Non-submodular model for group profit maximization problem in social networks

In social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions are made by simply majority, which means that most of the users in this group reach an agreement, such as US Presidential Elections. A group...

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Published in:	Computational social networks Vol. 8; no. 1
Main Authors:	Zhu, Jianming, Ghosh, Smita, Wu, Weili, Gao, Chuangen
Format:	Journal Article
Language:	English
Published:	Cham Springer International Publishing 07.01.2021 Springer Nature B.V
Subjects:	Algorithms Data-driven Science Database Management Election results Elections Industrial and Production Engineering Industrial Organization Lower bounds Mathematical Models of Cognitive Processes and Neural Networks Mathematics Mathematics and Statistics Maximization Media Sociology Modeling and Theory Building Optimization Presidential elections Profit maximization Social networks Upper bounds Group profit maximization Social networks Submodular–modular algorithm Non-submodular
ISSN:	2197-4314, 2197-4314
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Abstract	In social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions are made by simply majority, which means that most of the users in this group reach an agreement, such as US Presidential Elections. A group is called activated if β percent of users are influenced in the group. Enterprise will gain income from all influenced groups. Simultaneously, to propagate influence, enterprise needs pay advertisement diffusion cost. Group profit maximization (GPM) problem aims to pick k seeds to maximize the expected profit that considers the benefit of influenced groups with the diffusion cost. GPM is proved to be NP-hard and the objective function is proved to be neither submodular nor supermodular. An upper bound and a lower bound which are difference of two submodular functions are designed. We propose a submodular–modular algorithm (SMA) to solve the difference of two submodular functions and SMA is shown to converge to a local optimal. We present an randomized algorithm based on weighted group coverage maximization for GPM and apply sandwich framework to get theoretical results. Our experiments verify the efficiency of our methods.
AbstractList	In social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions are made by simply majority, which means that most of the users in this group reach an agreement, such as US Presidential Elections. A group is called activated if β percent of users are influenced in the group. Enterprise will gain income from all influenced groups. Simultaneously, to propagate influence, enterprise needs pay advertisement diffusion cost. Group profit maximization (GPM) problem aims to pick k seeds to maximize the expected profit that considers the benefit of influenced groups with the diffusion cost. GPM is proved to be NP-hard and the objective function is proved to be neither submodular nor supermodular. An upper bound and a lower bound which are difference of two submodular functions are designed. We propose a submodular–modular algorithm (SMA) to solve the difference of two submodular functions and SMA is shown to converge to a local optimal. We present an randomized algorithm based on weighted group coverage maximization for GPM and apply sandwich framework to get theoretical results. Our experiments verify the efficiency of our methods. In social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions are made by simply majority, which means that most of the users in this group reach an agreement, such as US Presidential Elections. A group is called activated if $$\beta$$ β percent of users are influenced in the group. Enterprise will gain income from all influenced groups. Simultaneously, to propagate influence, enterprise needs pay advertisement diffusion cost. Group profit maximization (GPM) problem aims to pick k seeds to maximize the expected profit that considers the benefit of influenced groups with the diffusion cost. GPM is proved to be NP-hard and the objective function is proved to be neither submodular nor supermodular. An upper bound and a lower bound which are difference of two submodular functions are designed. We propose a submodular–modular algorithm (SMA) to solve the difference of two submodular functions and SMA is shown to converge to a local optimal. We present an randomized algorithm based on weighted group coverage maximization for GPM and apply sandwich framework to get theoretical results. Our experiments verify the efficiency of our methods. In social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions are made by simply majority, which means that most of the users in this group reach an agreement, such as US Presidential Elections. A group is called activated if β percent of users are influenced in the group. Enterprise will gain income from all influenced groups. Simultaneously, to propagate influence, enterprise needs pay advertisement diffusion cost. Group profit maximization (GPM) problem aims to pick k seeds to maximize the expected profit that considers the benefit of influenced groups with the diffusion cost. GPM is proved to be NP-hard and the objective function is proved to be neither submodular nor supermodular. An upper bound and a lower bound which are difference of two submodular functions are designed. We propose a submodular–modular algorithm (SMA) to solve the difference of two submodular functions and SMA is shown to converge to a local optimal. We present an randomized algorithm based on weighted group coverage maximization for GPM and apply sandwich framework to get theoretical results. Our experiments verify the efficiency of our methods.
ArticleNumber	2
Author	Gao, Chuangen Ghosh, Smita Zhu, Jianming Wu, Weili
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References	Iyer R, Bilmes J. Algorithms for approximate minimization of the difference between submodular functions, with applications. arXiv preprint arXiv:1207.0560. 2012. Tagarelli A., Tong H, editors. Computational data and social networks. CSoNet 2019. Lecture notes in computer science, vol. 11917. Berlin: Springer. pp. 108–19. Ohsaka N, Akiba T, Yoshida Y, Kawarabayashi K-i. Fast and accurate influence maximization on large networks with pruned monte-carlo simulations. In: AAAI. 2014. pp. 138–44. BachFLearning with submodular functions: a convex optimization perspectiveFound Trends ® Mach Learn201362–31453731280.6800110.1561/2200000039 ZhuJGhoshSWuWGroup influence maximization problem in social networksIEEE Trans Comput Soc Syst201910.1109/TCSS.2019.2938575 Kempe D, Kleinberg J, Tardos É. Maximizing the spread of influence through a social network. In: Proceedings of the ninth ACM SIGKDD international conference on knowledge discovery and data mining, ACM. 2003. pp. 137–46. Borgs C, Brautbar M, Chayes J, Lucier B. Maximizing social influence in nearly optimal time. In: Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms. SIAM. 2014. pp. 946–57. NemhauserGLWolseyLAFisherMLAn analysis of approximations for maximizing submodular set functionsMath Program19781412652945038660374.9004510.1007/BF01588971 Leskovec J, Krause A, Guestrin C, Faloutsos C, VanBriesen J, Glance N. Cost-effective outbreak detection in networks. In: Proceedings of the 13th ACM SIGKDD international conference on knowledge discovery and data mining, ACM. 2007. pp. 420–9. ZhuJZhuJGhoshSWuWYuanJSocial influence maximization in hypergraph in social networksIEEE Trans Netw Sci Eng201810.1109/TNSE.2018.2873759 Fujishige S. Submodular functions and optimization. In: Of Annals Of discrete mathematics, vol. 47. 2008. OpsahlTTriadic closure in two-mode networks: redefining the global and local clustering coefficientsSoc Netw201335215916710.1016/j.socnet.2011.07.001 Tang Y, Shi Y, Xiao X. Influence maximization in near-linear time: a martingale approach. In: Proceedings of the 2015 ACM SIGMOD international conference on management of data, ACM. 2015. pp. 1539–54. Schoenebeck G, Tao B. Beyond worst-case (in)approximability of nonsubmodular influence maximization. In: International conference on web and internet economics. 2017. Zhu Y, Lu Z, Bi Y, Wu W, Jiang Y, Li D. Influence and profit: two sides of the coin. In: Data mining (ICDM), 2013 IEEE 13th international conference on, IEEE. 2013. pp. 1301–6. Lu W, Lakshmanan LV. Profit maximization over social networks. In: Data mining (ICDM), 2012 IEEE 12th international conference on, IEEE. 2012. pp. 479–88. LuWChenWLakshmananLVFrom competition to complementarity: comparative influence diffusion and maximizationProc VLDB Endow201592607110.14778/2850578.2850581 ZhuJGhoshSWuWRobust rumor blocking problem with uncertain rumor sources in social networksWorld Wide Web202010.1007/s11280-020-00841-8 ZhuJGhoshSZhuJWuWNear-optimal convergent approach for composed influence maximization problem in social networksIEEE Access2019714248814249710.1109/ACCESS.2019.2944207 Tang J, Tang X, Yuan J. Towards profit maximization for online social network providers. arXiv preprint arXiv:1712.08963. 2017. Nguyen HT, Thai MT, Dinh TN. Stop-and-stare: optimal sampling algorithms for viral marketing in billion-scale networks. In: Proceedings of the 2016 international conference on management of data, ACM. 2016. pp. 695–710. Tang Y, Xiao X, Shi Y. Influence maximization: near-optimal time complexity meets practical efficiency. In: Proceedings of the 2014 ACM SIGMOD international conference on management of data, ACM. 2014. pp. 75–86. DagumPKarpRLubyMRossSAn optimal algorithm for monte carlo estimationSIAM J Comput20002951484149617448331112.6530010.1137/S0097539797315306 DuNLiangYBalcanM-FGomez-RodriguezMZhaHSongLScalable influence maximization for multiple products in continuous-time diffusion networksJ Mach Learn Res201718214536257061433.91114 Zhu J, Ghosh S, Wu W, Gao C. Profit maximization under group influence model in social networks. In: International conference on computational data and social networks. Springer. 2019. pp. 108–19. Goyal A, Lu W, Lakshmanan LV. Simpath: an efficient algorithm for influence maximization under the linear threshold model. In: Data Mining (ICDM), 2011 IEEE 11th international conference on, IEEE. 2011. pp. 211–20. WuWLZhangZDuDZSet function optimizationJ Oper Res Soc China20183111 TangJTangXYuanJProfit maximization for viral marketing in online social networks: algorithms and analysisIEEE Trans Knowl Data Eng20183061095110810.1109/TKDE.2017.2787757 YangYLuZLiVOXuKNoncooperative information diffusion in online social networks under the independent cascade modelIEEE Trans Comput Soc Syst20174315016210.1109/TCSS.2017.2719056 Meeker M. Internet trends 2018-code conference. Glokalde. 2018;1(3). Chen W, Wang C, Wang Y. Scalable influence maximization for prevalent viral marketing in large-scale social networks. In: Proceedings of the 16th ACM SIGKDD international conference on knowledge discovery and data mining, ACM. 2010. pp. 1029–38. Cohen E, Delling D, Pajor T, Werneck RF. Sketch-based influence maximization and computation: scaling up with guarantees. In: Proceedings of the 23rd ACM international conference on conference on information and knowledge management, ACM. 2014. pp. 629–38. Aslay C, Lakshmanan LV, Lu W, Xiao X. Influence maximization in online social networks. In: Proceedings of the eleventh ACM international conference on web search and data mining, ACM. 2018. pp. 775–6. Forsyth DR. Group dynamics. 2018. p. 1. Narasimhan M, Bilmes JA. A submodular-supermodular procedure with applications to discriminative structure learning. arXiv preprint arXiv:1207.1404. 2012. NewmanMEThe structure of scientific collaboration networksProc Nat Acad Sci200198240440918126101065.0051810.1073/pnas.98.2.404 85_CR7 85_CR8 85_CR5 85_CR6 85_CR4 85_CR1 85_CR2 J Tang (85_CR16) 2018; 30 85_CR20 WL Wu (85_CR28) 2018; 3 J Zhu (85_CR13) 2018 85_CR9 W Lu (85_CR27) 2015; 9 Y Yang (85_CR11) 2017; 4 85_CR21 GL Nemhauser (85_CR31) 1978; 14 T Opsahl (85_CR34) 2013; 35 85_CR22 85_CR23 85_CR24 85_CR25 ME Newman (85_CR35) 2001; 98 N Du (85_CR10) 2017; 18 J Zhu (85_CR3) 2019 85_CR30 F Bach (85_CR26) 2013; 6 P Dagum (85_CR32) 2000; 29 85_CR18 85_CR19 J Zhu (85_CR29) 2020 85_CR33 85_CR12 85_CR36 J Zhu (85_CR14) 2019; 7 85_CR15 85_CR17
References_xml	– reference: Aslay C, Lakshmanan LV, Lu W, Xiao X. Influence maximization in online social networks. In: Proceedings of the eleventh ACM international conference on web search and data mining, ACM. 2018. pp. 775–6. – reference: Lu W, Lakshmanan LV. Profit maximization over social networks. In: Data mining (ICDM), 2012 IEEE 12th international conference on, IEEE. 2012. pp. 479–88. – reference: Narasimhan M, Bilmes JA. A submodular-supermodular procedure with applications to discriminative structure learning. arXiv preprint arXiv:1207.1404. 2012. – reference: Zhu J, Ghosh S, Wu W, Gao C. Profit maximization under group influence model in social networks. In: International conference on computational data and social networks. Springer. 2019. pp. 108–19. – reference: ZhuJGhoshSWuWGroup influence maximization problem in social networksIEEE Trans Comput Soc Syst201910.1109/TCSS.2019.2938575 – reference: DuNLiangYBalcanM-FGomez-RodriguezMZhaHSongLScalable influence maximization for multiple products in continuous-time diffusion networksJ Mach Learn Res201718214536257061433.91114 – reference: Tang Y, Shi Y, Xiao X. Influence maximization in near-linear time: a martingale approach. In: Proceedings of the 2015 ACM SIGMOD international conference on management of data, ACM. 2015. pp. 1539–54. – reference: Tagarelli A., Tong H, editors. Computational data and social networks. CSoNet 2019. Lecture notes in computer science, vol. 11917. Berlin: Springer. pp. 108–19. – reference: Tang Y, Xiao X, Shi Y. Influence maximization: near-optimal time complexity meets practical efficiency. In: Proceedings of the 2014 ACM SIGMOD international conference on management of data, ACM. 2014. pp. 75–86. – reference: Leskovec J, Krause A, Guestrin C, Faloutsos C, VanBriesen J, Glance N. Cost-effective outbreak detection in networks. In: Proceedings of the 13th ACM SIGKDD international conference on knowledge discovery and data mining, ACM. 2007. pp. 420–9. – reference: Fujishige S. Submodular functions and optimization. In: Of Annals Of discrete mathematics, vol. 47. 2008. – reference: DagumPKarpRLubyMRossSAn optimal algorithm for monte carlo estimationSIAM J Comput20002951484149617448331112.6530010.1137/S0097539797315306 – reference: Schoenebeck G, Tao B. Beyond worst-case (in)approximability of nonsubmodular influence maximization. In: International conference on web and internet economics. 2017. – reference: Borgs C, Brautbar M, Chayes J, Lucier B. Maximizing social influence in nearly optimal time. In: Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms. SIAM. 2014. pp. 946–57. – reference: LuWChenWLakshmananLVFrom competition to complementarity: comparative influence diffusion and maximizationProc VLDB Endow201592607110.14778/2850578.2850581 – reference: Kempe D, Kleinberg J, Tardos É. Maximizing the spread of influence through a social network. In: Proceedings of the ninth ACM SIGKDD international conference on knowledge discovery and data mining, ACM. 2003. pp. 137–46. – reference: OpsahlTTriadic closure in two-mode networks: redefining the global and local clustering coefficientsSoc Netw201335215916710.1016/j.socnet.2011.07.001 – reference: TangJTangXYuanJProfit maximization for viral marketing in online social networks: algorithms and analysisIEEE Trans Knowl Data Eng20183061095110810.1109/TKDE.2017.2787757 – reference: NemhauserGLWolseyLAFisherMLAn analysis of approximations for maximizing submodular set functionsMath Program19781412652945038660374.9004510.1007/BF01588971 – reference: Goyal A, Lu W, Lakshmanan LV. Simpath: an efficient algorithm for influence maximization under the linear threshold model. In: Data Mining (ICDM), 2011 IEEE 11th international conference on, IEEE. 2011. pp. 211–20. – reference: YangYLuZLiVOXuKNoncooperative information diffusion in online social networks under the independent cascade modelIEEE Trans Comput Soc Syst20174315016210.1109/TCSS.2017.2719056 – reference: WuWLZhangZDuDZSet function optimizationJ Oper Res Soc China20183111 – reference: Meeker M. Internet trends 2018-code conference. Glokalde. 2018;1(3). – reference: NewmanMEThe structure of scientific collaboration networksProc Nat Acad Sci200198240440918126101065.0051810.1073/pnas.98.2.404 – reference: Cohen E, Delling D, Pajor T, Werneck RF. Sketch-based influence maximization and computation: scaling up with guarantees. In: Proceedings of the 23rd ACM international conference on conference on information and knowledge management, ACM. 2014. pp. 629–38. – reference: ZhuJZhuJGhoshSWuWYuanJSocial influence maximization in hypergraph in social networksIEEE Trans Netw Sci Eng201810.1109/TNSE.2018.2873759 – reference: Chen W, Wang C, Wang Y. Scalable influence maximization for prevalent viral marketing in large-scale social networks. In: Proceedings of the 16th ACM SIGKDD international conference on knowledge discovery and data mining, ACM. 2010. pp. 1029–38. – reference: ZhuJGhoshSWuWRobust rumor blocking problem with uncertain rumor sources in social networksWorld Wide Web202010.1007/s11280-020-00841-8 – reference: BachFLearning with submodular functions: a convex optimization perspectiveFound Trends ® Mach Learn201362–31453731280.6800110.1561/2200000039 – reference: Iyer R, Bilmes J. 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Snippet	In social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions...
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SubjectTerms	Algorithms Data-driven Science Database Management Election results Elections Industrial and Production Engineering Industrial Organization Lower bounds Mathematical Models of Cognitive Processes and Neural Networks Mathematics Mathematics and Statistics Maximization Media Sociology Modeling and Theory Building Optimization Presidential elections Profit maximization Social networks Upper bounds
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Title	Non-submodular model for group profit maximization problem in social networks
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