Influence maximization under equilibrious groups in social networks

In a market, there are many groups caused by geographical location or other reasons, and consumers in groups have their own brand preferences for a type of products. The diversity of consumers' brand preferences will avoid the phenomenon of brand simplification and monopoly caused by consistent brand reference of consumers, and to some extent, promote the prosperity and development of various brands in each group. For brand diversification in a market, we hope the brand preferences of consumers in each groups are as equilibrious as possible, so Influence Maximization under Equilibrious Groups (IMEG) is proposed to select k nodes to maximize the number of equilibrious groups after information diffusion. This paper proves that the IMEG is NP-hard, and computing the objective function is #P-hard. It also proves that the objective function is neither submodular nor supermodular under Independent Cascade (IC) model and Linear Threshold (LT) model. Then, the Equilibrious Groups Maximization Solution (EGMS) algorithm is presented to solve our problem. And by comparing with baseline algorithms using different datasets (dolphins, wildbird, weaver and hamsterster), it can be found that the EGMS has obvious advantages in time complexity and spatial complexity. In particular, the running time of EGMS is about 3.7x10-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.7\times 10<^>{-2}$$\end{document}, 2.4x10-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.4\times 10<^>{-2}$$\end{document}, 1.1x10-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.1\times 10<^>{-1}$$\end{document} and 1.8x10-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.8\times 10<^>{-3}$$\end{document} times that of the fastest baseline algorithm in dolphins, wildbird, weaver and hamsterster respectively. And the experiments in small-world, scale-free, random and regular network verify the robustness of EGMS with the varying parameters, such as the number of nodes, the probability of adding edges and the number of adjacencies.