Using Node Centrality and Optimal Control to Maximize Information Diffusion in Social Networks

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.