Date: 2025-01-14
Designer: Hiro KATAOKA (University of Tsukuba)
Hypotheses: Figure 4 in Sasahara et al. (2020, doi: 10.1007/s42001-020-00084-7) can be produced with interactions between beliefs only
100 agents; 1 runs; 500 games
fixed variables: WORKFLOW TOPIC ATOM EPSILON PACTIVE PREWRITE NBITERATIONS NBAGENTS NBRUNS ALPHA MU REWRITE PREHOC
controled variables: DELTAS SEEDS INITS VALUES
dependent variables: mdb ngb anb
Before verifing each hypotheses, we define the measures needed to check them. Let $A$ be the set of agents.
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 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 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 of the data and calculate the measures.
We will use $0.01$ as the significance threshold.
We first try to produce Figure 4a in Sasahara et al. (2020) with beliefs (that is, the plots where the $x$-axis shows $\delta$ and $y$-axis shows the 'belief version' measures (e.g., maximal distance between beliefs instead of opinions))
The larger $\delta$ is, the less mdb is. Thus, this figure corresponds to the figure (especially Figure 4a) in Sasahara et al.
Since $B_0$ depends on values, we can see the difference between $V_a$ and $V_b$:
They show the same tendency.
mdb from experiments with $V_a$ and it from experiments with $V_b$ follow a normal distribution according to the Shapiro-Wilk test.
For $V_a$:
and for $V_b$:
Each follow the normal distributions. Thus we can perform paird $t$ test to see whether changing values has significant effect on mdb.
The $p$-value is larger than 0.01. Thus, we cannot conclude that changing values has such effects.
Similar analysis is performed compare mdb from experiments with same $\delta$ and same values.
From the table above, mdb follows a normal distribution if $\delta\in\{1,4,5\}$. Otherwise, it does not.
Thus, we apply the paired $t$-test for $\delta\in\{1,4,5\}$ and the Wilcoxon signed-rank test otherwise.
The table above shows the results. There are significant difference between two values if and only if $\delta=1$.
Using always the Wilcoxon signed-rank cannot change the conclusion as displayed below:
Next, we try to produce the 'belief version' of the Figure 4b in Sasahara et al., i.e., the relation between $\delta$ and ngb.
The same tendency with the figure in Sasahara et al. can be observed.
The table above shows that $\delta=4$ yields $2.3$ groups on average.
This result does not depend on values on average.
We perform the statistical tests following the same protocol as above to see whether changing values has an effect on ngb.
For $V_a$, the result of the Shapiro-Wilk test is:
For $V_b$:
Both of ngb from experiments which share $V_a$ and $V_b$ follow normal distributions. Thus we can perform the paired $t$-test.
But the $p$-value is larger than the significance level, we cannot conclude that changing value has an effect.
Same analysis is performed but each data are compared $\delta$-wise.
The table above shows whether the measure ngb from the experiments which share the same $\delta$ and the same values follows a normal distribution. From this table, when $\delta\in\{1,4,5\}$, they follow normal distributions; otherwise they do not.
The results from the statistical tests are the same as before. If we apply the Wilcoxon signed-rank test for all comparisons, this does not change the results:
Finally, we see whether the network has been segregated or not.
But unfortunately the larger $\delta$ is the smaller anb is... In other words, we cannot observe the V-shaped plot as we observe in 20241209-BROD.
This is beacuse agents are on the way to converge to single group. In the plots below, each circles correspond to each agents and the same colors show the same beliefs corresponding agents have.
We can draw the same plot with $V_b$:
The following networks are taken from the experiments which uses $\delta=4$.
The plots above show that we cannot observe network segregation ($\delta=1$ may be the exception but the network has not separated completely.).
Following plots show the same plots but data from experiments which end with a single community are excluded.
Excluding the data from experiments which end with the single community shows the similar effect.
To see whether changing values has the significant effect on anb, we perform statistical tests. With the Shapiro-Wilk test, anb from $V_a$ and $V_b$ follow the normal distribution.
For $V_a$:
and for $V_b$:
Thus we perform paird $t$ tests
The $p$-value is larger than the significance level (0.01). Thus, we cannot conclude that it has the effect.
Similar analysis is performed for each experiments which share the same $\delta$.
The table above shows that anb follows a normal distribution if and only if $\delta=5$. Thus, we perform the paired $t$ test for $\delta=5$ and the Wilcoxon Signed-Rank test for other data.
The table below shows the results if we apply the Wilcoxon Signed-Rank test for all experiments.
Thus, changing values has the significant effect if and only if $\delta=1$, which is the case when agents cannot interact enough.
We could produce the 'belief version' of the Figure 4 in Sasahara et al. (2020).
This file can be retrieved from URL https://sake.re/20241210-BROD