring strategy creates more echo chambers

100 agents; 2000 games

Variables¶

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

controlled variables: EPSILONS DELTAS SEEDS REWIRINGS

dependent variables: eo eb po pb uo ub

Values¶

Note. The difference between the rewiring strategies used here and what are used in [20260210-BROD] is:

• The removed agent from the set of neighbors is always discordant;
• With minnbo, the rewiring is performed if $a'\prec_a a''$ where $a'$ is the removed agent and $a''$ is the added one.

Other definitions are the same as before.

Note. This notebook does not contain the results and analysis of the experiments with the order opbel. This is a mistake when designing the experiments. Experiments with the order are performed in separate repository.

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:

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

po/pb (Population in opinion/belief echo chambers)¶

These measures count how many agents are in opinion/belief echo chambers, respectively.

uo (Unique opinions shared within opinion echo chambers)¶

This measure counts how many opinions are shared within opinion echo chambers. Let $A_O\subseteq A$ be the set of agents in opinion echo chambers. Then, this measure (roughly) returns $|\{O_a;a\in A_O\}|$.

To avoid applying strict equivalence check, i.e., $=$, toward several float numbers, we apply the hierarchical clustering with the allowed maximal distance $10^{-4}$.

ub (Unique belief shared within belief echo chambers)¶

This measure counts how many beliefs are shared within belief echo chambers. Let $A_B\subseteq A$ be the set of agents in belief echo chambers. Then, this measure returns $|\{B_a;a\in A_B\}|$.

Experiment¶

Date: 2026-06-10

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: 2c1b6fbeb6b101f8cf965f6f875f8e7f3e5a9087

Duration and Output¶

• Duration: 5 hours and 38 minutes
• Output: 22 GB

Raw Results¶

Raw results are available at Zenodo:

Analysis¶

We will use 0.01 for the significance threshold for the statistical tests.

Computing the measures¶

First, we compute the measures.

We reuse the obtained results in [20260210-BROD] for other rewiring strategies.

Do echo chambers coincide?¶

Before seeing all data, we test whether opinion and belief echo chambers coincide. This can be achieved by testing $eo=eb$ for each result.

But in fact, in some cases (482 out of 2800; here we consider the experiments with the rewiring strategies {min,max}n{ob,bo} only), they do not coincide:

The number of such experiments aggregated by $\varepsilon$, $\delta$, and the rewiring algorithms is:

The two kinds of echo chambers always coincide with the order opbel.

Opinion echo chambers ($eo$)¶

As expected, the strong rewiring strategy allows agents to form more echo chambers. The difference caused by the order between agents is small.

$$ \begin{aligned} \text{minbo2}\\ \text{minob2}\\ \text{no rewiring}\\ \textbf{minNbo} \end{aligned} < \begin{aligned} \text{random}\\ \textbf{maxNbo} \end{aligned} < \begin{aligned} \text{maxob} \\ \text{maxbo} \end{aligned} $$

Note: we show the additional results by bold font. Other results are obtained in [20260210-BROD].

It is interesting to see the difference between maxob, maxbo, maxNbo, and random. Taking the maximal discordant neighbors does not contribute to create more echo chambers.

We observe the similar tendencies if they are aggregated by $\delta$ except for that there is a cross of the maximal rewirings.

Hence, hereafter, we focus on the two pairs of experiments:

• maxNbo and random
• norewiring, minob2, minbo2, and minNbo

Comparisons between {norewiring, minob2, minbo2} and minNbo¶

We observe the same tables for no rewiring and minob2.

Comparison between {norewiring, minob2, minbo2, minNbo} and maxNbo¶

We can observe the same tendencies in all tables: in more than 1000 runs, maxNbo creates more echo chambers.

Comparison between random and maxNbo¶

There are no direct relations between random and maxnbo:

Comparison between maxNbo and {maxob, maxbo}¶

Same tendencies but no direct relations between them.

ANOVA¶

We apply ANOVA to $eo$ (each group shares the same rewiring strategy):

The effect of the rewiring strategy is significant. Hence, we apply the post hoc tests:

The differences are significant except for:

• maxbo-maxob;
• maxnbo-random;
• minbo2-minnbo, minbo2-minob2, minbo2-norewiring, minnbo-minob2, minnbo-norewiring, minob2-norewiring.

Hence, we cannot conclude that the rewiring strategy minNbo creates different results from norewiring etc.

Population in opinion echo chambers ($po$)¶

The plot below shows:

$$ \begin{aligned} \text{no rewire}\\ \text{minbo2} \\ \text{maxbo2} \\ \textbf{minNbo} \\ \end{aligned} < \begin{aligned} \text{maxbo} \\ \text{maxob} \\ \textbf{maxNbo} \end{aligned} \leq \text{random} $$

i.e., random rewiring can make the size of the population in echo chambers larger than oriented rewirings.

We observe this tendency if we aggregate by $\delta$. However, especially if $\delta$ is small, the maximal rewiring allows the size of echo chambers larger than the random rewiring.

Comparisons between {no rewire, minbo2, minob2, minNbo} and {maxbo, maxob, maxNbo}¶

In all cases, in more than 1000 runs out of 1400, {maxbo, maxob, maxNbo} create more echo chambers. However, there are no direct relation between them.

In more than 1200 runs, the size of population is the same.

Comparisons between {maxbo, maxob, maxNbo} and random¶

There are no direct relations.

Now we try the $t$-tests:

ANOVA¶

Now we perform ANOVA.

Changing the rewiring strategies has significant effect on $po$. Now we perform the post-hoc test:

Changing the rewiring strategies has significant effect on $po$, except for:

• maxbo-maxob;
• minbo2-minNbo-minob2-norewiring.

Unique opinions in opinion echo chambers ($uo$)¶

The same tendency as before, but maxob slightly creates more unique opinions than maxbo.

$$ \begin{aligned} \text{no rewire}\\ \text{minbo2} \\ \text{minob2} \\ \textbf{minNbo} \\ \end{aligned} < \begin{aligned} \text{random}\\ \textbf{maxNbo} \\ \end{aligned} < \text{maxbo} < \text{maxob} $$

We can observe the similar tendency as before when aggregated by $\delta$.

The differences between {norewire, minbo2, minob2, minNbo} and {random, maxNbo}¶

In more than 1000 runs, random or maxNbo create more echo chambers. However, there are no direct relation between them.

Comparisons between {random, maxNbo} and maxbo¶

We can observe the same tendencies.

ANOVA¶

Now we apply the statistical test (ANOVA).

Now we apply the post-hoc test:

Changing the rewiring strategies has significant effect on $uo$, except for:

• maxNbo-random;
• minbo2-minNbo-minob2-norewire.

Relationship with $eo$¶

Note that several opinion echo chambers can share the same opinions. The list of the runs with $eo\neq uo$ is as follows:

It contains >2000 experiments. In all the runs listed above, $eo>uo$. This means that several echo chambers can share the same opinions.

Belief echo chambers ($eb$)¶

The tendency is quite similar to opinion echo chambers. However, maxbo creates more echo chambers than maxob.

$$ \begin{aligned} \text{no rewire}\\ \text{minbo} \\ \text{minob} \\ \textbf{minNbo} \\ \end{aligned} < \begin{aligned} \text{random} \\ \textbf{maxNbo} \\ \end{aligned} \leq \text{maxob} \leq \text{maxbo} $$

We could observe the same tendencies as above if aggregated by $\varepsilon$.

Comparisons between {norewiring, minbo2, minob2, minNbo} and {random, maxNbo}¶

In more than 1000 runs out of 1400, the latter creates more echo chambers than the former.

This is the same as before.

Comparisons between {random, maxNbo} and maxob¶

The tendencies are the same as before. Still there are no direct relations between them.

ANOVA¶

Now we apply ANOVA:

Changing the rewiring strategies has significant effect on $eb$. Hence we apply the post-hoc tests:

Changing the rewiring strategies has significant effect on $uo$, except for:

• maxNbo-random;
• minbo2-minNbo-minob2-norewire.

Population of belief echo chambers ($pb$)¶

When $\delta$ is small, maxbo rewiring allows the size of the population to be the largest among such algorithms; otherwise the random rewiring is the maximal.

$$ \begin{aligned} \text{no rewire}\\ \text{minbo} \\ \text{minob} \\ \end{aligned} < \textbf{minNbo} < \begin{aligned} \text{maxbo} \\ \text{maxob} \\ \textbf{maxNbo} \end{aligned} \leq \text{random} $$

If aggregated by $\varepsilon$, the random rewiring (almost) always creates larger population than the maximal rewirings.

Comparisons between {no rewire, minbo, minob} and minNbo¶

In many cases (1200 out of 1400), both yield the same population of echo chambers.

Comparison between minNbo and {maxbo, maxob, maxNbo}¶

In more than 1000 runs out of 1400, the latter creates larger population. However, there are still no direct relation between them.

Comparison between {maxbo, maxob, maxNbo} and random¶

The tendencies are the same as before except for between maxNbo and random:

For the last table, the relationship between two $po$s are more unclear than the previous two tables.

ANOVA¶

Now we perform ANOVA:

Changing the rewiring strategies has significant effect on $pb$. Now we perform the post-hoc test:

Changing the rewiring strategies has significant effect on $pb$, except for:

• maxNbo-maxob;
• minbo2-minNbo-minob2-norewire.

Unique beliefs in belief echo chambers ($ub$)¶

The relationship is similar to #belief echo chambers.

$$ \begin{aligned} \text{no rewire}\\ \text{minbo2} \\ \text{minob2} \\ \textbf{minNbo} \\ \end{aligned} < \begin{aligned} \text{random}\\ \textbf{maxNbo} \\ \end{aligned} < \text{maxob} < \text{maxbo} $$

We could observe the same tendency:

Comparisons between {no rewire, minbo2, minob2, minNbo} and {random, maxNbo}¶

In more than 1000 runs out of 1400, the latter produces more unique beliefs than the former.

Comparisons between {random, maxNbo} and maxob¶

In around 1000 runs out of 1400, maxob produces more unique beliefs than random or maxNbo. There are still no direct relations between them.

ANOVA¶

Now we perform ANOVA:

The effect of changing the rewiring strategies is significant. Now we apply the post-hoc tests:

Changing the rewiring strategies has significant effect on $ub$, except for:

• maxNbo-random;
• minbo2-minNbo-minob2-norewire.

Relationship to $eb$¶

It is not necessarily that each echo chamber shares the unique beliefs. The list of runs with $eb\neq ub$ is as follows:

It contains >1200 runs.

Connected components¶

As a complement, we show the number of (weakly) connected components.

The maximal-oriented rewirings create more connected components than the others.

Table in the paper¶

Now we show the effect of $\varepsilon$, $\delta$, and the rewiring algorithms in tables (one for $eo$ and another for $eb$). From the observations above, it is enough to show the only three results:

• results from the minimal-oriented rewirings and the no rewiring;
• results from the random rewiring;
• results from the maximal-oriented rewirings.