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We consider a dynamic model for competition in a social network, where twostrategic agents have fixed beliefs and the non-strategic/regular agents adjusttheir states according to a distributed consensus protocol. We suppose that onestrategic agent must identify k+ target agents in the network in order tomaximally spread its own opinion and alter the average opinion that eventuallyemerges. In the literature, this problem is cast as the maximization of a setfunction and, leveraging on the submodular property, is solved in a greedymanner by solving k+ separate single targeting problems. Our main contributionis to exploit the underlying graph structure to build more refined heuristics.As a first instance, we provide the analytical solution for the optimaltargeting problem over complete graphs. This result provides a rule tounderstand whether it is convenient or not to block the opponent's influence bytargeting the same nodes. The argument is then extended to generic graphsleading to more accurate solutions compared to a simple greedy approach. As asecond instance, by electrical analogy we provide the analytical solution ofthe single targeting problem for the line graph and derive some usefulproperties of the objective function for trees. Inspired by these findings, wedefine a new algorithm which selects the optimal solution on trees in a muchfaster way with respect to a brute-force approach and works well also overtree-like/sparse graphs. The proposed heuristics are then compared to zero-costheuristics on different random generated graphs and real social networks.Summarizing, our results suggest a scheme that tells which algorithm is moresuitable in terms of accuracy and computational complexity, based on thedensity of the graphs and its degree distribution.

Graph structure based Heuristics for Optimal Targeting in Social Networks