  We study the evolution of opinions on a directed network with community
structure. Individuals update their opinions synchronously based on a weighted
average of their neighbors' opinions, their own previous opinions, and external
media signals. Our model is akin to the popular Friedkin-Johnsen model, and is
able to incorporate factors such as stubbornness, confirmation bias, selective
exposure, and multiple topics, which are believed to play an important role in
the formation of opinions. Our main result shows that, in the large graph
limit, the opinion process concentrates around its mean-field approximation for
any level of edge density, provided the average degree grows to infinity.
Moreover, we show that the opinion process exhibits propagation of chaos. We
also give results for the trajectories of individual vertices and the
stationary version of the opinion process, and prove that the limits in time
and in the size of the network commute. The mean-field approximation is
explicit and can be used to quantify consensus and polarization.

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