Opinion Optimization in Directed Social Networks

Shifting social opinions has far-reaching implications in various aspects, such as public health campaigns, product marketing, and political candidates. In this paper, we study a problem of opinion optimization based on the popular Friedkin-Johnsen (FJ) model for opinion dynamics in an unweighted directed social network with $n$ nodes and $m$ edges. In the FJ model, the internal opinion of every node lies in the closed interval $[0, 1]$, with 0 and 1 being polar opposites of opinions about a certain issue. Concretely, we focus on the problem of selecting a small number of $ k\ll n $ nodes and changing their internal opinions to 0, in order to minimize the average opinion at equilibrium. We then design an algorithm that returns the optimal solution to the problem in $O(n^3)$ time. To speed up the computation, we further develop a fast algorithm by sampling spanning forests, the time complexity of which is $ O(ln) $, with $l$ being the number of samplings. Finally, we execute extensive experiments on various real directed networks, which show that the effectiveness of our two algorithms is similar to each other, both of which outperform several baseline strategies of node selection. Moreover, our fast algorithm is more efficient than the first one, which is scalable to massive graphs with more than twenty million nodes.