Modeling and numerical simulations of the influenced Sznajd model

This paper investigates the effects of independent nonconformists or influencers on the behavioral dynamic of a population of agents interacting with each other based on the Sznajd model. The system is modeled on a complete graph using the master equation. The acquired equation has been numerically solved. Accuracy of the mathematical model and its corresponding assumptions have been validated by numerical simulations. Regions of initial magnetization have been found from where the system converges to one of two unique steady-state PDFs, depending on the distribution of influencers. The scaling property and entropy of the stationary system in presence of varying level of influence have been presented and discussed.

Figure 1

Effect of the population size $N$ on the steady state density $P_{ss}\left(m\right)$.

Figure 2

Dependency of mean steady-state magnetization on initial magnetization. (a) $N=50,I_{+}=6,I_{-}=5$, (b) $N=100$, $I_{+}=12$, $I_{-}=10$, (c) $N=200,I_{+}=24,I_{-}=20$, (d) $N=500,I_{+}=60,I_{-}=50$.

Figure 3

Convergence zone and stationary PDFs of a system with $N=100,I_{+}=12,I_{-}=10$. (a) Chosen initial values ($m_0$) for hypothesis testing, (b) $m_0=-0.04$, (c) $m_0=-0.7$, (d) $m_0=-0.8$, (e) $m_0=-0.9$, (f) $m_0=0.5$, (g) $m_0=0.7$, (h) $m_0=0.9$.

Figure 4

Dependency of mean steady-state magnetization on initial magnetization and control input for $N=100$, (a) $\lambda =0$, (b) $\lambda =0.2$, (c) $\lambda =0.4$, (d) $\lambda =0.6$, (e) $\lambda =0.8$, (f) $\lambda =1$.

Figure 5

(a) $m_{0_l}^{*}$ and $m_{0_h}^{*}$ marked on the 3D graph related to $\lambda =0.2$ for all possible values of $u$. (b) 2D representation of the boundaries on the $m_0\text{−}u$ plane.

Figure 6

Three-dimensional linear scaling property, (a) the base system ($N=50,\lambda =0.2$), (b) the scaled system ($N=100,\lambda =0.2$), (c) scalability zone for $m_0=\pm 1$ with $\lambda =0.2$.

Figure 7

Scalability zone for all $m_0\mathrm{s}$ for $N=100$, (a) $\lambda =0.2$, (b) $\lambda =0.4$, (c) $\lambda =0.6$, (d) $\lambda =0.8$, (e) $\lambda =1$.

Figure 8

Entropy of distributions for different values of $\lambda$ for $N=100$, (a) $\lambda =0$, (b) $\lambda =0.2$, (c) $\lambda =0.4$, (d) $\lambda =0.6$, (e) $\lambda =0.8$, (f) $\lambda =1$.