#
The KCOD Model on (3,4,6,4) and (3^{4},6) Archimedean Lattices

## Abstract

**:**

## 1. Introduction

## 2. Model and Simulations

_{4}), defined as

## 3. Results and Discussion

_{4}on the disorder parameter p, obtained from simulations on $(3,4,6,4)$ and $({3}^{4},6)$ AL with L ranging from $L=8$ to $L=128$. The shape of $O\left(p\right)$, $OF$, and ${O}_{4}$ curves for a given value of L indicate the occurrence of a second-order phase transition in the system. The phase transition occurs at the value of the critical disorder parameter ${p}_{c}$. This critical disorder parameter ${p}_{c}$ is estimated as the point where the curves of the Binder cumulant ${O}_{4}$ for different system sizes N intercept each other [24]. The corresponding value of ${O}_{4}$ is represented by ${O}_{4}^{*}$. Then, we obtained ${p}_{c}=0.085\left(6\right)$ and ${O}_{4}^{*}=0.605\left(9\right)$; ${p}_{c}=0.146\left(5\right)$ and ${O}_{4}^{*}=0.606\left(4\right)$ for $(3,4,6,4)$, and $({3}^{4},6)$ AL, respectively.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(Color online). The opinion O, $OF$, and ${O}_{4}$, as a function of the parameter p, for lattice size $L=8$, 16, 32, 64, and 128, and $N=6{L}^{2}$ sites for $(3,4,6,4)$ (

**a**–

**c**) and $({3}^{4},6)$ Archimedean lattice (AL) (

**d**–

**f**).

**Figure 3.**(Color online). The $-ln(1-\frac{3}{2}{O}_{4})$ as a function of the parameter p, for $L=8$, 16, 32, 64, and 128 lattice sizes, and $N=6{L}^{2}$ for $(3,4,6,4)$ and $({3}^{4},6)$ AL.

**Figure 4.**Log–log plot of the dependence of the opinion ${O}^{*}=O\left({p}_{c}\right)$ on the linear system size L. Fitting data, we obtained the estimate for the critical ratio $\beta /\nu $.

**Figure 5.**Log–log plot of the $O{F}^{*}=O{F}_{{p}_{c}}$ at ${p}_{c}$ versus L for $(3,4,6,4)$, and $({3}^{4},6)$ AL. Fitting data, we obtained the estimate for the critical ratio $\gamma /\nu $.

**Figure 6.**$O{F}^{max}=O{F}_{{p}_{O{F}_{max}}}\left(N\right)$ at ${p}_{O{F}_{max}}\left(N\right)$ versus L for $(3,4,6,4)$ and $({3}^{4},6)$, AL. Fitting data, we obtained another estimate for the critical ratio $\gamma /\nu $ .

**Figure 7.**Plot of $ln|{p}_{c}\left(L\right)-{p}_{c}|$ versus the linear system size L for $(3,4,6,4)$ and $({3}^{4},6)$ AL. Fitting data, we obtained the estimate for the critical ratio $1/\nu $.

**Figure 8.**(Color online) Data collapse of the opinion O, OF, and O

_{4}shown in Figure 3 for $L=32$, 64, and 128 $(3,4,6,4)$ (

**a**–

**f**) and $({3}^{4},6)$ (

**d**–

**f**) AL. The exponent ratios used here were $\beta /\nu =0.126\left(1\right)$, $\gamma /\nu =1.50\left(7\right)$, and $1/\nu =0.90\left(5\right)$ for $(3,4,6,4)$, and $\beta /\nu =0.125\left(3\right)$, $\gamma /\nu =1.54\left(6\right)$, and $1/\nu =0.99\left(3\right)$ for $({3}^{4},6)$ AL.

MVM | $(3,4,6,4)$ | $({3}^{4},6)$ | $\left({4}^{4}\right)$ Ising |
---|---|---|---|

${T}_{c}$ | 0.651(3) | 0.667(2) | ≈2.269 |

${O}_{4}^{*}$ | 0.603(9) | 0.608(4) | 0.61 |

$\beta /\nu $ | 0.105(8) | 0.113(2) | 0.125 |

$\gamma /{\nu}^{T={T}_{c}}$ | 1.48(11) | 1.60(4) | 1.75 |

$\gamma /{\nu}^{T={T}^{*}}$ | 1.44(4) | 1.66(2) | 1.75 |

$1/\nu $ | 1.16(5) | 0.84(6) | 1 |

${D}_{\mathrm{eff}}$. | 1.78(7) | 1.83(6) | 2 |

**Table 2.**Critical parameter (${p}_{c}$), exponents, and effective dimension for continuous opinion dynamic (KCOD) model on $(3,4,6,4)$ and $({3}^{4},6)$. For completeness, we cite data for KCOD model on $\left({4}^{4}\right)$ as well [16].

KCOD | $(3,4,6,4)$ | $({3}^{4},6)$ | $\left({4}^{4}\right)$ |
---|---|---|---|

${p}_{c}$ | 0.085(6) | 0.146(5) | 0.2266(1) |

${O}_{4}^{*}$ | 0.605(9) | 0.606(4) | 0.559(1) |

$\beta /\nu $ | 0.126(1) | 0.125(3) | 0.125(1) |

$\gamma /{\nu}^{p={p}_{c}}$ | 1.50(7) | 1.54(6) | 1.75(1) |

$\gamma /{\nu}^{p={p}^{*}}$ | 1.50(5) | 1.55(5) | |

$1/\nu $ | 0.90(5) | 0.99(3) | 1.01(1) |

${D}_{\mathrm{eff}}$ | 1.75(6) | 1.80(7) |

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**MDPI and ACS Style**

De Sousa Lima, F.W. The KCOD Model on (3,4,6,4) and (3^{4},6) Archimedean Lattices. *Entropy* **2017**, *19*, 459.
https://doi.org/10.3390/e19090459

**AMA Style**

De Sousa Lima FW. The KCOD Model on (3,4,6,4) and (3^{4},6) Archimedean Lattices. *Entropy*. 2017; 19(9):459.
https://doi.org/10.3390/e19090459

**Chicago/Turabian Style**

De Sousa Lima, Francisco W. 2017. "The KCOD Model on (3,4,6,4) and (3^{4},6) Archimedean Lattices" *Entropy* 19, no. 9: 459.
https://doi.org/10.3390/e19090459