Graph structure based Heuristics for Optimal Targeting in Social Networks.

We consider a dynamic model for competition in a social network, where two strategic agents have fixed beliefs and the non-strategic/regular agents adjust their states according to a distributed consensus protocol. We suppose that one strategic agent must identify k+ target agents in the network in order to maximally spread its own opinion and alter the average opinion that eventually emerges. In the literature, this problem is cast as the maximization of a set function and, leveraging on the submodular property, is solved in a greedy manner by solving k+ separate single targeting problems. Our main contribution is to exploit the underlying graph structure to build more refined heuristics. As a first instance, we provide the analytical solution for the optimal targeting problem over complete graphs. This result provides a rule to understand whether it is convenient or not to block the opponent's influence by targeting the same nodes. The argument is then extended to generic graphs leading to more accurate solutions compared to a simple greedy approach. As a second instance, by electrical analogy we provide the analytical solution of the single targeting problem for the line graph and derive some useful properties of the objective function for trees. Inspired by these findings, we define a new algorithm which selects the optimal solution on trees in a much faster way with respect to a brute-force approach and works well also over tree-like/sparse graphs. The proposed heuristics are then compared to zero-cost heuristics on different random generated graphs and real social networks. Summarizing, our results suggest a scheme that tells which algorithm is more suitable in terms of accuracy and computational complexity, based on the density of the graphs and its degree distribution.