Title: Opinion Dynamics with Stubborn Agents,
Abstract: We consider the problem of optimizing the placement of stubborn agents in a
social network in order to maximally impact population opinions.
We assume individuals in a directed social network each have a latent opinion
that evolves over time in response to social media posts by their neighbors.
The individuals randomly communicate noisy versions of their latent opinion to
their neighbors. Each individual updates his opinion using a time-varying
update rule that has him become more stubborn with time and be less affected by
new posts. The dynamic update rule is a novel component of our model and
reflects realistic behaviors observed in many psychological studies.
We show that in the presence of stubborn agents with immutable opinions and
under fairly general conditions on the stubbornness rate of the individuals,
the opinions converge to an equilibrium determined by a linear system. We give
an interesting electrical network interpretation of the equilibrium. We also
use this equilibrium to present a simple closed form expression for harmonic
influence centrality, which is a function that quantifies how much a node can
affect the mean opinion in a network. We develop a discrete optimization
formulation for the problem of maximally shifting opinions in a network by
targeting nodes with stubborn agents. We show that this is an optimization
problem with a monotone and submodular objective, allowing us to utilize a
greedy algorithm. Finally, we show that a small number of stubborn agents can
non-trivially influence a large population using simulated networks.