  We model information dissemination as a susceptible-infected epidemic process
and formulate a problem to jointly optimize seeds for the epidemic and time
varying resource allocation over the period of a fixed duration campaign
running on a social network with a given adjacency matrix. Individuals in the
network are grouped according to their centrality measure and each group is
influenced by an external control function---implemented through
advertisements---during the campaign duration. The aim is to maximize an
objective function which is a linear combination of the reward due to the
fraction of informed individuals at the deadline, and the aggregated cost of
applying controls (advertising) over the campaign duration. We also study a
problem variant with a fixed budget constraint. We set up the optimality system
using Pontryagin's Maximum Principle from optimal control theory and solve it
numerically using the forward-backward sweep technique. Our formulation allows
us to compare the performance of various centrality measures (pagerank, degree,
closeness and betweenness) in maximizing the spread of a message in the optimal
control framework. We find that degree---a simple and local measure---performs
well on the three social networks used to demonstrate results: scientific
collaboration, Slashdot and Facebook. The optimal strategy targets central
nodes when the resource is scarce, but non-central nodes are targeted when the
resource is in abundance. Our framework is general and can be used in similar
studies for other disease or information spread models---that can be modeled
using a system of ordinary differential equations---for a network with a known
adjacency matrix.
