The spread of opinions, memes, diseases, and "alternative facts" in a
population depends both on the details of the spreading process and on the
structure of the social and communication networks on which they spread. In
this paper, we explore how \textit{anti-establishment} nodes (e.g.,
\textit{hipsters}) influence the spreading dynamics of two competing products.
We consider a model in which spreading follows a deterministic rule for
updating node states (which describe which product has been adopted) in which
an adjustable fraction pHip of the nodes in a network are hipsters,
who choose to adopt the product that they believe is the less popular of the
two. The remaining nodes are conformists, who choose which product to adopt by
considering which products their immediate neighbors have adopted. We simulate
our model on both synthetic and real networks, and we show that the hipsters
have a major effect on the final fraction of people who adopt each product:
even when only one of the two products exists at the beginning of the
simulations, a very small fraction of hipsters in a network can still cause the
other product to eventually become the more popular one. To account for this
behavior, we construct an approximation for the steady-state adoption fraction
on k-regular trees in the limit of few hipsters. Additionally, our
simulations demonstrate that a time delay τ in the knowledge of the
product distribution in a population, as compared to immediate knowledge of
product adoption among nearest neighbors, can have a large effect on the final
distribution of product adoptions. Our simple model and analysis may help shed
light on the road to success for anti-establishment choices in elections, as
such success can arise rather generically in our model from a small number of
anti-establishment individuals and ordinary processes of social influence on
normal individuals.Comment: Extensively revised, with much new analysis and numerics The abstract
on arXiv is a shortened version of the full abstract because of space limit