Experiment 20260621-BROD-b¶

Experiment design¶

Date: 2026-08-27

Designer: Hiro KATAOKA (University of Tsukuba)

Hypotheses: The overall results are preserved with a threshold-based reinforcement

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¶

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 partitions of the network of agents at time $t$.

eo (Opinion echo chamber)¶

This measure counts the number of partitions such that:

• they are segregated, i.e., $L^t(C)\leq\theta$ where $0<\theta<1$;
• 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

$$ D_O^t(C)=\forall s\in [t_C,T),M_O^s(C)\geq M_O^{s+1}(C) + 10^{-4}. $$

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:

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

eb (Belief echo chamber)¶

This measure counts the number of partitions such that:

• they are segregated, i.e., $L^t(C)\leq\theta$ where $0<\theta<1$;
• 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

$$ D_B^t(C)=\forall s\in [t_C,T),M_B^s(C)\geq M_B^{s+1}(C)+10^{-4}. $$

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:

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

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: 2026-08-27

Performer: Hiro KATAOKA (University of Tsukuba)

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

Parameter file: params.sh

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

OS: Ubuntu 22.04.5 LTS x86_64

Raw Results¶

This experiment is a reanalysis of the existing experiment [20260210-BROD], [20260610-BROD], [20260615-BROD], and [20260621-BROD]. Hence, no new simulations (or runs of the SOBA simulator) are needed.

To perform the following analysis, obtain the raw results of the experiment from these repositories and move all files under the directory called 'results'.

Note that files required to perform the analysis are downloaded by executing the script file utils/unzip.sh (be careful, this script automatically access to Zenodo. Please do not execute the file frequently).

Analysis¶

First, we compute the measures.

Alternatively, you can use the script notebook.py which is based on this notebook so far to execute on machines with multiple cores.

For the statistical tests, we will use 0.01 as the significance threshold.

Effect on opinion echo chambers¶

The average number of opinion echo chambers in each rewiring strategy is:

Results from ANOVA:

As it reports the significant difference by changing the rewiring strategies, now we apply its post-hoc test (Tukey HSD):

Hence, there results show that:

$$ \begin{aligned} \text{minbo2}\\ \text{minob2}\\ \text{no rewiring}\\ \text{swapMinMaxNbo} \\ \text{swapMinMaxNob} \\ \end{aligned} < \text{maxNbo} \color{red}< \text{randomWCbo} \leq \begin{aligned} \text{maxNob}\\ \text{randomWCob} \end{aligned} < \text{maxob} \color{red}< \text{maxbo} $$

The red part is the difference from the previous results (it was regarded as no significant difference).

Effect on belief echo chambers¶

The average number of belief echo chambers in each rewiring strategy is:

The results from ANOVA is:

Now we apply Tukey HSD test as ANOVA reports that the effect of changing the rewiring strategies is significant:

These results show that:

$$ \begin{aligned} \text{no rewire}\\ \text{minbo} \\ \text{minob} \\ \text{minNbo} \\ \text{minNob} \\ \end{aligned} < \begin{aligned} \text{randomWCob} \\ \text{maxNob} \end{aligned} \color{red}< \begin{aligned} \text{randomWCbo} \\ \text{maxNbo} \\ \end{aligned} \leq \text{maxob} < \text{maxbo} $$

The only one difference is denoted in red (in the previous results $=$)

Table 1¶

Table 2¶

Table 3¶

For no rewiring, it is still reported that agents cannot create single large echo chambers in the end. For example, the list of the runs with $\varepsilon=0.5$ and $\delta=7$ (hence very tolerant agents) and no rewiring strategy such that no opinion echo chambers are detected is:

It contains 19 runs (out of 20 runs).

For example, in the run norewiring-0.5-7-756485, how the measure for opinion echo chambers worked is:

It still violates the reinforcement property (the difference is $2.8\times 10^{-4}$, which is larger than $10^{-4}$)

In that component, overall, opinions are converging:

However, while $\min O_a^{24}=\min O_a^{25}$, $\max O_a^{24}<\max O_a^{25}$ (yet the difference is tiny), it still violates the reinforcement. Possibly this is because of the asynchronous opinion/belief update, possibly because we can still improve the (mathematical) definition of this property.

In fact, with the reinforcement property, it is hard to create echo chambers with the no rewiring strategy. The list of runs with the strategy and ends with at least one opinion echo chamber is:

It contain only 9 runs (out of 1400).

Figure 4/5¶

All of them are the same as [20260621-BROD].

Figure 4a/5a¶

In this analysis, $eo=eb=0$ which is the same as before. Hence, the plot can be preserved.

They are the same as what is in the paper.

Figure 4b/5b¶

Figure 4c/5c¶

Figure 4d/5d¶

Conclusion¶

• Overall results on the effect of the rewiring strategies on the number of opinion/belief echo chambers are preserved from [20260621-BROD]
• On the other hand, the measures still do not identify echo chambers in the final networks with rewiring strategies that rarely modify the network, even if opinions (or beliefs) are converged (hence intuitively large single echo chamber can be found). This is because the maximal distance in opinions/beliefs does not decrease monotonically.