A Co\mbox{-}Operative Game Theoretic Approach for the Budgeted Influence Maximization Problem

Given a social network of users with selection cost, the \textsc{Budgeted Influence Maximization Problem} (\emph{BIM Problem} in short) asks for selecting a subset of the nodes (known as \emph{seed nodes}) within an allocated budget for initial activation to maximize the influence in the network. In this paper, we study this problem under the \emph{co\mbox{-}operative game theoretic} framework. We model this problem as a co\mbox{-}operative game where the users of the network are the players and for a group of users, the expected influence by them under the \emph{Maximum Influence Arborences} diffusion model is its utility. We call this game as \emph{BIM Game} and show this is `non-convex' and `sub-additive'. Based on the proposed game\mbox{-}theoretic model and using the solution concept called `Shapley Value', we propose an iterative algorithm for finding seed nodes. The proposed methodology is divided into mainly two broad steps: the first one is computing the approximate marginal gain in \emph{Shapley Value} for all the nodes of the network, and the second one is selecting seed nodes from the sorted list until the budget is exhausted. We also show that the proposed methodology can even be more effective when the community structure of the network is exploited. The proposed methodologies have been implemented, and an extensive set of experiments have been conducted with three publicly available social network datasets. From the experiments, we observe that the seed set selected by the proposed methodologies lead to more number of influence nodes compared to many standard and baseline methods from the literature with a reasonable computational overhead. In particular, if the community structure of the network is exploited then there is an increase upto $2 \%$ in number of influenced nodes.