Nonconsensus opinion model on directed networks

Dynamic social opinion models have been widely studied on undirected networks, and most of them are based on spin interaction models that produce a consensus. In reality, however, many networks such as Twitter and the World Wide Web are directed and are composed of both unidirectional and bidirectional links. Moreover, from choosing a coffee brand to deciding who to vote for in an election, two or more competing opinions often coexist. In response to this ubiquity of directed networks and the coexistence of two or more opinions in decision-making situations, we study a nonconsensus opinion model introduced by Shao et al. [Phys. Rev. Lett. 103, 018701 (2009)] on directed networks. We define directionality $\xi$ as the percentage of unidirectional links in a network, and we use the linear correlation coefficient $\rho$ between the in-degree and out-degree of a node to quantify the relation between the in-degree and out-degree. We introduce two degree-preserving rewiring approaches which allow us to construct directed networks that can have a broad range of possible combinations of directionality $\xi$ and linear correlation coefficient $\rho$ and to study how $\xi$ and $\rho$ impact opinion competitions. We find that, as the directionality $\xi$ or the in-degree and out-degree correlation $\rho$ increases, the majority opinion becomes more dominant and the minority opinion's ability to survive is lowered.

Figure 1

Schematic plot of the dynamics of the NCO model on a directed graph with 10 nodes.

Figure 2

Directionality-increasing rewiring (DIR).

Figure 3

(a) The degree-preserving rewiring for decreasing the directionality. (b) Plot of the minimal directionality ${\xi }_{\min }$ obtained by simulating ANC-DDR, for binomial networks (, $\langle k\rangle =4,{10}^5$ nodes) and SF networks (, $\lambda =2.63,{10}^5$ nodes) with 100 realizations $\left(M=\langle k\rangle N\right)$, and the theoretical minimum possible directionality ${\xi }_{\min }$ [Eq. (2)] for binomial networks (the solid line) and for SF networks (the dashed line) as a function of the in-degree and out-degree correlation $\rho$. (c) Plot of the minimal directionality ${\xi }_{\text{min}}$ obtained by simulating ANC-DDR with different values of the attempts $M$: $0.01*M_0$(),$0.1*M_0$(),$100*M_0$(), where $M_0=\langle k\rangle N$, and the theoretical minimum possible directionality ${\xi }_{\min }$ (the dashed line) for SF networks. All the results are the averages of 1000 realizations.

Figure 4

Plot of the normalized largest cluster $s_1$ of opinion ${\sigma }_{+}$ as a function of the initial fraction $f$ for different values of the directionality $\xi$: $0$(),$0.2$(),$0.4$(),$0.6$(),$0.8$()$,1.0$(), and for different networks with $N={10}^5$ nodes and $\langle k\rangle =4$: (a) RR, (b) binomial, (c) SF $\left(\lambda =2.63\right)$. In the upper insets of (a), (b), and (c), we plot $s_2$ as a function of $f$ with the same symbols and for the same networks as in the main figure. In the lower insets of (a) and (b), we plot the number of iterations to the steady-state NOI as a function of $f$. (d) Plot of the degree distribution of the nodes which keep the majority () and the minority opinion () in binomial networks (also with $N={10}^5$ nodes and $\langle k\rangle =4)$, when the directionality $\xi =0.0$ (the main figure) and $\xi =1.0$ (the inset). All results are based on averaging 1000 realizations.

Figure 5

(a) Plot of the critical threshold $f_c$ as a function of the directionality $\xi$ for different networks with $N={10}^5$ nodes and $\langle k\rangle =4$: RR (), binomial (), and SF () $\left(\lambda =2.63\right)$. All results are based on averaging 1000 realizations. (b) Plot of the critical threshold $f_c$ as a function of the variance of the degree sequence of the networks $(N={10}^5$ nodes and $\langle k\rangle =4)$ with different values of the directionality: $\xi =0$ () and $\xi =0.5$ (). All results are based on averaging 100 realizations.

Figure 6

Plot of the normalized largest cluster $s_1$ of opinion ${\sigma }_{+}$ as a function of the initial fraction $f$, when the directionality $\xi =0.6$, for different values of the in-degree and out-degree correlation $\rho$: $0$(),$0.5$(),$1$(), and for different networks with $N={10}^5$ nodes and $\langle k\rangle =4$: (a) binomial, (b) SF. In the insets, we plot $s_2$ as a function of $f$ with the same symbols and for the same networks as in the main figure. All results are based on averaging 1000 realizations.

Figure 7

Plot of the critical threshold $f_c$ as a function of the linear correlation coefficient $\rho$ and the directionality $\xi$ for binomial networks. All results are based on averaging 1000 realizations.

Figure 8

Plot of the average in-degree and out-degree of the nodes in the largest ${\sigma }_{+}$ and ${\sigma }_{-}$ clusters for binomial networks, when the initial faction $f$ of the opinion ${\sigma }_{+}$ equals $0.4$ as a function of $\rho$. The representations of the four lines are as follows: the average in-degree () and out-degree () of the nodes in the largest ${\sigma }_{+}$ cluster; the average in-degree () and out-degree () of the nodes in the largest ${\sigma }_{-}$ cluster. All results are based on averaging 100 realizations.