# Evolution towards Linguistic Coherence in Naming Game with Migrating Agents

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## Abstract

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## 1. Model

- The Speaker randomly selects a word from its inventory (or invents a new word if its inventory is empty) and transmits it to the Hearer. To invent a word, all agents have M different words at their disposal, and one of them is randomly selected.
- If the Hearer has the transmitted word in its inventory, the interaction is a success, and both players maintain only the transmitted word in their inventories.
- If the Hearer does not have the transmitted word in its inventory, the interaction is a failure, and the Hearer updates its inventory by adding this word to it.

## 2. Results

#### 2.1. State-Independent Migration

#### 2.2. State-Dependent Migration

#### 2.2.1. $\rho =0.3$

#### 2.2.2. $\rho =0.8$

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Time dependence of fraction x of bilingual agents for each set of parameters averaged over 100 independent runs. For $\rho \ge 0.1$, results were obtained for system size $N=2000$, and we checked that it was sufficiently large to avoid noticeable finite-size effects at the examined time scale. For $\rho =0.01$, results are shown for $N=5000$, and some late-time bending is still noticeable. Finite-size effects for $\rho =0.01$, $d=0.99$, and $M=2$ are illustrated in the inset. $N\ge 1000$ shows a transient slowdown in the decay of x, which we attribute to the formation of stripelike structures [32]. For increasing N, the influence of such stripes on fraction x seemed to diminish, and power-law decay set in.

**Figure 2.**Spatial distribution of agents and languages they use after $t={10}^{3}$ for $d=0.8$, and after $t={10}^{4}$ for $d=0.95$. After such transients, all agents form separated clusters, become monolingual, and thus immobile (see Figure 4). Simulations were performed for $N=200$ and $\rho =0.3$; different colours correspond to different languages.

**Figure 3.**Average size of surviving languages as function of mobility d. (inset) Same data on a logarithmic scale. Least-squares fit shows that numerical data follow power-law divergence $S\sim {({d}_{c}-d)}^{-\gamma}$ with $d={d}_{c}=0.995\left(1\right)$ and $\gamma =1.5\left(1\right)$ (dashed line).

**Figure 4.**Time dependence of fraction x of mobile agents calculated for $d=0.999$ (red), 0.95 (green), and 0.8 (blue). Simulations performed for $\rho =0.3$ and system size $N=200$ (continuous lines), 300 (dashed), and 500 (dotted). Presented results are averages over 30 independent runs. For $d=0.8$ and 0.95, the decay of x took place on a short time scale and was nearly size-independent. (inset) Same data but with logarithmic time axis.

**Figure 5.**Snapshots of spatial distribution of agents and languages they use for $d=0.999$, $\rho =0.3$, and $N=200$. Dominant language emerges (around $t=85000$) without any indication of Ising-like coarsening.

**Figure 6.**Final spatial distributions of agents and languages they use for $\rho =0.8$, $N=200$, $M=1000$, and $d=0.1$ (

**top**), $d=0.01$ (

**middle**), and $d=0.001$ (

**bottom**).

**Figure 7.**Time dependence of fraction x of bilingual agents calculated for $\rho =0.8$, $N=1000$, and several values of mobility d. Solid straight line has slope corresponding to $x\sim {t}^{-0.3}$, and some bending of our data indicates that asymptotic decay for $d=0$ might be even slower than that. Presented results were averaged over 100 independent runs.

**Figure 8.**$(\rho ,d)$ phase diagram as inferred from our simulations. In the largest portion of the phase diagram, the final state was quickly reached, comprising finite-size monolingual islands. We expected power-law coarsening only for $d=0$ and $\rho >{\rho}_{c}$, and for $\rho <1$, the coarsening was probably slower than ${t}^{1/2}$ due to random dilution. For d very close to 1, we expected the regime where the dominant language is formed via spontaneous fluctuation (SF, dashed line is not in scale).

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Lipowska, D.; Lipowski, A.
Evolution towards Linguistic Coherence in Naming Game with Migrating Agents. *Entropy* **2021**, *23*, 299.
https://doi.org/10.3390/e23030299

**AMA Style**

Lipowska D, Lipowski A.
Evolution towards Linguistic Coherence in Naming Game with Migrating Agents. *Entropy*. 2021; 23(3):299.
https://doi.org/10.3390/e23030299

**Chicago/Turabian Style**

Lipowska, Dorota, and Adam Lipowski.
2021. "Evolution towards Linguistic Coherence in Naming Game with Migrating Agents" *Entropy* 23, no. 3: 299.
https://doi.org/10.3390/e23030299