  Considerable efforts were made in recent years in devising optimization
algorithms for influence maximization in networks. Here we ask: "When do we
need optimization?" We use results from statistical mechanics and direct
simulations on ER networks, small-world networks, power-law networks and a
dataset of real-world networks to characterize the parameter-space region where
optimization is required. We show that in both synthetic and real-world
networks this optimization region is due to a well known physical phase
transition of the network, and that it vanishes as a power-law with the network
size. We then show that also from a utility-maximization perspective (when
considering the costs of the optimization process), for large networks standard
optimization is profitable only in a vanishing parameter region near the phase
transition. Finally, we introduce a novel constant-time optimization approach,
and demonstrate it through a simple algorithm that manages to give similar
results to standard optimization methods in terms of the influenced-set size,
while improving the results in terms of the net utility.
