Experiment 20260615-BROD¶

Experiment design¶

Date: 2026-06-15

Designer: Hiro KATAOKA (University of Tsukuba)

Hypotheses: The results obtained in [20260610-BROD] are preserved if different order (opbel) over agents 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 belop as this is a complement of [20260610-BROD] with the order opbel only.

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: 2026-06-15

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 31 minutes
• Output: 21 GB

Raw Results¶

Experimental 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] and [20260610-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:

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.

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

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.

Pairwise comparisons¶

We cannot observe direct relations by changing the rewiring strategies, while overall we can see the same tendencies as before:

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;
• between minNbo, minNob, minob2, minbo2, and norewiring

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

This results support the following relationship:

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

Here, it is interesting to see:

• minNob and minNbo produced roughly the same number of echo chambers, while
• maxNob produced more echo chambers than minNbo, which is roughly the same as random.

This may be because maxNob first evaluate the difference between opinions, which is more strict evaluation than measuring the difference between beliefs, as:

• different opinions should be supported by different beliefs, while
• different beliefs can support the same opinions

We also apply ANOVA after aggregating the data (computing the mean of the runs that share the same parameters except for seed):

The suggestion from ANOVA is slightly different:

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

Population in opinion echo chambers ($po$)¶

The plot below shows:

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

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.

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:

• between maxbo, maxNob, and maxob;
• minNbo, minNob, minob2, and norewiring.

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

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

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

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

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 >3000 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.

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

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

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.

This supports the following relationship:

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

We also apply ANOVA after aggregation:

Tendencies are slightly different:

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

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.

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

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

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-maxNob;
• minbo2-minNbo-minob2-norewire.

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

The relationship is similar to #belief echo chambers.

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

We could observe the same tendency:

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.

Summary so far¶

With opinion echo chambers ($eo$):

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

With the results from $po$, maxNob produced smaller echo chambers than maxNbo.

With belief echo chambers ($eb$):

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

With the results from $pb$, maxNob produced larger echo chambers than maxNbo.

The differences are:

• whether maxNob produces more echo chambers than random;
• whether maxbo produces more echo chambers than maxob.

Hence, apparently the effect of maximal-oriented rewirings is not monotonic: sometimes it does not produce more echo chambers than random rewiring, sometimes it creates opinion echo chambers but not belief echo chambers...

It seems that components are not well segregated. And some belief echo chambers are not opinion echo chambers.

But there are no direct relation.

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.

For the standard deviations are: