  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.
