  Influence propagation in social networks is a central problem in modern
social network analysis, with important societal applications in politics and
advertising. A large body of work has focused on cascading models, viral
marketing, and finite-horizon diffusion. There is, however, a need for more
developed, mathematically principled \emph{adversarial models}, in which
multiple, opposed actors strategically select nodes whose influence will
maximally sway the crowd to their point of view.
  In the present work, we develop and analyze such a model based on harmonic
functions and linear diffusion. We prove that our general problem is NP-hard
and that the objective function is monotone and submodular; consequently, we
can greedily approximate the solution within a constant factor. Introducing and
analyzing a convex relaxation, we show that the problem can be approximately
solved using smooth optimization methods. We illustrate the effectiveness of
our approach on a variety of example networks.
