Boosting node activity by recommendations in social networks

In a social network, the propagation of information has sparked intense research. Influence Maximization (IM) is a well-studied problem that asks for k nodes to influence the largest users in the social network. However IM is submodular at the most time. In recent years, many non-submodular problems have been proposed and researchers give a lot of algorithms to solve them. In this paper, we propose Activity Probability Maximization Problem without submodular property. For a given social network G, a candidate edge set E¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{E}}$$\end{document} and a constant k, the Activity Probability Maximization Problem asks for k edges in the candidate edge set that make the all nodes of G with highest probability of being activated under a pre-determined seed set S. Using the marginal increment, we give a general way to construct submodular lower bound and submodular upper bound functions of the non-submodular objective function at the same time. Interestingly, the optimal solution of upper bound is the same as that of lower bound. Therefore, we develop the Sandwich framework called Semi-Sandwich framework. Based on the same optimal solution of lower and upper bounds, we propose a Difference Minimizing Greedy (DMG) algorithm to get an approximation solution of the original problem. Through massive experiments, we show that the method and algorithm are effective.