Zealotry effects on opinion dynamics in the adaptive voter model

The adaptive voter model has been widely studied as a conceptual model for opinion formation processes on time-evolving social networks. Past studies on the effect of zealots, i.e., nodes aiming to spread their fixed opinion throughout the system, only considered the voter model on a static network. Here we extend the study of zealotry to the case of an adaptive network topology co-evolving with the state of the nodes and investigate opinion spreading induced by zealots depending on their initial density and connectedness. Numerical simulations reveal that below the fragmentation threshold a low density of zealots is sufficient to spread their opinion to the whole network. Beyond the transition point, zealots must exhibit an increased degree as compared to ordinary nodes for an efficient spreading of their opinion. We verify the numerical findings using a mean-field approximation of the model yielding a low-dimensional set of coupled ordinary differential equations. Our results imply that the spreading of the zealots' opinion in the adaptive voter model is strongly dependent on the link rewiring probability and the average degree of normal nodes in comparison with that of the zealots. In order to avoid a complete dominance of the zealots' opinion, there are two possible strategies for the remaining nodes: adjusting the probability of rewiring and/or the number of connections with other nodes, respectively.

Figure 1

Coefficient of variation of convergence time $V_t$ indicating a phase transition at $\varphi ={\varphi }_c$ where the giant component vanishes. The different parameter settings are (a) no zealots $n_z^0=0$, (b) zealot density $n_z^0=0.01$ with no excess degree $k_x=0$, and (c) zealot density $n_z^0=0.01$ with an excess degree of $k_x=20$. The red lines in (a) and (b) at ${\varphi }_c=0.45$ indicate the approximate positions of the phase transition, which is shifted towards ${\varphi }_c=0.505$ if the excess degree $k_x$ is nonzero (c). Distributions are computed from $n={10}^4$ runs on graphs with $N=800$ nodes and a mean degree (without the zealots' excess degree) of ${\overline{k}}_0=4$.

Figure 2

Frequency distributions of cluster size $P\left(s\right)$ at $t=t_c$ for all clusters (blue triangles) and for the fraction of convinced clusters (red squares) computed from $n={10}^4$ simulation runs on graphs with $N=800$ and ${\gamma }_0=10$. The distributions are computed for no zealots $n_z^0=0$ [(a), (d), and (g)], for zealot density $n_z^0=0.01$ and no excess degree $k_x=0$ [(b), (e), and (h)], and for zealot density $n_z^0=0.01$ and excess degree $k_x=20$ [(c), (f), and (i)]. The rows represent different values of the rewiring probability $\varphi :\varphi =0.04\ll {\varphi }_c$ (a)–(c), $\varphi ={\varphi }_c\left(\overline{k}\right)$ (d)–(f), and $\varphi =0.96\gg {\varphi }_c$ (g)–(i). Blue solid and red dashed vertical lines indicate ${\gamma }_0=10$ and initial number of zealots $N{n}_z^0$, respectively. Note the black arrows indicating the giant component.

Figure 3

(a)–(c) Fraction of convinced nodes at the final state $n_z\left(t_c\right)$ and (d)–(f) the former minus $n_{z\overline{k}}\left(t_c\right)$ in which the $k_{x}N{n}_z^0$ additional edges are distributed homogeneously among all nodes (no zealot hubs) resulting in $\mathrm{\Delta }{n}_z\left(t_c\right)=n_z\left(t_c\right)-n_{z\overline{k}}\left(t_c\right)$, both as a function of excess degree $k_x$ and zealot density $n_z^0$. The rewiring probability is increased from (a) and (d) $\varphi =0.4$ via (b) and (e) $\varphi =0.5$ to (c) and (f) $\varphi =0.75$. The black [(b), (c), (e), and (f)] and white [(c) and (f)] lines indicate contours where $\overline{k}={\overline{k}}_c$ and ${\overline{k}}_z={\overline{k}}_c$, respectively, highlighting the fragmentation transition, with ${\overline{k}}_z$ being the mean degree of $g_z$-nodes. All results have been obtained from 200 simulation runs on networks with $N=800$ nodes and a mean degree without zealots of ${\overline{k}}_0=4$.

Figure 4

Results of microscopic ensemble simulations (symbols) and macroscopic approximation (solid lines) using Eqs. (1, 2, 3). Mean values have been estimated from 200 simulation runs. (a) ZOS efficiency $n_z\left(t_c\right)$, (c) density of links between convinced nodes $m_{zz}\left(t_c\right)$, and (e) density of links between unconvinced nodes $m_{oo}\left(t_c\right)$ at convergence time $t_c$ with $N=800$ and varying ${\gamma }_0$ as indicated in (c). (b), (d), and (f) Same as for (a), (c), and (e) but with varying $N={\gamma }_0$ and therefore only one opinion different from $g_z$. The blue dashed vertical line marks the approximate phase transition ${\tilde{\varphi }}_c$ [Eq. (9)]. For all runs, $n_z^0=0.01$ and $k_x=20$ have been kept fixed. In (a), (c), and (e), ${\overline{k}}_0=4$, while in (b), (d), and (f), ${\overline{k}}_0=15$. Note the different scales along the horizontal axes.

Figure 5

Coefficient of variation of convergence time $V_t$ indicating a phase transition at $\varphi ={\varphi }_c$ where the giant component vanishes. Passive (a) and heterophilic (b) zealots are compared. The red line in (a) marks ${\varphi }_c=0.45$ and in (b) ${\varphi }_c=0.77$. Distributions are computed from $n={10}^4$ runs on graphs with $N=800$ nodes, a zealot density of $n_z^0=0.01$ with no excess degree $k_x=0$ and a mean degree of ${\overline{k}}_0=4$.