Date: 2025-01-14
Designer: Hiro KATAOKA (University of Tsukuba)
Hypotheses: Echo chambers are reinforced with opinions-beliefs interactions and fixed threshould for beliefs
100 agents; 1 runs; 500 games
fixed variables: WORKFLOW TOPIC ATOM DELTA PACTIVE PREWRITE NBITERATIONS NBAGENTS NBRUNS ALPHA MU REWRITE PREHOC
controled variables: EPSILONS SEEDS INITS VALUES
dependent variables: mdo mdb ngo ngb ano anb
Before verifing each hypotheses, we define the measures needed to check them. These measures are the same as 20241209-BROD and 20241210-bROD.
This measure shows the maximal distance between opinions: $$ \max_{a,a'\in A}|O_a-O_{a'}| $$ or equivalently, $$ \max_a O_a - \min_{a'} O_{a'}. $$
Intuitively, the more mdo is, the more polarized agents' opinions are.
This measure shows the maximal distance between beliefs: $$ \max_{a,a'\in A}d(B_a,B_{a'}) $$ where $d$ is the Hamming distance between the sets of models $m$: $$ d(B,B')=|\{m;m\models B\oplus m\models B\}|, $$ with $\oplus$ as the exclusive or.
Intuitively, the more mdb is, the more polarized agents' beliefs are.
This measure shows the number of groups of agents based on opinions.
The $i$-th group $G_i$ is a nonempty subset of $A$ such that $G_i\cap G_j=\emptyset$ if $i\neq j$. Each $G_i$ can be defined by performing hierarchial clustering with complete linkage: two clusters $C$ and $C'$ are merged iff $$\max_{a\in C,a'\in C'}|O_a-O_{a'}|\leq 0.1$$
Intuitively, the more ngo is, the more polarized agents' opinions are.
This measure shows the number of groups of agents based on opinions.
The $i$-th group $G_i$ is a nonempty subset of $A$ such that $G_i\cap G_j=\emptyset$ if $i\neq j$. Each $G_i$ can be defined as the maximal subset of $A$ such that all members share the same beliefs, i.e., $$ \forall i,\forall a,a'\in G_i,B_a\equiv B_{a'}. $$ Or equivalently, this measure can be calculated by $|\{B_a;a\in A\}|$.
Intuitively, the more ngb is, the more polarized agents' beliefs are.
This measure shows the average number of neighbors in another communities. Let $G_1,\ldots,G_n$ be the groups based on opinions. Let $E(\subseteq A\times A)$ be the set of edges in the network agents form. Then, this measure is defined as follows: $$ \frac{1}{n}|\{(a,a')\in E; \exists ij.a\in G_i,a'\in G_j,i\neq j\}|. $$
Intuitively, the less ano is, the more segregated the network is.
This measure shows the average number of neighbors in another communities. Let $G_1,\ldots,G_n$ be the groups based on beliefs. Let $E(\subseteq A\times A)$ be the set of edges in the network agents form. Then, this measure is defined as follows: $$ \frac{1}{n}|\{(a,a')\in E; \exists ij.a\in G_i,a'\in G_j,i\neq j\}|. $$
Intuitively, the less anb is, the more segregated the network is.
Date: 2025-01-14
Performer: Hiro KATAOKA (University of Tsukuba)
The whole experiment, from scratch, can be executed through:
Hardware: AMD EPYC 7302P, Memory 128GB
OS: Ubuntu 22.04.5 LTS x86_64
Nim version: 2.2.0
Simulator version: 747763f895c41497a0f7066177762ba2d7cb93ff
Before showing plots, we read all data and calculate measures.
Results from this experiment ($\delta=4$) are compared with them from the experiment for H0 (20241209-BROD, $\delta=8$).
We use $0.01$ as the significance level.
The standard deviation is quite larger than H0. Moreover, even if $\varepsilon$ becomes larger, mdo does not decrease drastically.
For smaller $\varepsilon$, mdo is smaller than it from H0 because opinions less than $0.2$ have disappeared after the first $15$ interactions because no beliefs support such lower opinions.
The two plots below are generated from the experiment 1439-0.1-15596-a:
For the middle $\varepsilon$, there can be some communities. For example, when $0.25$ is used for $\varepsilon$ instead of $0.1$, the evolution of opinions and the final network will be as follows:
For larger $\varepsilon$, agents' opinions are either about $0.3$ or $1.0$. This is because of the beliefs: it is very unlikely that a belief $B$ supports the opinion $1.0$ ($B$ should be $10000000$).
For example, when $\varepsilon=0.4$:
Increasing $\varepsilon$ helps to merge communities.
These points can be observed in the experiments with different cultural values $V_b$.
The table above shows that mdo from experiments for H2 and H0 follow normal distributions. Thus we can perform the paired $t$-test to see the difference between the two data.
Yes, we see the significance difference.
Here, changing values has significant effect on mdo!
We perform the same analysis for each $\delta$.
From the table above, we can apply the paired $t$-test for all cases.
We see the significant difference only if $\varepsilon=0.05$.
It is expected that something similar can be observed for ngo.
Experiments except for the following one end with a single community (i.e., no segregation).
There are more groups compared with before if $0.05<\varepsilon$. This coincides with what is observed in the previous section.
In most of the cases, ano is larger than what is observed in the previous experiment. However, this is not the case for $\varepsilon=0.05$.
The example of evolution of opinions when $\varepsilon=0.05$:
The same example but when $\varepsilon=0.1$:
From the table above, we can apply the paired $t$-test since both follow normal distributions.
We see the significant differences.
We perform the same analysis to see the difference between two cultural values.
From the table above, we can apply the paired $t$-test.
We do not see the significant differences.
We perform the same analysis for each $\varepsilon$.
The tables above show that we can apply the paired $t$-test for all of the cases.
If it is checked for each $\varepsilon$, there are the significant differences if $\varepsilon=0.15$.
The following heatmap shows the average ano when experiments are aggregated with $\varepsilon$ and ngo.
The same plot can be generated by excluding results from the experiments which yields the single community:
For $\varepsilon\leq 0.2$, we can observe less segregated network. For example, when $\varepsilon=0.15$:
The final distribution of opinions is as follows:
For example, it is difficult for agents whose opiinons are between $0.5$ and $0.55$ to follow other concordant agents because such agents are rare in this example.
Following list shows the direct edges between different communities:
For example, agent 19's neighbors and their opinions and beliefs are:
The output above shows the id of the neighbors, their beliefs (represented as 8-bit integers), and their opinions, respectively.
And agent 19 has:
Thus there are no concordant neighbors for the agent.
In most of the interactions it also did not have such neighbors:
For $0.2<\varepsilon$, we can observe more segregated network (such plot is generated when $\varepsilon=0.15$ for example in H0). For example, when $\varepsilon=0.3$:
The table above shows that ano from the experiments for H0 and H2 follow normal distributions. Thus we can apply the paired $t$-test:
Significant differences are observed.
Again we see the statistical difference.
We perform the same analysis for each $\delta$.
From the tables above, ano from the experiment follow if $\delta\in\{0.3,0.4\}$. Thus, we will apply the paired $t$-test for these two $\delta$ and the Wilcoxon signed-rank test otherwise.
But for pairwise, we do not see...
This file can be retrieved from URL https://sake.re/20241211-BROD