How Social Network Structure Impacts the Ability of Zealots to Promote Weak Opinions

Abstract

Social networks are often permeated by agents who promote their opinions without allowing for their own mind to be changed: Understanding how these so-called `zealots' act to increase the prevalence of their promoted opinion over the network is important for understanding opinion dynamics. In this work, we consider these promoted opinions to be `weak' and therefore less likely to be accepted relative to the default opinion in the network. We show how the proportion of zealots in the network, the relative strength of the weak opinion, and the structure of the network impact the long-term proportion of the those in the network who subscribe to the weak opinion.

Figures (4)

• Figure 1: (a) Illustration of the effects of changing relative weak opinion strength, $F$, the proportion of the network who are zealots, $z$, and the mean degree of nodes in an Erdős - Rényi graph, $C$. (b) Underlying modified voter model applied to an example graph. The possible updates during the next simulation step are illustrated by dashed arrows, annotated with the corresponding probability of each event occurring. (c) The mean long-term frequency of the weak opinion over the entire graph (of size $N=10^4$) for varying $C$, for $F=0.3$ and $z=0.3$, averaged over the last $10\%$ of $10^9$ simulations steps of $10$ repeats of the modified voter model. All free nodes are initially of the strong variety.
• Figure 2: (a) Comparison between the analytical steady-state solution given by Eqn. \ref{['SteadyState']} (solid line) and the normalised frequency density of a simulation (dotted lines) for a variety of $(N,F,Z)$ combinations over $10^8$ simulation steps. Starting with $(1000,0.9,10)$ (black) we see how increasing $F$ (magenta), increasing $Z$ (green) or decreasing $N$ (yellow) acts to increase the overall long-term proportion of weak opinion. Decreasing $Z$ and increasing $F$ to preserve the equilibrium number of nodes subscribed to the weak opinion (given by Eqn. \ref{['CompleteGraph']}) is also compared (cyan). (b) Analytical steady-state solution given by Eqn.\ref{['SteadyState']} for a varying number of nodes, $N$, for the cases of $(F,z) = (0.8,0.05)$ (solid) and $(F,z) = (0.7,0.35)$ (dashed). The $N\rightarrow\infty$ vertical lines corresponds to the results of Eqn. \ref{['CompleteGraph']}.
• Figure 3: (a) The long-term fate of trees containing a zealot and tree-like subgraphs bound by zealots in the giant component is consensus to the weak opinion. (b) Visualization of how removing zealots from a graph creates a reduced graph. (c) How simulation (crosses) and theory (solid line, given by Eqn. \ref{['ER_SUPERCRITICAL_TREES']}) agree for the long-term proportion of weak opinion over a graph of size $N=10^4$ as a function of $C^*$ for $z = 0.001$ (magenta), $z = 0.1$ (cyan), $z = 0.5$ (yellow). In all cases, $F=1$, and the simulation results are the median of $100$ repetitions of the steady state obtained.
• Figure 4: How simulation (crosses) and theory (solid line, given by Eqn. \ref{['FULL']}) agree for constant $z=0.3$ and different $F$ (top) and for constant $F=0.3$ and different $z$ (bottom), each for varying $C$. Vertical dashed lines correspond to $C=1/(1-z)$ for each value of $z$: These are identical for the upper graph and given in black. Simulation results are averaged over the last $10\%$ of $10^9$ simulations steps of $10$ repeats of the modified voter model on a network of size $N=10^4$. All free nodes are initially of the strong variety.