Experiment 20250527-BROD¶

Experiment design¶

Date: 2025-05-27

Designer: Hiro KATAOKA (University of Tsukuba)

Hypotheses: Echo chambers are observed if opinions evolve independently of beliefs

100 agents; 1 runs; 3000 games

Variables¶

fixed variables: DELTA WORKFLOW TOPIC ATOM PACTIVE PREWRITE NBITERATIONS NBAGENTS NBRUNS ALPHA MU REWRITE VALUE PREHOC ORDER

controled variables: EPSILONS SEEDS

dependent variables: eo

Values¶

Measures¶

Before verifying 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 $t$. Let $T$ be the number of interactions (i.e., 3000).

eo (Opinions echo chamber)¶

This measure counts the ratio of communities (i.e., strongly connected components) such that:

• they are segregated, i.e., $L^t(C)\leq 0.5$;
• opinions are homogeneous, i.e., $M_O^t(C)\leq 10^{-3}$;
• opinions have been reinforced, i.e., $D_O^t(C)$

to all of the communities, 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:

$$ eo^t=|\{C\in\mathcal S^t;L^t(C)\leq 0.5\land M_O^t(C)\leq 10^{-3}\land D_O^t(C)\}|. $$

In complement, we compute the proportion of agents involved in echo chambers.

Experiment¶

Date: 2025-05-27

Performer: Hiro KATAOKA (University of Tsukuba)

The whole experiment, from scratch, can be executed through:

Parameter file: params.sh

Executed command (script.sh):

Hardware: AMD EPYC 7302P (16) @ 3.000GHz, Memory 128GB

OS: Ubuntu 22.04.5 LTS x86_64

Nim version: 2.2.0

Simulator version: ca6b226c686026a0d615dd968fae0b25a0017a5d

Duration and Output¶

• Duration: 38 minutes
• Output: 1.9 GB

Raw Results¶

Raw results are available at Zenodo:

Analysis¶

Stabilization check¶

Before computing the measures, we test whether opinions and the strongly connected components are stable or not. We regard them as stable iff they do not change during the last 20 interactions: $\forall t\in [T-20,T],\forall a\in A,|O_a^t-O_a^T|\leq 10^{-3}$ for opinions and $\forall t\in [T-20,T],\mathcal S^t=\mathcal S^T$ for the network. This means that agents do not change their opinions and neighbors during the last 10 times (in expectation) they are active.

Opinions¶

The table above shows the distribution of the minimal time $T'$ such that $\forall t\in [T',T],\forall a\in A,|O_a^t-O_a^T|\leq 10^{-3}$. It shows that for all experiments $T'<T-20$. Thus, opinions are stable.

Network¶

The table above shows the distribution of the minimal time $T'$ such that $\forall t\in [T',T],\forall a\in A,\mathcal S^t=\mathcal S^T$. It shows that for all experiments $T'<T-20$. Thus, the network is stable.

Computing the measures¶

Since opinions and the network are stable, now we compute the measure.

The table above shows some statistics of $eo$, agents in opinions echo chambers, $|\mathcal S^T|$, and the number of singletons ($|\{C\in\mathcal S^T;|C|=1\}|$). It shows that in all experiments $0<eo^T$.

It is noteworthy that in all experiments $2<eo^T$. This means that agents never form single component.

Some pltos¶

The list of experiments where agents create $11(=\min eo^T)$ echo chambers is as follows:

The final networks colored with respect to agents' opinions from these experiments are as follows:

Plots from the experiments with $\varepsilon=0.5$ are as follows. Here, mdo means the maximal distance between opinions.

Even if $\varepsilon=0.5$, they will be split into several components. Moreover, agents' opinions are not homogeneous among $A$.

Time evolution¶

When $t=30$, agents are roughly split into several weakly connected components (WCCs) with respect to their opinions. After that, within each WCC, agents are again split into various components.

We look at some components. Here is the list of agents such that $O_a^{30}>0.7$ (this corresponds to the set of members in the components colored with red)

At $t=30$, $eo^t$ is equal to or less than $3$ in the subgraph. After that, they form 6 echo chambers at time $t=80$.

The final network¶

In the plots below, echo chambers are surrounded by solid lines; other components are surrounded by dashed lines.

Different echo chambers can share the same opinions. For example, the list of opinions in the two echo chambers at the bottom of the plot is:

They are almost the same.

The initial network¶

We show the initial network of the same experiment as before.

Obtaining the same plot as Figure 3 in Sasahara et al. (2020)¶

We produce the same plot as Figure 3 in Sasahara et al. (2020).

Message diversity¶

Evolution of opinions¶

Evolution of the network¶

These three plots are at $t=0$, $t=15$, and $t=30$, respectively.

Conclusion¶

The hypothesis is supported: agents always form echo chambers.