  We study the optimal referral strategy of a seller and its relationship with
the type of communication channels among consumers. The seller faces a
partially uninformed population of consumers, interconnected through a directed
social network. In the network, the seller offers rewards to informed consumers
(influencers) conditional on inducing purchases by uninformed consumers
(influenced). Rewards are needed to bear a communication cost and to induce
word-of-mouth (WOM) either privately (cost-per-contact) or publicly (fixed cost
to inform all friends). From the seller's viewpoint, eliciting Private WOM is
more costly than eliciting Public WOM. We investigate (i) the incentives for
the seller to move to a denser network, inducing either Private or Public WOM
and (ii) the optimal mix between the two types of communication. A denser
network is found to be always better, not only for information diffusion but
also for seller's profits, as long as Private WOM is concerned. Differently,
under Public WOM, the seller may prefer an environment with less competition
between informed consumers and the presence of highly connected influencers
(hubs) is the main driver to make network density beneficial to profits. When
the seller is able to discriminate between Private and Public WOM, the optimal
strategy is to cheaply incentivize the more connected people to pass on the
information publicly and then offer a high bonus for Private WOM.
