  Consider a directed network where each node is either red (using the red
product), blue (using the blue product), or uncolored (undecided). Then in each
round, an uncolored node chooses red (resp. blue) with some probability
proportional to the number of its red (resp. blue) out-neighbors. What is the
best strategy to maximize the expected final number of red nodes given the
budget to select $k$ red seed nodes? After proving that this problem is
computationally hard, we provide a polynomial time approximation algorithm with
the best possible approximation guarantee, building on the monotonicity and
submodularity of the objective function and exploiting the Monte Carlo method.
Furthermore, our experiments on various real-world and synthetic networks
demonstrate that our proposed algorithm outperforms other algorithms.
Additionally, we investigate the convergence time of the aforementioned process
both theoretically and experimentally. In particular, we prove several tight
bounds on the convergence time in terms of different graph parameters, such as
the number of nodes/edges, maximum out-degree and diameter, by developing novel
proof techniques.
