Note. It has been found that the code implementing rewiring was not taking into account the current opinions and beliefs but those at the beginning of the workflow instead. Although this does not impact the main results of the notebook, the data is not reliable anymore. These experiments have been corrected (see [20260209-BROD] and [20260217-BROD]).
Date: 2025-04-10
Designer: Hiro KATAOKA (University of Tsukuba)
Hypotheses: Echo chambers are observed if opinions evolve independently of beliefs
100 agents; 1 runs; 5000 games
fixed variables: DELTA WORKFLOW TOPIC ATOM PACTIVE PREWRITE NBITERATIONS NBAGENTS NBRUNS ALPHA MU REWRITE VALUE PREHOC
controled variables: EPSILONS SEEDS
dependent variables: eo
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 strongly connected components of the network of agents at time $t$.
This measure counts the number of communities (i.e., strongly connected components) 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:
$$ eo^t=|\{C\in\mathcal S^t;L^t(C)\leq 0.5\land M_O^t(C)\leq 10^{-4}\land D_O^t(C)\}|. $$
Date: 2025-04-10
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: 8244359cd4a790a216e5c162fe0faa9f6ffc0754
Before we analyze the results, we test whether everything are stable. To test this, we compute the distribution of $T_O'$ and $T_N'$ such that:
Clearly, this shows that opinions have been stable in all experiments. However, the network does not have to stabilize.
In 65 experiments out of 200, $T'$ is larger than $4500$:
This is because for some agents,
$$ |N_a^T|\geq |\{a'\in A;|O_a^T-O_{a'}^T|\leq 10^{-4}\}| $$
and thus $a$ has to find other concordant agent $a'$ who is not in $N_a$ while this is impossible. We call such agents as unstable agents.
We test whether the two sets of the experiments are the same:
This means that unstable agents prevent the network from stabilizing. Thus, the network has been stable after $T$ iterations if this is possible.
We also show the distribution of $T_S'$ such that $T_S'$ is the minimal and $\forall t\in [T_S',T]$, $\mathcal S^t=\mathcal S^T$.
There are 19 experiments whose $T_S'$ is larger than $4500$:
As checked before, everything has been stable if this is possible. Now we compute the measures.
Following table shows some statistic of $eo$:
It shows that in all experiments, $eo^T>0$.
The number of components such that they are singletons, segregated, homogeneous, reinforcing, and opinion echo chambers are as follows:
And the number of agents in components with each characteristic is:
Below we show all of the final networks.
The hypothesis is supported: we can observe echo chambers.
This file can be retrieved from URL https://sake.re/20250410-BROD
