Date: 2026-08-26
Designer: Hiro KATAOKA (University of Tsukuba)
Hypotheses: The overall results are preserved with corrected implementation of the measures
100 agents; 2000 games
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
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 $t$.
This measure counts the number of partitions such that:
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:
$$ eo^t=|\{C\in\mathcal S^t;L^t(C)\leq\theta\land M_O^t(C)\leq 10^{-4}\land D_O^t(C)\}| $$
This measure counts the number of partitions such that:
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:
$$ eb=|\{C\in\mathcal S^t;L^t(C)\leq\theta\land M_B^t(C)=0\land D_B^t(C)\}| $$
These measures count how many agents are in opinion/belief echo chambers, respectively.
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 $10^{-4}$.
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 $|\{B_a;a\in A_B\}|$.
Date: 2026-08-26
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
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).
First, we compute the measures.
Note that the implementation of the measures has been corrected from [20260621-BROD].
Alternatively and to run the script stand alone, use notebook.py (which is generated from the notebook so far).
For the statistical tests, we will use 0.01 as the significance threshold.
The average number of opinion echo chambers in each rewiring strategy is:
This results show that: $$ \begin{aligned} \text{minbo2}\\ \text{minob2}\\ \text{no rewiring}\\ \text{swapMinMaxNbo} \\ \text{swapMinMaxNob} \\ \end{aligned} < \text{maxNbo} < \text{maxNob} < \text{randomWCbo} < \text{randomWCob} < \text{maxbo} < \text{maxob} $$
This is different from what has been obtained so far:
$$ \begin{aligned} \text{minbo2}\\ \text{minob2}\\ \text{no rewiring}\\ \text{swapMinMaxNbo} \\ \text{swapMinMaxNob} \\ \end{aligned} < \begin{aligned} \text{maxNbo}\\ \text{randomWCbo} \end{aligned} \leq \begin{aligned} \text{maxNob}\\ \text{randomWCob} \end{aligned} < \begin{aligned} \text{maxob} \\ \text{maxbo} \end{aligned} $$
Now we test ANOVA. It shows that changing the rewiring strategy has significant effect on the number of opinion echo chambers:
Hence, we perform the post-hoc test:
This result supports the results from the comparisons of the average number of echo chambers.
The average number of belief echo chambers in each rewiring strategy is:
This results suggest:
$$ \begin{aligned} \text{no rewire}\\ \text{minbo} \\ \text{minob} \\ \end{aligned} \leq \begin{aligned} \text{minNbo} \\ \text{minNob} \\ \end{aligned} < \begin{aligned} \text{randomWCbo} \\ \text{randomWCob} \\ \text{maxNbo} \\ \text{maxNob} \end{aligned} \leq \text{maxob} < \text{maxbo} $$
Compared to what is reported before (reposted below), minNbo and minNob can be classified in different groups from norewiring, minobo, and minob:
$$ \begin{aligned} \text{no rewire}\\ \text{minbo} \\ \text{minob} \\ \text{minNbo} \\ \text{minNob} \\ \end{aligned} < \begin{aligned} \text{randomWCbo} \\ \text{randomWCob} \\ \text{maxNbo} \\ \text{maxNob} \end{aligned} \leq \text{maxob} < \text{maxbo} $$
Now we perform ANOVA:
The results from ANOVA shows that the effect of changing the rewiring strategies on the number of belief echo chamber is significant.
Then we perform the Tukey HSD post hoc test:
From this results:
min{bo,ob}2 or norewiring and swapMinMaxN{bo,ob};maxNbo/randomWCbo and maxNob/randomWCob should be classified as different groups.Hence,
This results suggest:
$$ \begin{aligned} \text{no rewire}\\ \text{minbo} \\ \text{minob} \\ \text{minNbo} \\ \text{minNob} \\ \end{aligned} < \begin{aligned} \text{randomWCob} \\ \text{maxNob} \end{aligned} < \begin{aligned} \text{randomWCbo} \\ \text{maxNbo} \\ \end{aligned} < \text{maxob} < \text{maxbo} $$
The largest difference is that when agents follow no rewiring they do not produce opinion (and belief in principle) echo chambers.
$\varepsilon=0.5$ and $\delta=7$ correspond to very tolerant agents.
For example, in the run norewiring-0.5-7-544371, the measure to detect opinion echo chambers works as follows:
It identifies no echo chambers because the largest component violates the reinforcement property by the small difference in the maximal distance between $t=121$ and $t=122$ ($2.2\times 10^{-15}$ and $2.3\times 10^{-15}$, respectively).
What happens if we continue testing the reinforcement even if the reinforcement is the only one reasons not to be an echo chamber? If the largest component showed above violates it several times, all of the information should be printed below:
It violates several times while the difference is not large. For example, the difference in the maximal distance between $t=31$ and $t=32$ is around $1.3\times 10^{-7}$.
Hence, it is interesting to see how testing the reinforcement property can be relaxed.
This file can be retrieved from URL https://sake.re/20260621-BROD-a