Raw results are available at Zenodo:
The analysis can be performed by simply uncompressing (unzip) the results file from Zenodo in the current directory.
Date: 2025-07-10
Designer: Hiro KATAOKA (University of Tsukuba)
Hypotheses: Opinions have some effects on beliefs and vice versa; Opinions and beliefs can still converge/stabilize if they are connected
100 agents; 1 runs; 1000 games
fixed variables: EPSILON DELTA PACTIVE PREWRITE ATOM REWRITE PREHOC
controled variables: WORKFLOWS TOPICS SEEDS ALPHAS VALUES
dependent variables: Oc Bc Od Bd Bu
We use following measures. Let $A=\{1,2,\ldots,n\}$ be the set of agents. Let $T$ be the set of topics. Let $\mathcal L$ be the set of propositional formula expressed in Full Disjunctive Normal Form (FDNF). Each agent $a\in A$ has its opinions $O_a^t\colon T\to [0,1]$ and its beliefs $B_a^t\in\mathcal L$.
This measure counts the number of agents whose opinions have changed at time $t$:
$$ oc^t = \left|\left\{a\in A; d_O(O_a^t,O_a^{t-1})>\varepsilon\right\}\right| $$
such that $\varepsilon=10^{-5}$ and $d_O$ is the distance between two opinions $O$ and $O'$:
$$ d_O(O,O')=\sum_{\phi\in T}|O(\phi)-O'(\phi)|. $$
This measure counts the number of agents whose beliefs have changed at time $t$:
$$ bc^t = |\{a\in A; B_a^{t-1}\not\equiv B_a^t\}|. $$
This measure shows the maximal distance between agents' opinions at time $t$:
$$ od^t = \max\{d_O(O_a^t,O_{a'}^t);a,a'\in A\}. $$
This measure shows the maximal distance between agents' beliefs at time $t$:
$$ bd^t = \max\{d_B(B_a^t,B_{a'}^t); a,a'\in A\} $$
such that $$ d_B(B,B')=|\{m\in\mathcal M(\top);m\models B\oplus m\models B'\}| $$ where $\mathcal M(\top)$ is the set of all interpretations and $\oplus$ is exclusive or.
This measure counts the number of unique beliefs at time $t$:
$$ bu^t = |\{B_a^t;a\in A\}|. $$
Date: 2025-07-10
Performer: Hiro KATAOKA (University of Tsukuba)
The whole experiment, from scratch, can be executed through:
Hardware: AMD EPYC 7302P (16) @ 3.000GHz, Memory 128GB
OS: Ubuntu 22.04.5 LTS x86_64
Nim version: 2.2.0
Simulator version: f3237ada231042bec069ae39f2134442541625b7
Raw results are available at Zenodo:
The analysis can be performed by simply uncompressing (unzip) the results file from Zenodo in the current directory.
Before we start analyzing the results, we first compute the measures. In addition to the measures defined above, we compute following measures for opinions:
where $i=0,1$ indicates which topic is considered, either $p$ if $i=0$ or another topic, if $i=1$.
The table below shows the experiments in which agents' opinions did not converge, i.e., $od>10^{-5}$:
It contains only 20 experiments out of 1080.
In particular, no experiment uses the workflow indep.
This means that evolving opinions and beliefs together can prevent opinions from converging.
This result coincides with the convergence property of the DeGroot model. In fact, in all of the adjacency matrix $W=(w_{i\to j})_{i,j}$ of the network, there exists $k\in\mathbb N$ such that $A^k$ contains one strictly positive column where $w_{i\to j}$ is the weight of agent $i$'s opinions when they are transmitted to agent $j$:
Similarly, the table below shows the experiments in which agents' beliefs did not converge, i.e., $bd>0$:
It contains 23 experiments out of 1080.
There are 3 experiments with the workflow indep in it.
The other 20 experiments are the same as before.
Even in these experiments, more than 80 agents share the same beliefs (below are their list and the number of agents per belief set):
For the experiments with the topics nooverlap (i.e., $\{p,\lnot p\}$), there are only two beliefs.
As a summary, the number of experiments such that the same workflow is used and opinions/beliefs/opinions and beliefs converge is as follows:
Similarly, we test the stability of agents' opinions. The table below shows the list of experiments such that agents' opinions are not stable, i.e., $oc>0$ during the last 20 interactions.
It contains only one experiment. In this experiment, opinions for the topic $p$ have evolved during the last 50 iterations as follows:
Opinions are oscillating!
Note that in all of the experiments such that opinions converge, they are also stable. The table below shows the list of experiments such that $od\leq 10^{-5}$ and opinions are not stable. It contains no experiments.
The list of experiments such that opinions are stable but do not converge is as follows:
It contains 19 experiments.
The table below shows the list of experiments such that $bc>0$ during the last 20 interactions:
The same experiment is listed as above (analysis of opinion stability)
Note that in all of the experiments in wich beliefs have converged, they are also stable. The table below shows the list of experiments such that $bd=0$ and beliefs are not stable. It contains no experiments.
The list of experiments such that beliefs are stable but did not converge is as follows:
It contains 22 experiments (3 from indep and 19 from other two workflows).
The obtained results so far can be summarized as follows:
In the table above,
o means opinions only; b means beliefs only; no such prefix means opinions and beliefs.oconv means the number of experiments which satisfy $od\leq 10^{-5}$, bconv uses $bd=0$, and conv means od\leq 10^{-5}\land bd=0.Their ratio is as follows:
Now we perform ANOVA.
The results presented above show that changing workflow has an effect on opinions but has no effect on $od$, $bd$ and $bu$. Now we perform the post-hoc test (the Tukey HSD test) on the opinions.
For the opinions toward $p$, we can observe the difference between indep-bfirst and indep-ofirst.
For the opinions toward $\lnot p$ or $q$, we can observe the difference between bfirst and ofirst.
Note that this result compares opinions toward different topics.
We perform a similar analysis on a0 and a1 but using the results from the experiments with the topic nooverlap (i.e., $T=\{p,\lnot p\}$) only.
Now we could not see a clear difference on $O(p)$ by the workflow.
However, we can see the difference between indep-bfirst and indep-ofirst.
However, when we apply the analysis to the results from the experiments with the topic overlap (i.e., $T=\{p,q\}$) , we obtain different results:
indep-ofirst and indep-bfirst;indep-bfirst and ofirst-bfirst.These results show that the settings of the topic matters.
On average, if the workflow is changed from indep to bfirst,
overlap and $V_2$ make opinions higher; other values make opinions lower;nooverlap and $V_1$ make opinions lower except for the opinion toward $q$ and $\alpha=0.75$; $V_2$ make opinions lower if $\alpha=0.75$; $V_3$ make opinions lower if $\alpha\in\{0.5,0.75\}$Only opinions toward $p$ show a significant difference.
Changing the topic has an effect. In this case, $\mathcal M(p)=\mathcal M(\lnot p)=\mathcal M(q)=4$ since $\mathcal P=\{p,q,r\}$. This means that the number of 'common' models across topics matters.
The table below shows the average of the minimal, maximal, and the average of $O^\text{nooverlap}(p)-O^\text{overlap}(p)$ where $O^w$ is the opinions with the workflow $w$.
With $V_2$, overlap leads to higher opinions than nooverlap; otherwise it makes lower ones.
For opinions toward $p$, $V_1$-$V_3$ and $V_2$-$V_3$ is significant.
For opinions toward $\lnot p$ or $q$, only $V_1$-$V_2$ is significant.
In most of the cases, $V_1$ makes opinions toward $p$ higher than $V_3$.
In case of nooverlap-0.25-bfirst, the effect is not clear.
In most of the cases, $V_2$ makes opinions toward $p$ higher than $V_3$.
In case of nooverlap-0.25-bfirst, the effect is not clear.
The difference is not significant for opinions themselves. Instead, $\alpha$ affects the maximal distance between opinions and beliefs.
In both cases, changing from $\alpha=0.25$ to $\alpha=0.5$ has an effect. However, we cannot conclude that changing $\alpha$ from $0.5$ to $0.75$ has an effect.
Testing the effect of the seed makes sense because they control the initial states (agents' initial opinions/beliefs and the initial network).
The distributions of the average of final opinions are not normal in general. Hence we use the Kruskal-Wallis test.
The test shows that changing the initial condition significantly affects the final average opinions, $bd$, and $bu$.
For $od$, we apply ANOVA as their distributions are normal.
We could not see any significant difference for $od$.
Now we apply the post-hoc test (the Dunn test):
Which pair of the seeds makes the difference is not clear even if the last one ($bu$):
The following plots contain all non opinion converging experiments.
Actual beliefs and amounts are displayed above.
By plotting side by side the same experiments with bfirst and ofirst, it is possible to observe that although they give most of the time similar results, this is not always the case.
Three plots of experiments (seed: 94821) with (a) a non converging process with respect to beliefs with indep, (b) the beliefs of its network (different colors = different belief sets) at iteration 1000 and two fully converging processes with (c) ofirst and (d) bfirst.
Three plots of experiments (seed: 909432) with (a) a non stabilizing process with $V_2$, (c) a non converging process with $V_1$ and (d) a converging process with $V_3$. The network (b) shows the beliefs (different colors = different belief sets) with $V_2$ at iteration 1000.
Four plots (seed: 501765) which show the variability of the impact of connecting beliefs and opinions generated by different workflows and values.
Measures for these experiments used to draw the plots are as follows:
From the results of overlap-0.25-2-bfirst-909432
The hypotheses are supported: connecting opinions and beliefs affects the resulting opinions and beliefs. Moreover, opinions and beliefs can converge even if they are connected.