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: 1) scientific collaboration; 2) Slashdot; and 3) Facebook. The optimal strategy targets central nodes when the resource is scarce, but noncentral 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.
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