Discord in the voter model for complex networks

TL;DR

A method to calculate the probability of discord between any two agents in the multistate voter model with and without zealots is introduced and the existence of a unique equilibrium solution is proved, which can be computed via an iterative algorithm.

Abstract

Online social networks have become primary means of communication. As they often exhibit undesirable effects such as hostility, polarisation or echo chambers, it is crucial to develop analytical tools that help us better understand them. In this paper, we are interested in the evolution of discord in social networks. Formally, we introduce a method to calculate the probability of discord between any two agents in the multi-state voter model with and without zealots. Our work applies to any directed, weighted graph with any finite number of possible opinions, allows for various update rates across agents, and does not imply any approximation. Under certain topological conditions, their opinions are independent and the joint distribution can be decoupled. Otherwise, the evolution of discord probabilities is described by a linear system of ordinary differential equations. We prove the existence of a unique equilibrium solution, which can be computed via an iterative algorithm. The classical definition of active links density is generalized to take into account long-range, weighted interactions. We illustrate our findings on real-life and synthetic networks. In particular, we investigate the impact of clustering on discord, and uncover a rich landscape of varied behaviors in polarised networks. This sheds lights on the evolution of discord between, and within, antagonistic communities.

Figures (6)

• Figure 1: Dependency between opinions. Nodes $0$ and $1$ are the 0- and 1-zealot respectively. Numbers along the arrows denote edge weights. In (a) there is a path from $i$ to $j$. In (b) there is none, but $i$ and $j$ have a common ancestor $k$. In both cases, Eq. \ref{['qij_dep']} gives $\rho_{ij}=1/4$, while Eq. \ref{['qij_indep']} gives $\rho_{ij}=1/2$.
• Figure 2: Moving average for $\rho_{ij}$ (dotted blue line) and error percentage of $\tilde{\rho}_{ij}$ (orange line) for all dependent agent pairs, function of (a) path strength, (b) ancestry similarity, and (c) and total zealousness. All curves differ in their range of values, which we don't precise on the y-axes for the sake of clarity. On the right are shown average errors and standard deviation for the distribution of errors.
• Figure 3: Impact of clustering coefficient on $\langle\rho\rangle$. For each set of parameters $(k,p)$ we generate $m=30$ Watts-Strogatz networks of size $N=100$, average degree $k$ and rewiring probability $p$. We average $\langle\rho\rangle$ over all realisations. Shaded areas cover $\pm 1$ standard deviation. Half of the agents support opinion 0 $(z^{(0)}>0)$, the other half opinion 1 $(z^{(1)}>0)$. (a) With homophily: the majority of edges are between agents supporting the same opinion. (b) Without homophily: no correlation between edges and opinion support. (c) Average clustering coefficient (independent from homophily). Note the different scales on the y-axes.
• Figure 4: Network of $N=100$ agents with two communities $\mathcal{C}_0$ and $\mathcal{C}_1$. The $s$-zealot exert a total influence $z^s$ on each agent in $\mathcal{C}_s$ and none on others. In-group edge probability is fixed at $p_{in}=0.1$, while the out-group edge probability $p_{out}$ and zealousness $z=(z^0,z^1)$ vary. Results are averaged over 20 agent graphs generated under the Stochastic Block Model, for the whole network (a), within each community (b,c) and between them (d). Top: generalized active links density. Middle: Opinion difference. Bottom: Support for opinion 0. The plots have different scales for clarity, but the purpose is to focus on the qualitative dynamics rather than exact values.
• Figure 5: Unweighted degree distributions for each toy dataset, logarithmic scale. Zachary and Football are undirected so both distributions are the same. Top: in-degrees. Bottom: out-degrees.

Theorems & Definitions (1)

• Lemma 1: Azimzadeh, 2018, Lemma 2.1 azimzadeh2018